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Designing and running Olympic venues in legacy
Creating the perfect blend
Shaun Dawson, Chief Executive of Lee Valley Regional Park Authority
London is generally held to be the best planned legacy from any Olympic Games – a million miles away from the sadly numerous examples around the world of grand unused sporting edifices from showpiece events like the Olympics. In this article Shaun Dawson will share how balancing community use and commercial opportunity has been the golden thread leading to the three London 2012 legacy venues for which he is responsible being popular and successful exemplars of well-executed legacy.
As Chief Executive of Lee Valley Regional Park Authority, I oversee an eclectic 26 mile, 10,000 acre (42 km / 4,000 ha) park which starts on the banks of the River Thames in a constantly changing area of east London, stretches through what is now Queen Elizabeth Olympic Park, and out to Hertfordshire.
Photos: Lee Valley Regional Park Authority and Eleanor Bentall, England Hockey
Author: Shaun Dawson, Chief Executive of Lee Valley Regional Park Authority, United Kingdom
The Authority is an unusual beast. It was set up by an Act of Parliament 50 years ago when the Lee Valley was largely derelict, ignored and unloved – known as "London's privy, workshop and backyard". The Lee was London's working river and the Valley home to railway sidings, rubbish dumps, gravel pits, sewage works and polluting industry. Over the last half century Lee Valley Regional Park Authority has, with partners, transformed this unloved corridor into glorious award-winning open spaces and world-class sports venues that now attract more than seven million visits a year.
We have always aimed high and operate with a community focused, commercially driven business ethos. Back in the 1970s we built Europe's largest leisure centre and in the 1980s created Britain's first swimming pool with a wave machine.
While we were retiring our first generation of what had become outdated sports venues in the 1990s, we were developing a strategy for a "Zone of Sporting Excellence" in the Lee Valley. This included a VeloPark and a White Water Centre, with advanced plans for both developed in partnership with the sports' National Governing Bodies.
So when the idea emerged of London bidding for the 2012 Games, partly using our land in the Lower Lee Valley, we were enthusiastic partners. As owners of a quarter of the land on which the Olympic Park was planned and owning strategic land holdings nearby, we were able to contribute expertise and passion along with worked-up plans for two venues. And crucially, before the bid was written we were the obvious "legacy client" or eventual owner of three of the venues planned for our land – what would become Lee Valley VeloPark, Lee Valley Hockey and Tennis Centre, and Lee Valley White Water Centre.
With this ownership came the responsibility – and pressure – to ensure that they would be successful in the long term. Our name would be above the door and we had a duty to make sure they worked from day one. This role made us the protector and advocate of their legacy. We were on the panels which selected architects, and made all key design decisions, championing the post-Games use throughout.
Photo: Lee Valley Regional Park Authority and Eleanor Bentall, England Hockey
We learned from other Olympic experiences by journeying to Athens, Barcelona, Sydney and Beijing plus examining how Commonwealth cities like Manchester and Melbourne successfully transformed post Games. While the Olympic designers were focused on 17 days of the greatest sport in the world, our attention was on post-Games day-to-day essentials such as catering provision and sufficient toilets, plus building in commercial potential. We worked on detailed business plans to ensure that these three venues were community assets which had community use first and foremost through their 25-year plus life span, but could host major events and generate commercial income and ensure sufficient income to guarantee their long-term future.
We now run three successful legacy venues. Financially, two make an operational surplus. We have outsourced their running to a Leisure Trust with the aim of increasing commercial development and lessening costs, while maintaining their focus on community sports activity. The three have different dynamics, visitor demographics and uses, but share the same nine ingredients for success apply:
1 Have a long-term vision and philosophy as an owner to determine what the venues will be.
Our business philosophy was clear from the outset – we are community focused and commercially driven. This twin approach placed community use at the heart of the three venues alongside a laser-like focus on commercial opportunities to guarantee their financial future. This means that while sport is still king, our venues have hireable spaces built-in, are used as filming locations, host meetings and conferences and evolve to meet changing leisure trends. For example, a large area used for overlay during the Games at Lee Valley White Centre now hosts a huge inland beach with volleyball, beach football and an inflatable play park. None of these are linked to canoeing or rafting, but they broaden the venue's demographic enormously.
2 Build in as much transformation as possible at the planning stage.
London 2012 needed only a velodrome and BMX track, but our vision for the post-Games venue was a "VeloPark" where all four Olympic cycling disciplines would be available. Adding in a one-mile road circuit and five miles of mountain bike trails would have been impossible if it had not been factored in from the start. Similarly, we learnt from other white water centres that a second, gentler white water course would broaden our market and provide a sporting pathway for people new to canoeing. We led a consortium which raised GBP 6 million to create this and integrated it into the London 2012 design.
3 Getting venues right is difficult.
Planning 25 years ahead is hard, do not expect to get everything right.
4 Expect to compromise.
Major sports venues are almost always delivered through joint working and an array of stakeholders. It is natural to have tensions between a group made up of organisations with different agendas – the only way to deal with this is through compromise.
5 Be flexible, adapt and expect venues to change over time.
We have added considerable infrastructure to Lee Valley Hockey and Tennis Centre, but only now are we poised to maximise the entire potential of the site with further leisure development, taking advantage of a large development platform.
6 Prioritise the programme and business plan.
We have achieved a blended programme with around 80% of activity reserved for schools, clubs and community, and the rest packed with a range of commercial activities such as private hires, teambuilding and corporate events.
7 Remove barriers to participation.
Activities at our venues start at GBP 5 – and for disadvantaged groups our Community Access Fund can pay for travel and coaching. We have partnered with providers to run all-ability cycling and tennis programmes for wheelchair users, people with learning disabilities, sight impairments and mental health issues. We invested in 243 bikes, which means anyone can turn up and have a go, and visitors need no specialist equipment.
8 Bring back the magic of big events.
We have hosted four World Cups, two World Championships and ten other major international events since London 2012. In 2022, Lee Valley VeloPark will make history becoming the only venue in the world to host an Olympics, World Championships and Commonwealth Games in the same sport. We work closely with our national governing bodies, international federations and commercial promoters and invest time, money and expertise in major events.
9 Embrace change.
The market is continually evolving. Be prepared to face constant challenges of meeting customer demand, keeping audiences engaged, creating new products and pushing the boundaries of what is possible at each venue.
Curious to read all features in sb 2/2019?
Read complete issue
Cor Van Raay YMCA at the ATB Centre in Lethbridge Alberta
The ATB Centre has within its scope a very large aquatics program, a fitness centre for strength training and aerobics, a large field house with six gymnasia and a 250-metre running track surrounding multiple studios, multi-purpose and meeting rooms, a youth room, children's indoor playground, a childcare, physiotherapy, and retail food vendors.
Activity area on top of a waste-to-energy plant
CopenHill, also known as Amager Bakke, is opening as a new breed of waste-to-energy plant topped with a ski slope, hiking trail and climbing wall, embodying the notion of hedonistic sustainability while aligning itself with Copenhagen's goal of becoming the world's first carbon-neutral city by 2025.
Pixeland activity zone in Mianyang
Pixeland is a public space project comprising a combination of different outdoor facilities in a single space, such as landscape features, playground features for children and leisure features for adults.
Concord Community Pop-up Park in Vancouver BC
From a sea of asphalt to a green oasis, Concord Community Pop-up Park has re-animated the former Expo '86 lands in Vancouver, BC as a vibrant, highly programmed destination.
Innovative strategy at Smale Riverfront Park's Destination Playscapes in Cincinnati
As a part of a larger masterplan, the city of Cincinnati played host to an innovative and nuanced park project. The resulting design activated a wealth of multidisciplinary research and theory to create an engaging and challenging play space for children and families alike.
Wonder Wood schoolyard and forest activity zone in Rebild
In recent years there has been an increased focus on what happens in the schoolyard between lessons, and what play, exercise and motor skills can do for learning. In an area of 12,000 m² in the Rebild municipality, VEGA landskab has created an animating activity zone connecting the schoolyard with the nearby forest.
New sb: Activating Urban Spaces
Activating Urban Spaces is the focus of this edition of sb and a critical part of what makes our cities and urban centres liveable.
Baakenpark in Hamburg
Opening in May 2018, Baakenpark is considered the district's green centrepiece. On a 1.6 ha artificial peninsula, the park displays comfort with its "island sofa", elevation with its 15 m high Himmelsberg hill and excitement with its flotsam playground.
Indoor swimming pool with a view of Lake Ägeri
Its breath-taking view of Lake Ägeri invites you to swim and relax. The glazing of the swimming pool on three sides offers swimmers a magnificent view and walkers a glimpse of swimming action.
Extension and modernisation of Val Blu outdoor pool and sauna in Bludenz
Thanks to the placement of the clearly shaped poolscape at the lowest point of the site, the terrain now forms a kind of arena around the water surfaces. These consist of a competition pool with diving tower, a large fun pool and a wide slide.
The impact of climate in indoor pools on the architectural design
More advanced building technologies with improved insulation and air tightness allow for a new approach in the design. A better understanding on how disinfected biproducts in the water are formed, about the evaporation of gas from the water into the room, and the possible impact of these substances on health, call for a ventilation concept where the swimmer in the water plays a more important role.
Inauguration of the Olympic House
The headquarter is one of the most sustainable buildings in the world.
Aaniin Community Centre and Library in Markham
The Aaniin Community Centre and Library combines an extensive multi-use program including a district library, aquatics centre, fitness area, multi-use rooms, double gym, and performance space in addition to various change room, food concession and support spaces.
New sb: Pools and wellness facilities
The swimming pools and sports facilities presented in this issue of "sb" are also characterised by their response to the needs of users.
Montigalà Stadium in Badalona
The stand seating displays a variety of shades of blue. Monigalà Stadium in Badalona, designed by AIA Salazar-Navarro architects, has created a community focus for the residents of the nearby residential area.
Felipe Alou Baseball Academy in Boca Chica
The Felipe Alou Baseball Academy designed by jones I haydu consists of three major league-size playing fields, batting tunnels, pitching mounds, and two substantial buildings.
"La Doce" sports ground and pavilion in Valle de Chalco
"La Doce" is not only a place for soccer, but also a meeting place for the neighbourhood. All Arquitectura has created an urban landmark in the socially deprived district of Valle de Chalco in Mexico.
Dojo Kùkí in Épinay-sur-Seine
Kùkí means atmosphere. Nomade Architectes have created a dojo consisting of two martial arts rooms with their changing rooms, one fitness room and three offices.
'The Heart' multi-purpose building and activity park in Ikast
C.F. Møller Architects is behind a major project next to Ikast-Brande International School with a much-awaited expansion comprising several halls, and multi-functional and educational facilities.
Great Plains Recreation Facility for ice sport in Calgary by MJMA
The Great Plains recreation facility features two multi-purpose rinks and brings together ice sport enthusiasts from across the city for hockey, sledge hockey, ringette, and figure skating. Flipped geometry
Varkenoord ClubHouse in Rotterdam by NL Architects
Three sports clubs with different backgrounds and 'club cultures' are joining forces on the new Rotterdam sports campus: two soccer clubs (FC IJsselmonde and Overmaas) and Feyenoord Hockey Club are sharing a common facility. NL Architects are positioning the building between the two main soccer fields and the adjoining pitches. The clubhouse attempts to exploit its specific position.
Streetmekka street sports and cultural centre in Viborg by Effekt
Streetmekka is a new cultural destination offering a wide variety of facilities for self-organized sports like parkour, skate, bouldering, basketball, trial and a series of customized workshop areas for music production, DJ'ing, an animation studio, fabrication lab and various artist studios and wood- and metalworkshops. Architecture firm Effekt has transformed an abandoned windmill factory into a vibrant hub for street sports and youth culture.
Arena du Pays d'Aix by Auer Weber Architects
Arena du Pays d'Aix, opened in September 2017.
Domino Park in New York City by James Corner Field Operations
New York's newest waterfront recreational public space at the Domino Sugar Factory site.
Public saltwater swimming pool in Mazatlán by Colectivo Urbano
The renovation of Mazatlán's historic Carpa Olivera ocean-fed public swimming pool is steeped in history while being free of charge and open to the public.
Thomas Kick
Marketing and event management / Editor "sb" and graphics
kick@iaks.sport


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David Cameron's real foreign policy
Iain Dale: The time has come to dig up Cameron's green tree
Scruton and the Tories. "A prophet is not without honour, but in his own country."
Bob Seely: Why the Government should listen to our allies and say: no way, Huawei
We need strategic clarity. What poses the greater danger to Britain – Iranian aggression abroad or Sunni extremism here?
Trump and Iran. What's the plan?
By Paul Goodman
Follow Paul on Twitter
If all one knew of foreign policy was gleaned from institutes, think-tanks, centres and specialist events, one might well conclude that it is shaped by theory – by rational people gathering together to apply grand designs to the world. It would follow that David Cameron's speech to the United Nations must therefore be seen in that way – as the product of the application of ideas to circumstances, judiciously weighed over time. So it's worth looking back at the history of the Prime Minister's engagement with world affairs, from which today's speech has emerged, in order to work out whether or not foreign policy emerges in this considered way.
Cameron was drawn deeper into it after his patron, Michael Howard, became leader of the party and leaned on his former special adviser for help. Iain Duncan Smith had backed the Iraq war enthusiastically. No weapons of mass destruction had been found. Tony Blair was in political difficulty as a result. The inclination of part of the party was that Howard should proclaim that Britain should never have fought the war; that of another to disagree; that of a third part to believe that it would look weak to first support the war and then oppose it. But there was agreement that the policy pendulum between isolation and intervention had swung too far towards the latter.
George W.Bush didn't care for this shift, and Howard was never received at the White House. The latter was thereby snubbed by a political ally, which he understandably resented. Cameron had been sceptical about the Iraq War as a up-and-coming backbencher, and the flavour of his foreign policy outlook as Leader of the Opposition was influenced by this fact, by Howard's treatment by Bush, by the political opportunities which Blair's plight offered, by the shift of mood against interventionism, by his relationship with both an older generation of senior Tories – of which his constituency predecessor Douglas Hurd provides the best example – and with his chief of staff, Ed Llewellyn.
Llewellyn is not an isolationist, having served as Chief of Staff both to Chris Patten in Hong Kong and Paddy Ashdown in Bosnia. And Cameron doesn't share Hurd's Euro-enthusiast instincts: if not Euro-sceptic, he is at the very least Euro-cool, regarding the EU and the controversies that spring from it as a menace to party management. But he shares with both men a temperamental calm and detachment very different from the passion and fever with which Blair approached Iraq – and, previously, Kosovo. The best-known line in a Cameron foreign policy opposition speech was about not being able to drop democracy out of an aeroplane at 40,000 feet. It caught the tone of his approach: politically adroit and interventionism-sceptic.
Some of his best political friends and other senior colleagues have never shared this view – George Osborne, Liam Fox and Michael Gove fall into one of these categories or both. What has been seen to shift the Prime Minister back towards their more activist vision of foreign policy has not so much been Afghanistan (where caution has won out) but Libya. Who knows why Cameron tilted so decisively in favour of intervention. Was his stance heart-led, a horrified response to impending massacre in Benghazi? Was it head-led, a colder calculation that, with the "Arab Spring" taking place, Britain would be blamed if it did nothing? Did he believe that Libya was more like Kosovo than Iraq – a case in which the consequences of not intervening would be worse than doing so?
We can all offer our own answer, and mine is: I doubt if he knows himself. But even I'm wrong, there is a moral in this history – that foreign policy isn't usually shaped by reflection built on theory, but by events, eruptions, happenstance, prejudices – above all, by that marvellously Burkean word, circumstances. The Prime Minister's text will be plundered in search of meanings it doesn't possess. New "Cameron doctrines" will be unearthed. New "Cameron worldviews" will be proclaimed. Sure, there is something there: liberal intervention, the U.N permitting – what emerged, as much by luck as judgement, from Libya – the policy of "it depends". But be certain that it will be qualified by events, don't be too sure it will last, and I mean the Prime Minister no disrespect by saying: don't take it all too seriously.
David Cameron Foreign affairs
19 comments for: David Cameron's real foreign policy

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Headline: Queen's Theatre
Thur. to Saturday, Nov. 6th to 8th at 2.30, 5.15, 7.15 & 9.15 p.m., Dorothy Gish in 'The Ghost in the Garret.' Commencing Sunday, Nov. 9th at 6 & 9.15 p.m., Sylvia Breamer in 'The Face Between,' a Metro classic. Ordinary Prices. Tel. C. 4636. November 7 Remarks: On…
Mon. Nov. 10th only at 5.15 & 9.15 p.m., farewell performances by Miss Bessie Lester. Also showing Sylvia Breamer in 'The Face Between,' a Metro classic. Price details included.
Headline: Queen's Theatre: Miss Bessie Lester
There was a large attendance at the Queen's Theatre for the performance of Miss Bessie Lester, and the Metro feature 'The Face Between,' featuring Sylvia Breamer.
Thursday to Saturday, November 6th, 7th and 8th at 2.30. 5.15. 7.15 and 9.15 p.m., Dorothy Gish in 'The Ghost in the Garret.' Commencing Sunday, November 9th at 6.00 and 9.15 p.m., Sylvia Breamer in 'The Face Between,' a Metro classic. Ordinary prices. Tel. C. 4636.
Today only at 5.15 and 9.15 p.m., special engagement of Miss Bessie Tester in character, songs, originalities and American novelties. Also showing 'The Ghost in the Garret,' a Paramount comedy. Tomorrow at 6.00 and 9.15 p.m., entire change of programme, Miss Bessie Tester in new comic…
Monday, November 10th only at 5.15 and 9.15 p.m., farewell performances by Miss Bessie Lester, featuring new songs and new novelties. Also showing Sylvia Breamer in 'The Face Between,' a Metro classic. Prices: at 5.15, $1.20, $1.00 and 60 cents; at 9.15, $1.50, $1.00 and 60 cents.
Headline: Queen's Theatre: American Vaudeville Comedienne
Big crowds attended on Saturday and Sunday both to the matinee and evening performances, at the Queen's Theatre. A young American comedienne, Miss Bessie Lester made her first appearance. Her work is clever and she sings superbly. In the same programme, the Metro feature 'The Face…
Thursday to Saturday, November at 6, 7 and 8 at 2.30, 5.15, 7.15 and 9.15 p.m., Dorothy Gish in 'The Ghost in the Garret.' Commencing Sunday, November 9 at 6 and 9.15 p.m., Sylvia Breamer in 'The Face Between,' a Metro classic. Ordinary prices. Tel. C. 4636.
Monday, November 10th only at 5.15 and 9.15 p.m., farewell performances by Miss Bessie Lester featuring new songs, new novelties. Also showing Sylvia Breamer in 'The Face Between,' a Metro classic. Prices: at 5.15, $1.20, $1.00 and 60 cents; 9.15, $1.50, $1.00 and 60 cents.
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All 120 cards are confirmed to appear.
Speaking of Zero Collection, GameFaqs today made a poll including the Zero Collection in it. As with my vote for it, Zero Collection has the second place with 4001 votes (17.97%).
So, where'd the timeline from the old site go?
It's gone! Why Capcom hates detailed timelines? ;_; They should release a OCW-style book with more information and less artwork!
Here, I translated the first part of Hironouchi's part.
I've finally been able to report to everyone about Rockman Zero Collection.
I responded to those passionate feelings by trying out working on Zero Collection!
and I think Zero's action and great world hasn't gotten dull even now.
Something that caught my attention was the fact that he said that the series actually did END. Well, for the longest time, they never confirmed Zero's death. They never said he didn't die, but they never said he did die either. I just think that he might just be implying that. Anyway, you can wait for a full translation by someone better than I.
Oh! He also says that if it gets a sequel though, he'll be happy. Maybe implying the chance for a sequel?
No. No no no. No sequel, please. Zero 4's ending was perfect, and it was crystal clear that Zero did in fact die in the Ragnarok crash. Sad, yes, but incredibly poignant when you consider the sacrifice he made for Ciel and everyone else.
It was this very ending that allowed me to take Zero seriously as a character again. Up to that point he had become something of a joke with all his dying and resurrecting, but bringing him to this point to have him, like X before him, sacrifice his life for the promise of a better world was a beautiful thing, and the PERFECT way to end the Zero series.
A sequel would absolutely ruin that and just make Zero a joke again.
It doesn't matter that it was implied. It's never been confirmed Zero's dead anyway. Just look at the Rockman Zero Official Complete Works. It specifically says his whereabouts are currently unknown. Yes, he may have died, but we don't know that.
Fair enough, but in the name of good storytelling and not making Zero a damn joke, let's hope Zero is dead and that Zero 4 completes the circle. It would be so much more poetic that way.
Okay, I could get behind Zero 5 being a prequel. I still think that would be more an X game and not a Zero game, though, since it's before the start of what we generally consider the start of Zero's story. Everything up to and including the Elf Wars is really more X's domain; it seems like Zero is really just "along for the ride" there. It is X who finally puts an end to the Maverick Wars, after all.
My problem is, I don't think Zero 5 would be a prequel. I don't think Capcom cares enough about the story to make a prequel chapter to a spinoff series. It'd be neat to be wrong in this case, though.
Beneath Alouette's room: "System. Sound"
X: "Oh my god! You killed Zero!"
X: "Oh my god! Zero is alive!"


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The Boys from Biloxi: A Legal Thriller (Hardcover)
By John Grisham
McLean & Eakin
#1 NEW YORK TIMES BESTSELLER • Two families. One courtroom showdown. • John Grisham's most gripping thriller yet. • "A legal literary legend." —USA Today
John Grisham returns to Mississippi with the riveting story of two sons of immigrant families who grow up as friends, but ultimately find themselves on opposite sides of the law. Grisham's trademark twists and turns will keep you tearing through the pages until the stunning conclusion.
For most of the last hundred years, Biloxi was known for its beaches, resorts, and seafood industry. But it had a darker side. It was also notorious for corruption and vice, everything from gambling, prostitution, bootleg liquor, and drugs to contract killings. The vice was controlled by small cabal of mobsters, many of them rumored to be members of the Dixie Mafia.
Keith Rudy and Hugh Malco grew up in Biloxi in the sixties and were childhood friends, as well as Little League all-stars. But as teenagers, their lives took them in different directions. Keith's father became a legendary prosecutor, determined to "clean up the Coast." Hugh's father became the "Boss" of Biloxi's criminal underground. Keith went to law school and followed in his father's footsteps. Hugh preferred the nightlife and worked in his father's clubs. The two families were headed for a showdown, one that would happen in a courtroom.
Life itself hangs in the balance in The Boys from Biloxi, a sweeping saga rich with history and with a large cast of unforgettable characters.
John Grisham is the author of forty-seven consecutive #1 bestsellers, which have been translated into nearly fifty languages. His recent books include The Judge's List, Sooley, and his third Jake Brigance novel, A Time for Mercy, which is being developed by HBO as a limited series.
Grisham is a two-time winner of the Harper Lee Prize for Legal Fiction and was honored with the Library of Congress Creative Achievement Award for Fiction.
When he's not writing, Grisham serves on the board of directors of the Innocence Project and of Centurion Ministries, two national organizations dedicated to exonerating those who have been wrongfully convicted. Much of his fiction explores deep-seated problems in our criminal justice system.
John lives on a farm in central Virginia.
Fiction / Thrillers / Suspense
Fiction / Thrillers / Legal
Kobo eBook (October 18th, 2022): $14.99
Paperback, Large Print (November 1st, 2022): $32.00
CD-Audio (October 18th, 2022): $45.00
Publisher: Doubleday
Praise for John Grisham:
"A legal literary legend." —USA Today
"John Grisham is about as good a storyteller as we've got in the United States these days." —The New York Times Book Review
"Grisham's work — always superior entertainment — is evolving into something more serious, more powerful, more worthy of his exceptional talent." —The Washington Post
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CA Mainframe Security Products Under Review for U.S. Government Clearance
CA announced that three of its leading mainframe security management software products-CA ACF2, CA Top Secret r14 and CA Compliance Manager for z/OS r1-are officially in evaluation for EAL4+ certification under the Common Criteria International security standard. Common Criteria evaluation of security products is mandated for commercial information security products purchased by the U.S. government for use in national security systems.
The three CA products have cleared the first, pre-evaluation stage of the certification process and have advanced to the "in-evaluation" stage where they will be tested for specific security functionality against a set of predetermined requirements. Common Criteria certification is granted when a Common Criteria testing laboratory determines that a product meets a measure of security. The certification addresses product functionality, development environment, documentation and product testing measures.
The Common Criteria status is a benchmark that can be referenced by private-sector businesses as well, Kirk Willis, vice president of mainframe security management at CA, tells 5 Minute Briefing. "The Common Criteria is an international framework for security that has been recognized by the consuming countries in the Common Criteria Recognition Agreement," he says. "Some financial institutions have started adding CC as a "nice to have" in RFPs and RFIs. With the private-sector adoption of DoD standards as documented in the Security Technical Implementation Guides (STIG), we have had inquiries from a few of our larger mainframe security sites. We expect this trend to continue as the DoD standards are becoming more widely adopted for regulatory mandates, audits and compliance."
Common Criteria status is a holistic look at a system security environment, Willis adds. "Common Criteria only applies to the Target of Evaluation (TOE) defined for the specific evaluation," he says. "At the EAL4+ level of evaluation, tools and techniques for developing the product, as well as, source code are considered as part of the evaluation. The solution's deployment process, as described in the documentation, is also evaluated. CA strongly recommends choosing the highest security settings within each product. We also offer a best practice guide that helps customers properly implement these controls. In addition, we offer our customers implementation reviews to confirm that they have implemented CA ACF2 or CA Top Secret to our recommended settings without compromising performance."
Common Criteria is recognized by governments in more than 26 countries. CA ACF2 and CA Top Secret provide comprehensive access control for IBM z/OS resources across operating systems, subsystems, third-party software and databases which includes externalized security controls for CICS, DB2, Unix System Services (USS) and IMS. They enable organizations to monitor and adjust their security policies and accommodate virtually all organizational structures.
CA Compliance Manager for z/OS is the first platform-resident solution to provide real-time policy management of security and compliance events across the z/OS environment and mainframe security subsystems. It consolidates real-time and historical monitoring of select system events and security events to safeguard IT environments. CA ACF2 and CA Top Secret also work with CA Compliance Manager to provide a single view of compliance for the mainframe.
All three solutions are part of CA's Mainframe 2.0 initiative, which is greatly simplifying mainframe ownership and facilitating the generational shift in mainframe management staffs.
CA SiteMinder, CA eHealth Performance Manager and CA NSM are among the CA products most recently evaluated under the Common Criteria. Additional CA products currently in evaluation include CA Access Control and CA Identity Manager.
The evaluation of CA software will be performed by Booz Allen Hamilton's Common Criteria Testing Laboratory (CCTL). For more information, visit the CA website.


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Compare the Cheapest Car Insurance Quotes in Modesto, CA
Car insurance premiums are never static because they're based on many different factors such as age, gender, driving record and location, among others. If you're looking for car insurance in Modesto, it's important to get a personalized quote so you can get the most accurate insurance rates based on your profile.
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The Best Cheap Car Insurance in Modesto
Average Cost of Car Insurance in Modesto
Modesto$1,320
California$1,429
U.S.$1,424
Aside from being influenced by specific factors in your personal profile, car insurance rates also vary across locations. Average car insurance rates in Modesto are cheaper than both the state at $1,429 and national averages at $1,424.
Drivers in Modesto should expect to pay roughly 7.6% less for car insurance than others do in California. Compared to the rest of the country, Modesto residents generally pay 7.3% less for car insurance at $1,320. Though these may not look like significant amounts, every single dollar saved definitely counts.
Compare Car Insurance Rates in Modesto
Ensure you're getting the best rate for your car insurance. Compare quotes from the top insurance companies.
The Best Cheap Car Insurance Companies in Modesto
If you are looking for cheap car insurance in Modesto, then you'll want to compare prices from several companies. On average, the most affordable company is Progressive. The most expensive, on the other hand, is Travelers. You can expect a 65.5% average difference in prices between these two companies.
1. Progressive
In 2021, Progressive ranked #1 in Modesto for the cheapest car insurance for the average driver.
2. Esurance
In 2021, Esurance ranked #2 in Modesto for the cheapest car insurance for the average driver.
In 2021, USAA ranked #3 in Modesto for the cheapest car insurance for the average driver.
4. GEICO
In 2021, GEICO ranked #4 in Modesto for the cheapest car insurance for the average driver.
5. CSAA
In 2021, CSAA ranked #5 in Modesto for the cheapest car insurance for the average driver.
In 2021, MetLife ranked #6 in Modesto for the cheapest car insurance for the average driver.
In 2021, Allstate ranked #7 in Modesto for the cheapest car insurance for the average driver.
8. Mercury
In 2021, Mercury ranked #8 in Modesto for the cheapest car insurance for the average driver.
In 2021, State Farm ranked #9 in Modesto for the cheapest car insurance for the average driver.
10. Hartford
In 2021, Hartford ranked #10 in Modesto for the cheapest car insurance for the average driver.
11. Farmers
In 2021, Farmers ranked #11 in Modesto for the cheapest car insurance for the average driver.
12. Allied
In 2021, Allied ranked #12 in Modesto for the cheapest car insurance for the average driver.
13. Travelers
In 2021, Travelers ranked #13 in Modesto for the cheapest car insurance for the average driver.
Personalized Quotes to Get You the Cheapest Car Insurance in Modesto
Car insurance companies in Modesto will also look at specific factors when computing your insurance premium. Some of these factors include ZIP code, moving violations, car make and model and coverage level, among others. The more high-risk your profile, the higher premium you'll need to pay.
Drivers with no offenses, plenty of experience and a favorable location should expect cheaper quotes. Since some of these factors change over time, make sure you get regular quotes for comparison. You may even find out that you qualify for discounts or rebates.
Cheapest Car Insurance in Modesto by Coverage Amount
How much car insurance coverage you purchase in Modesto also affects how much your policy will cost. Coverage includes required liability coverage and optional items like collision, comprehensive, and deductibles.
Visit these guides to learn more:
Auto Insurance buying guide
California auto insurance liability minimums
California Minimum Liability Requirements
Bodily Injury and Property Damage $15,000 per person, $30,000 per accident and $5,000 for property damage. A more protective level is 100/300/100.
The three numbers (for example, 50/100/50) in your car insurance plan stand for the limits of your policy: per person injury, per vehicle injury and property damage liability. You may opt to get the state minimum or increase the coverage for greater protection. Take note, however, that liability-only coverage only covers damage to the other driver(s).
You may also opt to get full coverage, which provides a higher level of protection, including covering your own expenses. For car insurance in Modesto, differences between state minimum and 50/100/50 full-coverage policies range from less than $300 to nearly $700, depending on which company you choose. If you can afford the difference, getting full coverage can give you more peace of mind.
Average Car Insurance Premiums by Coverage Type
Liability Only
State Minimumper year
50 / 100 / 50per year
1. Progressive $436 $600
2. Esurance $562 $670
3. USAA $472 $611
4. GEICO $501 $688
5. CSAA $516 $665
6. Mercury $557 $767
7. MetLife $669 $907
8. Allstate $737 $851
9. State Farm $691 $925
10. Hartford $718 $914
11. Farmers $783 $918
12. Travelers $629 $866
13. Allied $755 $972
100 / 300 / 100per year
3. USAA $1,051 $1,118
4. GEICO $1,077 $1,185
5. CSAA $1,201 $1,268
6. Mercury $1,228 $1,321
7. MetLife $1,181 $1,272
8. Allstate $1,199 $1,273
9. State Farm $1,263 $1,405
10. Hartford $1,326 $1,434
11. Farmers $1,376 $1,479
12. Travelers $1,536 $1,646
13. Allied $1,442 $1,563
Cheapest Car Insurance in Modesto If You Have Tickets
Car insurance costs more for people with violations. In Modesto, Esurance policyholders with no violations pay an average of $997 per year, but a speeding ticket increases the average rate by $265. Meanwhile, an at-fault accident bumps up your costs by an average of $565, and a DUI will add roughly $1,480. Make sure to keep your record clean if you want to score cheap car insurance in Modesto.
Speeding Ticketper year
At-Fault Accidentper year
DUIper year
1. Esurance $1,262 $1,562 $2,477
2. Progressive $1,893 $2,058 $2,362
3. USAA $1,222 $1,952 $3,163
4. Mercury $1,731 $2,300 $2,168
5. GEICO $1,749 $1,997 $2,780
6. Farmers $2,030 $2,537 $2,537
7. CSAA $1,591 $1,591 $4,524
8. Allstate $1,797 $2,265 $4,011
9. State Farm $2,071 $2,157 $4,321
10. Allied $2,189 $2,513 $3,775
11. Travelers $2,453 $2,716 $3,404
12. Hartford $2,251 $3,162 $5,288
13. MetLife $1,650 $2,142 $7,994
Average Cost of Car Insurance by Postal Code
For many cities in the United States, car insurance prices can fluctuate by hundreds of dollars depending on the driver's ZIP code. However, this is not the case for Modesto. The difference between the most and least expensive ZIP codes in the city is $22.
95350 $1,309
top ratedCar Insurance agentsModesto2021
Top Agents Offering Affordable Car Insurance in Modesto
1. TopFlight Insurance Services901 N Carpenter Rd, Ste 37, Modesto, CA, 95351
2. Allstate Insurance: David Ngo3330 Mchenry Ave, Modesto, CA, 95350
3. Wylie Insurance1729 Tully Rd, Ste 6B, Modesto, CA, 95350
4. The Source Auto Insurance Agency822 9th St, Ste 1, Modesto, CA, 95354
5. Toste Insurance Services3501 Coffee Rd, Modesto, CA, 95355
6. JCO Insurance2909 Coffee Rd, Ste 14, Lapata International, Modesto, CA, 95355
7. Corina Hodges - State Farm Insurance Agent2209 Coffee Rd, Ste E, Modesto, CA, 95355
8. Gingerich Insurance817 Coffee Rd, Ste C-2, Modesto, CA, 95355
9. Cost-U-Less Insurance123 Burney St, Modesto, CA, 95354
10. SaveDirect Insurance Agency1800 Tully Rd, Ste G, Modesto, CA, 95350
11. Pam Shelton-Allen - State Farm Insurance Agent3516 Oakdale Rd, Ste A, Modesto, CA, 95357
12. Nick Winters - State Farm Insurance Agent3931 Coffee Rd, Ste A, Modesto, CA, 95355
13. Lamoure's Insurance Service1367 Standiford Ave, Modesto, CA, 95350
14. Omega Pacific Insurance Solutions1016 12th St, Ste A, Modesto, CA, 95354
15. TSM Insurance Services1317 Oakdale Rd, Ste 910, Modesto, CA, 95355
16. Gladiator Insurance Agency118 McHenry Ave, Modesto, CA, 95354
Insurance Providers That Serve Modesto
Factors That Impact Insurance Costs in Modesto
In addition to individual driver needs and history, location-specific factors like the percentage of uninsured motorists and the rate of fatal accident also contribute to car insurance rates in Modesto. How Modesto ranks compared to the 300 most populous cities in the United States can have a significant impact on premium costs.
Most Accidents With Fatalities38th/300
Most Car Thefts301st/300
Most Uninsured Motorists by State12th/50
Personal Injury Protection RequiredNo
Most Annual Rainfall245th/300
Most Rainy Days235th/300
Shopping around for car insurance in Modesto could save you $325, which is enough to buy:
108Cups of Coffee
25Months of Your Favorite Streaming Service
Methodology and Assumptions
MoneyGeek's average premium rates for Modesto are based on 100/300/100 comprehensive collision coverage with a $1,000 deductible for a 40-year-old male driver. The average driver profiled drives a 2010 Toyota Camry LE and has no tickets or accidents on their record. Additional premiums are added based on overage type, location and driving record. Learn more about MoneyGeek's data and methodology here.


Select your Star rating above in each Category and Click "Submit Your Rating" Write a Review of your Visit of Aspenglen Campground. .
Aspenglen Campground is located on U.S. Highway 34 just west of the Fall River Entrance Station has 54 sites.
Opens at noon May 26, 2010; closes at noon September 25, 2010.
Recreational vehicle and trailer length limit is 30 feet.
A Loop: tents only, including walk-ins; B Loop: no generators allowed; C Loop: generators allowed.


text:
Leonardo DiCaprio Net Worth and Interesting Facts You May Not Know: Leonardo Soon to Sell His Malibu Home!
Leonardo DiCaprio is a 42-year-old American film producer and actor. Leonardo DiCaprio made his big Hollywood breakout in the year 1997 when he starred alongside Kate Winslet in the widely successful film 'Titanic'. This was however not Leonardo DiCaprio's first acting role; before 'Titanic', Leonardo DiCaprio also starred in 'The Basketball Diaries' and 'Romeo and Juliet'. He also played roles in 'What's Eating Gilbert Grape', 'This Boy's Life', 'Critters 3', 'Growing Pains' and 'Santa Barbara'. In recent years, Leonardo DiCaprio has featured in many major films such as 'The Aviator', 'Blood Diamond', 'Catch Me If You Can', 'Inception' and 'The Great Gatsby'. Leonardo DiCaprio has dated a few models including Giselle Bundchen, Emma Miller and Bijou Philips. These are well-known facts about Leonardo DiCaprio, and here are interesting facts you may not know about Leonardo DiCaprio.
Leonardo DiCaprio Malibu Home Just Went On The Market
Leonardo DiCaprio owns a few different residents including a home in Palm Springs, the Hamptons, Belize, as well as a luxury apartment in Greenwich Village and two more in Battery Park City. Alongside these homes, Leonardo DiCaprio also owns a four-bedroom house in Malibu which he just recently put up for sale. The four bedroom house has 2 bathrooms, a large swimming pool, a built-in bar, gorgeous mountain views and large glass windows. Leonardo DiCaprio has placed his house on the market for $2.2 Million.
He is Named After Leonardo Da Vinci
Leonardo DiCaprio's mother named him after Leonardo Da Vinci because the first time she ever felt him kick when she was pregnant was when she was in a museum and looking at a Leonardo Da Vinci painting. Leonardo DiCaprio was advised to change his name to something more 'American-sounding' when he came to Hollywood, but he decided to keep his name.
He Was Trained By One of The Best
Leonardo DiCaprio was trained by one of the best acting trainers in Hollywood; Thell Reed. Leonardo DiCaprio shares the same acting trainer with Brad Pitt, Val Kilmer, Edward Norton, Russell Crowe and Ben Foster.
He's a Father
Not many people know that Leonardo DiCaprio is, in fact, a father to a little girl. She is not actually his biological daughter; she is an adopted child. While Leonardo DiCaprio was in South Africa in order to film the movie 'Blood Diamond', he adopted a little girl from an orphanage who he sends a check to every month. Leonardo DiCaprio also checks up on his little girl and often speaks to her on the phone.
Leonardo DiCaprio has a net worth of around $217 Million.
VIA: TheGuardian


Sir Arthur C. Clarke (1917-2008) wrote a hundred books and more than a thousand short stories and essays covering science fiction and science fact in a career spanning more than six decades. Among his bestselling novels are Childhood's End, 2001: A Space Odyssey, and Rendezvous with Rama.
In 1945, he proposed global broadcasting via communication satellites in geostationary orbit. One of his short stories inspired the World Wide Web, while another was expanded into 2001: A Space Odyssey, which he cowrote with Stanley Kubrick.
Born in Somerset, England, Clarke was educated at King's College, London. He worked in the British civil service and the Royal Air Force before turning full-time author in 1950. The recipient of dozens of awards, fellowships, and honorary doctorates, Clarke had both an asteroid and dinosaur species named after him. Queen Elizabeth II gave him a knighthood in 1998.
Clarke lived in Sri Lanka since 1956, engaged in diving, astronomical observations, and underwater tourism.


text:
Globe Advisor Six ways investors can play a recovering energy sector
ADVISOR CONTEXT
Six ways investors can play a recovering energy sector
Chevron Corp. is one company whose performance is reflected in energy-oriented exchange-traded funds.
David Paul Morris/Bloomberg
SHIRLEY WON
Special to The Globe and Mail
Published October 31, 2017 Updated November 1, 2017
Amid a rally in crude oil prices and signs of global economic growth, investors might be tempted to bet on a recovering energy sector.
Because the price of crude has been volatile in its climb to around $54 (U.S.) a barrel recently in New York from less than $30 early last year, it may be easier and more prudent to invest in exchange-traded funds (ETFs) rather than individual stocks.
Given uncertainty about OPEC production cuts and continuing debate about the rally's sustainability, investors might also consider a tactical approach aimed at shorter-term potential profits. However, investors who have a big weighting in Canadian equity funds may already have sufficient energy exposure.
We asked three ETF experts for their top picks to play an energy recovery.
Daniel Straus, ETF analyst at National Bank Financial Inc., Toronto
The pick: VanEck Vectors Oil Refiners ETF
Management expense ratio: 0.59 per cent
This global oil-refiner ETF has diversification potential because it can do well when energy prices rise, or when they fall, says Mr. Straus. Refineries transform crude oil into gasoline, jet fuel and other products. The "crack spread," which is the price difference between crude and petroleum products, has been rising this year, he said. "Our energy analysts [at National Bank] believe the refineries are still in a recovery mode." Refiners, however, can also benefit when the price of crude falls, since it is their key raw material, he said. The dynamics of the energy market have worked in this ETF's favour as it gained 40 per cent for the year ended Sept. 30, he said. This fund is suited to aggressive or tactical investors because buyers face a risk that oil prices could rise faster than the price of petroleum products, he added.
The pick: BMO Junior Oil ETF
MER: 0.59 per cent
This junior oil-and-gas ETF has more upside potential than its larger-cap peers when commodity prices rise, but also more downside when they fall, says Mr. Straus. The ETF, which traded at around $30 a unit in 2014, is now hovering around $12. It is suited to investors who are "confident in their bullish attitude towards the price of oil," he said. This fund invests in oil and gas explorers and producers, as well as equipment, service and drilling firms. Its 50 holdings include Crescent Point Energy Corp. and Whitecap Resources Inc. Large-cap companies may be a safer way to play the oil market, but they tend to have many business lines and commodity hedges into the future, he said. "That complicates their relationship to the price of crude."
Denise Davids, ETF and mutual fund analyst, Industrial Alliance Securities Inc., Toronto
The pick: Energy Select Sector SPDR ETF
This ETF, which tracks energy stocks in the S&P 500 index, is a low-cost way for investors to tap into the space, says Ms. Davids. "The United States is one of the leading oil-producing countries in the world, and this ETF provides a good way to gain access to its energy sector." It holds 32 securities. Because this fund tracks a market-cap-weighted index, the percentage weighting of the firms in the ETF depends on their market value. The portfolio is top heavy in Exxon Mobil Corp. and Chevron Corp., which make up 40 per cent of the ETF. Negative news affecting these two industry giants, such as serious disruptions at major refineries or a lack of production growth, could hurt this fund, she noted. This easily tradeable ETF is suited to investors who can tolerate moderate risk, she said.
The pick: iShares S&P/TSX Capped Energy Index ETF
Investors will get diversified exposure to the Canadian energy sector in this fund, but its fee is "pretty high" compared with that of its U.S. ETF peer, says Ms. Davids. This fund tracks 39 names that make up about 20 per cent of the S&P/TSX composite index, and it's the only broad Canadian energy ETF available, she noted. The fund holds oil and gas explorers, producers, integrated companies and drillers as well as oil-service firms. The portfolio is top heavy, with names such as Suncor Energy Inc. and Canadian Natural Resources Ltd. making up about 45 per cent of the ETF. About 13 per cent is invested in small-cap stocks, which tend to be more sensitive to commodity-price movements and can "exhibit higher bouts of volatility." This fund is suited to higher-risk investors, she said.
David Kletz, ETF analyst and portfolio manager, Forstrong Global Asset Management Inc., Toronto
The pick: Global X MSCI Norway ETF
This Norwegian equity ETF, which has 26 per cent in energy companies, offers a diversified and less volatile way to play a recovery in the industry, says Mr. Kletz. Among oil-dependent nations, Norway is a standout, he said. "With geopolitical tensions flaring up worldwide, Norway's high degree of political stability is important." But other areas of the Norwegian economy, which are represented in the fund, "have a high degree of sensitivity to the energy sector as well," he noted. Norwegian financials, which make up 24 per cent of the ETF, carry a large loan-book exposure to energy firms. The fund's largest names are Statoil ASA, DNB ASA and Telenor ASA. The risk comes from the fact that the top 10 holdings represent more than two-thirds of this unhedged ETF, he said.
The pick: SPDR S&P Oil & Gas Exploration & Production ETF
This ETF, which tracks U.S. oil-and-gas explorers and producers, should outperform more broad-based energy-sector ETFs when commodity prices rise, says Mr. Kletz. This group has more upside potential than its peers involved in the transportation, storage and refining of crude. This ETF is suited to aggressive investors with a high-risk tolerance and long-term horizon because 35 per cent of the holdings are in the more volatile, smaller-cap companies, he noted. The ETF's 64 equal-weighted names include Carrizo Oil & Gas Inc., Whiting Petroleum Corp. and Marathon Oil Corp. "While oil prices have been relatively stable since mid-2016, it is not uncommon during a bear market for prices to appreciate and depreciate rapidly," he cautioned. Forstrong, he said, is not expecting a renewed bull market soon.
The days of double digit returns are over. Rob Carrick, personal finance columnist, lays out what you can expect given your investment risk profile. The Globe and Mail
Country or region? Pick focused ETFs wisely Globe Advisor content
Passion bested intimidation for young portfolio manager Globe Advisor content
Do your investments measure up? How to assess them Globe Advisor content
How to invest a sudden cash windfall Globe Advisor content
Seven ETFs to counter a Canadian portfolio bias Globe Advisor content
Companies serving seniors ride demographic tailwind Globe Advisor content
Optimism and ambivalence for investors in sub-Saharan Africa Globe Advisor content


text:
Phone : +1 (218)-288-1755
Why People Think They Might Hasten Their Death When Faced With Irremediable Health Conditions Compared to Why They Actually Do so
Post Author:wayneboyd112
Robert R. Blake, Charlie BlakeFirst Published July 23, 2021 Research Article https://doi.org/10.1177/00302228211033368
This study surveys the differences of relatively healthy proponents of end-of-life choices and people with irremediable health conditions having already made the decision to hasten their deaths on what each group considers important in influencing a desire to hasten death. Psychosocial factors were more important than physical ones for both groups; but those contemplating what might influence them to hasten their deaths in the future thought pain and feeling ill would be much bigger factors than they turned out to be for those deciding to do so. Those having decided to hasten their deaths cited the lack of any further viable medical treatments and having to live in a nursing home as bigger factors. Identifying these psychosocial factors influencing a desire for a hastened death suggests that caregivers and medical providers may want to review what compassionate understanding and support looks like for people wanting to hasten their death.Keywords end of life decisions, desire to hasten death, assisted suicide, elder suicide, terminally ill
Since Derek Humphry's best-selling book in 1991, Final Exit: The Practicalities of Self-Deliverance and Assisted Suicide, public opinion on terminally ill patients seeking a hastened death has moved from wildly controversial to enormous public support, including now the majority of physicians (Humphry, 1991; Kane, 2020). After initial concerns about potential harmful consequences for vulnerable populations proved unfounded (Lindsay, 2009; Albaladejo, 2019), more and more countries and states are passing legislation to support the public's interest in being able to have control over the manner and timing of their own death when faced with irremediable health conditions. Final Exit Network (FEN) is one of the organizations in the United States, alongside ten (and growing) state approved medical-aid-in-dying options, and other domestic and international groups and countries, that support end of life options. Examples of these organizations include Compassion and Choices, Death with Dignity, Dignitas, Exit International, and the World Federation of Right to Die Societies. These programs continue to grow in number and provide resources for those seeking information about how to achieve a peaceful death rather than a drawn-out process of increasing frailty, violent gun options, or loss of quality of life and autonomy due to health conditions over which they otherwise have little or no control (Bellamy, 2017). Seniors ages 85 and older had the second-highest suicide rate in the United States in 2018, and firearms were used in 70.0% of them among seniors aged 65 and older (America'sHealthRankings, 2021; Mertens & Sorenson, 2012). State-sponsored physician-aid-in-dying programs and informational organizations like the ones we have mentioned above aim to offer elders and others with intractable health conditions peaceful options when it comes to controlling their deaths.
There have been multiple studies seeking information from doctors, families, and patients assessing why people who use state sanctioned physician-aid-in-dying programs have chosen to hasten their deaths as well as studies of people's emotions who are dying of terminal illnesses without such choices (Ganzini et al., 2008b; Hendry et al., 2013; Hudson et al., 2006; Singer et al., 1999). Central to these investigations has been the value of a person's autonomy over both their life and death decisions, complicated sometimes by mismatches between provider and patient knowledge, lack of effective communication, familial, financial, demographic and social factors, or outside constraints on patients' real options (Meier & Morrison, 2002; Winzelberg et al., 2005). There have been efforts to identify cognitive, emotional, relational, and values-based needs that go into autonomous decision making at the end of life, and our current study further adds to that identification process (Callahan, 2002). Our current study further delineates the factors that go into what previous studies have generally concluded are the main reasons people wish they could hasten their deaths in the face of progressive illness or frailty. These factors included not only the sense of loss of self and dignity associated with autonomy, but also an inability to participate in activities that make life enjoyable and a generalized global feeling of intolerable or unbearable suffering (Chochinov et al., 2005; Dees et al., 2011; Ganzini et al., 2009; Rodríguez-Prat et al., 2017).
Our current study compares the view of relatively healthy people who are professed supporters of end-of-life options with irremediably ill people who have already decided to hasten their deaths. We wanted to know whether those who are actually suffering from intractable health conditions and faced with end-of-life decision making may differ in their views from relatively healthy people who are thinking about the questions hypothetically. We sought to understand what factors were most important to each of these groups by asking each group to rate the importance of several physical, cognitive, social, and emotional factors in influencing their current thoughts on why they would hasten their deaths.
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text:
\section{Introduction}
  {One of the central questions in knot theory is whether two knots are topologically equivalent.
    Currently, the most effective way to
    determine this is to compute some polynomial knot invariant for each knot:
    if the invariants differ, knots are for sure not topologically equivalent.
    Complementary statement is not true: if invariants are the same it does not mean
    that the knots are equivalent. So, in knot theory one seeks stronger and stronger
    knot invariants that have the power to distinguish more and more knots.
  }

  {Perhaps, the most widely known polynomial knot invariants (knot polynomials) are
    the so-called Jones \cite{JonesPolynomialDef} and HOMFLY \cite{HOMFLYPolynomialDef,PTPolynomialDef} polynomials.
    One of the ways to compute (and define) them is quite elementary
    -- one just repeatedly applies skein relations

    \begin{align} \label{eq:skein-relation}
    \begin{picture}(300,40)(-50,-10)
      \put(0,0){\put(-12,6){$A$} \thicklines \crossingPosUp}
      \put(40,7){$-$}
      \put(80,0){\put(-18,6){$A^{-1}$} \thicklines \crossingNegUp}
      \put(120,7){$=$}
      \put(190,0){\put(-48,6){$\left(q - q^{-1}\right)$} \thicklines
        \arcRight \put(0,20){\vector(-1,1){0}}
        \put(20,0){\arcLeft} \put(20,20){\vector(1,1){0}}
      }
    \end{picture}
    \end{align}

    \noindent and Reidemeister moves to express knot polynomial for a given planar diagram
    through polynomials for simpler diagrams.

  }

  {
    {

      A surprising and indirect consequence of the above naive definition is the existence
      of the so-called \textit{evolution method} for both Jones and HOMFLY polynomials
      (see, for instance, \cite[Introduction]{AnoM}; this story is also well-explained in \cite{MMM-evo-diff-colored}).
      This is the statement that, whenever
      an $m$-strand braid is present somewhere in the planar diagram $\mathcal{K}$ of some knot,
      the knot polynomial $P$ (either Jones or HOMFLY)  depends on the number $n$ of windings of the braid
      in quite a simple way

      \begin{align} \label{eq:evolution-method}
        P^{\mathcal{K}}(n) = \sum_{\lambda \vdash m} C^{\mathcal{K}}(\lambda) \left((\pm)_\lambda q^{\text{power}(\lambda)}\right)^n
      \end{align}

      \noindent Here sum is over partitions $\lambda$ of the number of strands $m$ (i.e. over Young diagrams),
      and sign~'$(\pm)_\lambda$'
      and the exponent~'$\text{power}(\lambda)$' of an eigenvalue are universal; they depend only on partition~$\lambda$
      but not on the diagram $\mathcal{K}$.
      In fact, they are simple
      combinatorial expressions of the shape of~$\lambda$.
      The \textit{evolution coefficients}~$C^{\mathcal{K}}(\lambda)$, on the contrary,
      depend both on the knot diagram~$\mathcal{K}$ and the partition~$\lambda$.
    }

      A subtle feature of the above formula is that number of windings $n$ can be
      arbitrary integer -- the evolution coefficients $C^{\mathcal{K}}(\lambda)$
      \textbf{do not change} as one goes from negative number of crossings
      (i.e. crossings in a different direction)
      to zero crossings to positive number of crossings
      in the braid. The formula seamlessly interpolates between these three cases.
      This feature is, in fact, a manifestation of another, more conceptual and deep,
      definition of Jones and HOMFLY polynomials
      -- through the so-called Reshetikhin-Turaev (RT) formalism \cite{RTpaper}.
    }

    {
      In the RT-formalism one associates a certain linear operator,
      the so-called $\mathcal{R}$-matrix,
      to each positive crossing
      and to each negative crossing its inverse. The factors $\left((\pm)_\lambda q^{\text{power}(\lambda)}\right)$
      in the formula \eqref{eq:evolution-method}
      are then naturally reinterpreted as eigenvalues of the $\mathcal{R}$-matrix in the irreducible representation
      corresponding to the Young diagram $\lambda$. The knot polynomial itself
      is just a tensor contraction of several $\mathcal{R}$-matrices,
      in the order dictated by the planar diagram $\mathcal{K}$.
    }

    { RT-formalism allows one to establish a link between knot theory and physics:
      knot polynomials turn out to be Wilsonian averages in Chern-Simons theory
      (see, for instance, \cite{Ano-r-matrix-approaches} for a review).
      This is most easily seen in the so-called \textit{temporal gauge} \cite{SmirnovMorozovTempGauge}.
      The gauge theory point of view immediately leads to the generalization
      of both Jones and HOMFLY polynomials to the \textit{colored} case, where one decorates the knot with some
      representation of the gauge group. The skein relation description is not available in the colored case
      and from the naive skein relations \eqref{eq:skein-relation} it is not at all easy to guess, that
      the colored generalization should even exist.
    }
  }

  {
    {The (uncolored, or fundamental)
      Jones and HOMFLY polynomials have a \textit{homological} generalization -- Khovanov \cite{KhovanovDefinition}
      and Khovanov-Rozansky \cite{KhovanovRozanskyDefinition1,KhovanovRozanskyDefinition2}
      polynomials, respectively. The definition of these polynomials (especially the Khovanov-Rozansky one)
      seems much more elaborate than definition of their non-homological analogs -- one needs to calculate
      the homology groups of
      a certain differential complex, built out of the planar diagram $\mathcal{K}$.
      While in the case of Khovanov polynomial at least the construction of the complex is well-understood
      \cite{BarNatan},
      in the Khovanov-Rozansky case even explicit construction of the linear spaces in the complex
      presents difficulties (each space is a factor of an infinite-dimensional space of so-called foams
      over infinitely many relations \cite{RobertWagnerSymmetricKhR}).
      Explicit construction of maps between the spaces is even more difficult.
      Hence, one is tempted to search for alternative definitions for Khovanov and Khovanov-Rozansky
      polynomials \cite{DolMorI,DolMorII,DolMorIII,AnoMorTowardsRMatKhov,AnoSolCount},
      partly motivated by the empirical observation that \textit{answers}
      for both Khovanov and Khovanov-Rozansky polynomials (in known examples)
      seem much simpler than their cumbersome definitions would lead one to expect.
    }

    {A natural question on this path is
      whether the relevant generalization of the RT formalism
      exists for the Khovanov and Khovanov-Rozansky polynomials.
      This homological RT formalism is believed to be related
      to the so-called \textit{refined} \cite{ArthShakDAHA} Chern-Simons theory,
      which is at the moment defined only for the simplest examples of knots.
    }
  }

  {
    {Even more naively, one may wonder, whether something like evolution formula \eqref{eq:evolution-method}
      exists for Khovanov and Khovanov-Rozansky polynomials.
      In \cite{AnoM} this was investigated for the simplest case of torus knots.
      Surprisingly, it was observed that the symmetry between positive and negative crossings is broken
      -- an essential difference with the Jones and HOMFLY case.
      Moreover, from studies of superpolynomials (the $N \rightarrow \infty$ stable component of Khovanov-Rozansky polynomials)
      for torus knots (see \cite{DBMMSS-superpolys-toric})
      one knows that for them the symmetry between positive and negative crossings is \textit{not} broken,
      which makes the observed breaking at finite $N$ even more unexpected.
    }
  }

  {
    {In this short note we look at what happens for a slightly
      more complicated family of knots (see Section~\ref{sec:fig-eight-like-knots}).
      This family is inspired by the figure 8 knot (the first non-torus knot)
      and one is allowed to insert arbitrary two-strand braids in place of its upper and lower pairs of crossings.}

    {We concentrate only on Khovanov polynomials and completely ignore Khovanov-Rozansky ones.
      First of all, Khovanov polynomials are easier to calculate:
      there is a computer program readily available in the ``KnotTheory'' package for Wolfram Mathematica,
      which is available on Katlas~\cite{katlas}, though there are some caveats (see Section~\ref{sec:caveats}).
      Second of all, non-trivial new phenomena are seen already in Khovanov case, so this generalization
      is sufficiently interesting.
    }

    {
      {We find the following picture (see Section~\ref{sec:phase-diagram}, Theorem~\ref{thm:evolution-method}).
        For some suitable subfamily (see Section~\ref{sec:fig-eight-like-knots}) we observe that:
        \begin{itemize}
        \item evolution is preserved in the \textit{chambers} on the parameter plane;
          there are also 2 isolated points, where the knot becomes two unlinked unknots,
        \item if one interprets the formulas as coming from some sort of RT-formalism,
          one concludes that \textit{fundamental} $\mathcal{R}$-matrix must have one more eigenvalue:
          $(-1) t q^3$, in addition to $t q^3$ and $q$. The coefficient in front of this
          eigenvalue is always proportional to $(t+1)$ and vanishes when one goes to
          the Jones limit $t \rightarrow -1$,
        \item transitions between chambers are tricky: some coefficients in front of eigenvalues
          just get multiplied by some constants, while others re-glue in a more elaborate way.
        \end{itemize}
      }
    }

    { We prove our evolution formulas (Theorem~\ref{thm:evolution-method}) in Section~\ref{sec:proof}. The proof relies
      on the fact that all knots and links in the family we consider are alternating, and on the reconstruction
      theorem \cite[Theorem 4.5]{Lee-endomorphism} that allows one to completely determine Khovanov polynomial
      from the Jones polynomial for the alternating knot/link, provided the signature is known.
      The symmetry breaking in the evolution method is ultimately traced to the jumps in the signature.
    }

    { Forward references throughout the introduction provide sufficient description
      of the organization of the paper, so ``this paper is organized as follows'' paragraph
      is really unnecessary.
    }

  }

  {\subsection*{Acknowledgements} We would like to thank A.Anokhina, A.Morozov, Y.Zenkevich
    and M.Khovanov
    for stimulating discussions.
    The research of A.P. is supported in part by
    Vetenskapsr\r{a}det under grant \#2014-5517,
    by the STINT grant,
    by the grant  ``Geometry and Physics"  from the Knut and Alice Wallenberg foundation
    and by RFBR grants 16-01-00291 and 18-31-20046 mol\_a\_ved.
    The research of P.D.-B. is supported in part by RFBR grants 18-01-00461 A and 18-31-20046 mol\_a\_ved.
  }
}

\newcommand\figureEightLikeTemplate{
  {
    \put(0,20){\arcSmallLeft}
    \put(60,20){\arcSmallRight}
    \put(0,30){\slashPos}
    \put(60,30){\slashNeg}
  }
  {
    \put(-10,0){\arcSmallDown}
    \put(-20,20){\bendDownRight}
    \put(-20,20){\line(0,1){70}}
    \put(-20,90){\bendUpRight}
    \put(-10,110){\arcMedUp}
  }
  {
    \put(60,0){\arcSmallDown}
    \put(80,20){\bendDownLeft}
    \put(80,20){\line(0,1){70}}
    \put(80,90){\bendUpLeft}
    \put(40,110){\arcMedUp}
  }
}

\newcommand\theBottomBraid{
  {
    \linethickness{0.5mm}
    \put(0,0){\line(0,1){20}}
    \put(0,0){\line(1,0){60}}
    \put(0,20){\line(1,0){60}}
    \put(60,0){\line(0,1){20}}
    \put(27,6){$b$}
  }
}

\newcommand\theTopBraid{
  {
    \linethickness{0.5mm}
    \put(0,0){\line(0,1){60}}
    \put(0,0){\line(1,0){20}}
    \put(0,60){\line(1,0){20}}
    \put(20,0){\line(0,1){60}}
    \put(7,26){$a$}
  }
}

{\section{The figure-eight-like knots} \label{sec:fig-eight-like-knots}
  {We consider the following family of planar diagrams, inspired by the figure-8 knot

    \begin{picture}(300,130)(-20,-5)

      \put(0,0){
        \thicklines
        \figureEightLikeTemplate

        \put(0,0){
          \put(0,0){\crossingNegDown}
          \put(20,0){\arcDown}
          \put(20,20){\arcUp}
          \put(40,0){\crossingNegUp}
        }
        \put(20,50){
          \put(0,0){\crossingPosLeft}
          \put(0,40){\crossingPosRight}
          \put(0,20){\arcLeft}
          \put(20,20){\arcRight}
        }
      }

      \put(100,50){$\longrightarrow$}

      \put(160,0){
        \thicklines
        \figureEightLikeTemplate

        \theBottomBraid
        \put(20,50){\theTopBraid}
      }

      \put(250,0){
        \put(20,30){
          \theTopBraid
          \thicklines
          \put(0,0){\line(-1,-1){10}}
          \put(20,0){\line(1,-1){10}}
          \put(0,60){\line(-1,1){10}}
          \put(20,60){\line(1,1){10}}
        }
      }
      \put(310,55){$=$}
      \put(320,0){
        \put(20,30){
          \thicklines
          \put(0,0){\line(-1,-1){10}}
          \put(20,0){\line(1,-1){10}}
          \put(0,60){\line(-1,1){10}}
          \put(20,60){\line(1,1){10}}
          \put(0,0){\prePositiveCrossing}
          \put(0,40){\prePositiveCrossing}
          \put(3,30){$\dots$}

          \put(25,28){$\Bigg\}$}
          \put(35,30){$\substack{a \\ \text{times}}$}
        }
      }
    \end{picture}

\noindent i.e. we allow arbitrary 2-strand braid to be inserted in place of higher or lower pair of crossings
of the figure eight knot. This family was also considered in \cite{MMM-evo-diff-colored}, where it was called
``double braid'' family of knots, and where in particular a conjecture was made about explicit form of
colored HOMFLY polynomials for this family in any symmetric representation $[r]$.
  }

  {This family is interesting to consider, since it includes both torus and non-torus knots
    (so we will not observe something that is specific just to torus case).
    It is also two-parametric, which allows us to see, how evolutions w.r.t the
    two braid-winding parameters $a$ and $b$ interplay.
    Moreover, it was instrumental in finding an explicit relation between so-called
    inclusive and exclusive Racah matrices (see \cite{MMMS-racah-hidden-integrability})
  }

  {Depending on the parity of the braid parameters~$a$ and $b$, strands are oriented differently
    (so $a$- and $b$- braids can be both parallel and antiparallel).
    Moreover, in case both $a$ and $b$ are odd the diagram is a link, so orientations of its two
    components can be chosen independently.

  }

  {In what follows we choose a particular orientation of strands

    \begin{picture}(300,130)(-20,-5)
      \put(160,0){
        \thicklines
        \figureEightLikeTemplate

        \theBottomBraid
        \put(20,50){\theTopBraid}

        {
          \put(60,0){\put(2,-2){\vector(-1,0){0}}}
          \put(0,20){\put(-2,9){\vector(0,1){0}}}
          \put(0,0){\put(-2,-2){\vector(1,0){0}}}
          \put(60,20){\put(2,9){\vector(0,1){0}}}
        }
        {
          \put(20,50){\put(-2,-2){\vector(1,1){0}}}
          \put(40,50){\put(2,-2){\vector(-1,1){0}}}
          \put(20,110){\put(-8,7){\vector(-1,1){0}}}
          \put(40,110){\put(8,7){\vector(1,1){0}}}
        }
      }
    \end{picture}

    \noindent With this choice $a$ can be arbitrary integer, while $b$ can only be odd.
    On the example of the $a$-braid we are able to see what happens when we add a single crossing to the diagram.
    If the representation theory is at all applicable to Khovanov polynomials,
    then in some sense we must have, that in the $a$-braid there is the representation $[1]\otimes[1] = [1,1] \oplus [2]$ running,
    while in the $b$-braid there is the representation $[1]\otimes\overline{[1]} = \emptyset \oplus \text{\textit{adjoint}}$ running.
    Of course, it would be interesting in future to consider a family of knots that allows
    to add single crossings to more that one of its braids and see whether this produces some interesting effects.
}

  {\section{The phase diagram} \label{sec:phase-diagram}
  {We find (see Section~\ref{sec:proof} for a proof) that the space of parameters splits into 4 regions and 2 isolated points

    \begin{picture}(300,170)(0,-10)
      \put(200,75){

          \put(-80,0){\vector(1,0){160}}
          \put(0,-80){\vector(0,1){160}}
          \put(80,-5){$a$}
          \put(5,78){$b$}
        }
        {
          \put(10,-2){\line(0,1){4}}
          \put(-2,10){\line(1,0){4}}
          \put(9,-6){${}_1$}
          \put(-7,9){${}_1$}
        }
        {
          \put(10,10){\circle*{4}}
          \put(-10,-10){\circle*{4}}
        }
        {

          \multiput(-70,10)(10,0){15}{\circle*{2}}
          \multiput(-70,30)(10,0){15}{\circle*{2}}
          \multiput(-70,50)(10,0){15}{\circle*{2}}
          \multiput(-70,70)(10,0){15}{\circle*{2}}
          \multiput(-70,-10)(10,0){15}{\circle*{2}}
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          \multiput(-70,-50)(10,0){15}{\circle*{2}}
          \multiput(-70,-70)(10,0){15}{\circle*{2}}
        }
        {
          \thicklines
          \put(-75,5){\line(1,0){75}}
          \put(0,0){\qbezier(0,5)(5,5)(5,10)}
          \put(5,10){\line(0,1){65}}
        }
        {
          \thicklines
          \put(20,10){\qbezier(-5,0)(-5,-5)(0,-5)}
          \put(20,5){\line(1,0){55}}
          \put(10,30){\qbezier(-5,0)(-5,-8)(0,-10)}
          \put(10,10){\qbezier(5,0)(5,8)(0,10)}
        }
        {
          \thicklines
          \put(0,-5){\line(1,0){75}}
          \put(0,-10){\qbezier(0,5)(-5,5)(-5,0)}
          \put(-5,-10){\line(0,-1){65}}
        }
        {
          \thicklines
          \put(-20,-10){\qbezier(5,0)(5,5)(0,5)}
          \put(-20,-5){\line(-1,0){55}}
          \put(-10,-30){\qbezier(5,0)(5,8)(0,10)}
          \put(-10,-10){\qbezier(-5,0)(-5,-8)(0,-10)}
        }
        \put(-5,-10){
          \put(-50,35){\color{gray}{\Huge{UL}}}
          \put(25,35){\color{gray}{\Huge{UR}}}
          \put(25,-35){\color{gray}{\Huge{LR}}}
          \put(-50,-35){\color{gray}{\Huge{LL}}}
        }
      }
    \end{picture}

    \noindent The isolated points are $(1,1)$ and $(-1,-1)$ and in what follows we will refer to the
    four regions as UL, UR, LL and LR for upper-left, upper-right, lower-left and lower-right, respectively.
  }

  {Dependence of Khovanov polynomial on the number of braid windings $a$ and $b$ is quite remarkable:
    \begin{theorem}\label{thm:evolution-method}
    In each of the regions UL, UR, LL and LR dependence of the Khovanov polynomial on the braid parameters $a$ and $b$
    is consistent with the evolution method

    \begin{align} \label{eq:evolution-formula}
      Kh_{\substack{a \hfill \\ b=2 k+1}} =
      \left(\begin{array}{cc} (1)^b & (t q^2)^b \end{array}\right)
      \left(\begin{array}{ccc} M_{1,1} & M_{1,2} & M_{1,3} \\ M_{2,1} & M_{2,2} & M_{2,3} \end{array}\right)
      \left(\begin{array}{c} q^{-a} \\ (t q^3)^{-a} \\ (-t q^3)^{-a} \end{array}\right)
    \end{align}

    \noindent but the matrices of coefficients $M$ are different among the four regions

    \begin{align} \label{eq:evolution-matrices}
      M_{UL} & = \left(\begin{array}{ccc}
        \frac{q^8 t^3+q^6 t^3-q^4 t^2-q^4 t+q^2+1}{\left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{- q^5 t^2-q t}{\left(q^2 t-1\right)^2 \left(q^2 t+1\right)}
        & 0
        \\ \frac{-q^8 t^3+q^6 t^2-q^4 t^2-q^4 t+q^2 t-1}{\left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{2 q^8 t^3+q^6 t^3-q^6 t^2-q^2 t+q^2+2}{2 \left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{- q t-q}{2 q^2 t+2}
      \end{array}\right) \\
      M_{UR} & = \left(\begin{array}{ccc}
        \frac{q^8 t^4-2 q^6 t^3-q^6 t^2+q^2 t-1}{q t \left(q^2 t-1\right)^2 \left(q^2 t+1\right)}
        & \frac{q^5 t+q}{\left(q^2 t-1\right)^2 \left(q^2 t+1\right)}
        & 0
        \\ \frac{q^8 t^3-q^6 t^2+q^4 t^2+q^4 t-q^2 t+1}{q t \left(q^2 t-1\right)^2 \left(q^2 t+1\right)}
        & \frac{q^8 t^4-q^8 t^3-2 q^6 t^3-q^4 t^2-q^4 t+2 q^2 t-2}{2 q t \left(q^2 t-1\right)^2 \left(q^2 t+1\right)}
        & \frac{-q^3 t-q^3}{2 q^2 t+2}
      \end{array}\right) \\ \notag
      M_{LL} & = \left(\begin{array}{ccc}
        \frac{-q^8 t^4+q^6 t^3-q^2 t^2-2 q^2 t+1}{\left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{q^5 t^3+q t^2}{\left(q^2 t-1\right)^2 \left(q^2 t+1\right)}
        & 0
        \\ \frac{q^8 t^4-q^6 t^3+q^4 t^3+q^4 t^2-q^2 t^2+t}{\left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{-2 q^8 t^4+2 q^6 t^3-q^4 t^3-q^4 t^2-2 q^2 t-t+1}{2 \left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{-t-1}{2 \left(q^3 t+q\right)}
      \end{array}\right) \\ \notag
      M_{LR} & = \left(\begin{array}{ccc}
        \frac{q^8 t^3+q^6 t^3-q^4 t^2-q^4 t+q^2+1}{\left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{-q^5 t^2-q t}{\left(q^2 t-1\right)^2 \left(q^2 t+1\right)}
        & 0
        \\ \frac{-q^8 t^3+q^6 t^2-q^4 t^2-q^4 t+q^2 t-1}{\left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{2 q^8 t^3+q^6 t^3-q^6 t^2-q^2 t+q^2+2}{2 \left(q^2 t-1\right)^2 \left(q^3 t+q\right)}
        & \frac{-q t-q}{2 q^2 t+2}
      \end{array}\right)
    \end{align}
    \end{theorem}

    We give a proof of this theorem in Section~\ref{sec:proof}.
  }

  {
    {The fact that the space of parameters splits into chambers with phase transitions between them
      is striking by itself, however, evolution formula \eqref{eq:evolution-formula} has
      another notable feature: there are \textit{three} eigenvalues for the evolution w.r.t $a$-parameter, whereas
      in the HOMFLY (and Jones) case there were only two. What is the meaning of the third eigenvalue
      in the relevant $t$-deformation of the representation theory
      (where, naively, from the decomposition $[1]\otimes[1] = [2]\oplus[1,1]$ one expects \textit{two} eigenvalues)
      is an interesting question for future research.
    }

    {The coefficient in front of extra eigenvalue turns out to be always proportional
      to $(t+1)$, so it never contributes to the Jones ($t \rightarrow -1$) limit.
    }
  }

  {
    {One immediately sees that some matrix elements in \eqref{eq:evolution-matrices} transform
      quite simply when one goes between the chambers -- they are just multiplied by some scalars.
      Namely, one has the following relations
      \begin{align}
        M_{UR\ 1,2} & = (-t^{-1}) M_{UL\ 1,2}; \ \ M_{UL\ 1,2} = M_{LR\ 1,2} = (-t^{-1}) M_{LL\ 1,2}
        \\ \notag
        M_{UR\ 2,1} & = (-t^{-1}) M_{UL\ 2,1}; \ \ M_{UL\ 2,1} = M_{LR\ 2,1} = (-t^{-1}) M_{LL\ 2,1}
        \\ \notag
        M_{UR\ 2,3} & = q^2 M_{UL\ 2,3}; \ \ \ \ \ \ \ \ M_{UL\ 2,3} = M_{LR\ 2,3} = q^2 M_{LL\ 2,3}
      \end{align}
      \noindent Note that the scalar factor is the same when going from LL-chamber to LR or UL
      and when going from LR or UL to UR. At the moment we don't know why is this so and whether it is always so.
    }

    {On the other hand, coefficients $M_{1,1}$ and $M_{2,2}$ transform in a more complicated way.
      If one plots the (Laurent) polynomials that result from multiplication of the $M_{1,1}$ matrix elements
      by $\left(q^2 t-1\right)^2 \left(q^3 t+q\right)$ on the $(q,t)$ Newton plane one sees the following
      picture (when two points are drawn close to the same integer point that means that they are both
      at this integer point, just are drawn this way so as not to clash)

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          \put(0,80){\put(5,0){$t$-power}}
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        {
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          }
        }
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            }
            \put(0,0){
              \put(0,0){\color{blue} \qbezier(-2,-2)(-10,-10)(-2,-18)}
              \put(-2,-18){\color{red}\vector(1,-1){0}}
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              }
            }
          }
        }
      \end{picture}

      \noindent One of the possible ways of how different monomials ``flow'' when one goes from LL-chamber
      to LR-chamber to UR-chamber is denoted with arrows. It's interesting that this flow features multiplication
      by same coefficients $q^2$ and $-t^{-1}$ that appear in phase transition of the less complicated matrix elements
      $M_{1,2}$, $M_{2,1}$ and $M_{1,3}$. The picture of the Newton plane for the elements $M_{2,2}$ is similar.
      It is not clear, what is the generic rule of transformation of the matrix elements of the evolution method
      -- one needs to study more complicated families of knots to even make a guess.
  }

  {
    {Transitions between regions, where evolution is valid, seem to happen when
      something ``essential'' happens to the planar diagram as a result of adding/removing a particular
      crossing.}

    {For instance, when we go from the point $(1,1)$ to the point $(2,1)$ the knot goes from two unknots
      which are not mutually linked to one unknot

      \begin{picture}(300,130)(-60,-5)
        \put(0,0){
          \thicklines
          \figureEightLikeTemplate
          \put(20,70){
            \crossingPosUp
            \put(0,-20){\line(0,1){20}}
            \put(0,20){\line(0,1){20}}
            \put(20,-20){\line(0,1){20}}
            \put(20,20){\line(0,1){20}}
          }
          \put(20,0){
            \crossingNegUp
            \put(-20,0){\line(1,0){20}}
            \put(20,0){\line(1,0){20}}
            \put(-20,20){\line(1,0){20}}
            \put(20,20){\line(1,0){20}}
          }
        }
        \put(120,50){$\longrightarrow$}
        \put(200,0){
          \thicklines
          \figureEightLikeTemplate
          \put(20,70){
            \put(0,-20){\crossingPosUp}
            \put(0,20){\crossingPosUp}
            \put(0,0){\line(0,1){20}}
            \put(20,0){\line(0,1){20}}
          }
          \put(20,0){
            \crossingNegUp
            \put(-20,0){\line(1,0){20}}
            \put(20,0){\line(1,0){20}}
            \put(-20,20){\line(1,0){20}}
            \put(20,20){\line(1,0){20}}
          }
        }
      \end{picture}
    }

    {\noindent and in transition from $(1,3)$ to $(0,3)$
      Hopf link goes into unknot

      \begin{picture}(300,130)(-60,-5)
        \put(0,0){
          \thicklines
          \figureEightLikeTemplate
          \put(20,70){
            \crossingPosUp
            \put(0,-20){\line(0,1){20}}
            \put(0,20){\line(0,1){20}}
            \put(20,-20){\line(0,1){20}}
            \put(20,20){\line(0,1){20}}
          }
          \put(0,0){\crossingNegUp}
          \put(20,0){\crossingNegDown}
          \put(40,0){\crossingNegUp}
        }
        \put(120,50){$\longrightarrow$}
        \put(200,0){
          \thicklines
          \figureEightLikeTemplate
          \put(20,70){
            \put(0,-20){\line(0,1){60}}
            \put(20,-20){\line(0,1){60}}
          }
          \put(0,0){\crossingNegUp}
          \put(20,0){\crossingNegDown}
          \put(40,0){\crossingNegUp}
        }
      \end{picture}

      \noindent it is not at all clear where and when the phase transitions occur in general
      -- one needs to study more examples.
     }
  }

}

  {\section{Caveats in using KnotTheory package to calculate Khovanov polynomials}
    \label{sec:caveats}

    Even though the program ``Kh'' that is included in the ``KnotTheory'' package for Wolfram Mathematica
    calculates most Khovanov polynomials very fast and answers are correct,
    it somehow makes mistakes for the diagrams that contain a small number of crossings.

    For instance, it incorrectly calculates Khovanov polynomial of an unknot whose diagram contains just one crossing.
    Since we are no experts in Java, in which the program is actually written, and cannot debug it,
    we've used a workaround: we inserted an extra two-strand braid in our diagram, that contained alternating
    positive and negative crossings. Thanks to the second Reidemeister move such planar diagram is equivalent
    to the original diagram, so the Khovanov polynomial should not change. But it does have more crossings
    and the ``Kh'' program does not err on it.
  }

  {\section{Proof} \label{sec:proof}

    In this section we prove the Theorem~\ref{thm:evolution-method}.

    First of all, note, that all knots and links in family that we consider are alternating.

    Second, for alternating knots and links, there is a theorem (\cite[Theorem 4.5]{Lee-endomorphism})
    that allows one to express Khovanov polynomial through the Jones one.
    For completeness, we reproduce it here.
    \begin{theorem}\cite{Lee-endomorphism} \label{for-thm:alternating-reconstruction}
      For an $n$ component oriented nonsplit alternating link $L$ with its components
      $S_1, \dots, S_n$ and linking numbers $l_{jk}$ of $S_j$ and $S_k$,
      \begin{align}
        Kh(L)(q,t) = q^{-\sigma(L)}
        \left ( (q + q^{-1})
        \left (\sum_{E \subset \{2,\dots n\}} (t q^2)^{\sum_{j \in E, k \notin E} 2 l_{jk}} \right)
        + \left ( q^{-1} + t q^2 \cdot q \right ) Kh'(L)(t q^2)
        \right )
      \end{align}
      for some polynomial $Kh'(L)$.
    \end{theorem}

    In our case link has at most two components.
    The signature, depending on the region of the parameter plane, equals
    \begin{align} \label{eq:signature-for-quadrants}
      \sigma_{UR} & = a - 2 \\ \notag
      \sigma_{UL} & = a \\ \notag
      \sigma_{LR} & = a \\ \notag
      \sigma_{LL} & = a + 2
    \end{align}

    For even $a$ our diagram is a knot, whereas for odd $a$ it is a two-component link,
    with the components' linking number $(b-a)/2$, so, for the sum over
    subsets~$E \subset \{2,\dots n\}$
    we can write
    \begin{align}
      1 + \frac{1}{2} \left ( 1 -(-1)^a \right) (t q^2)^{b-a}
    \end{align}

    So, for Khovanov and Jones polynomials in, say, UL region
    (one needs to adjust the value of the signature for the other regions)
    we can write
    \begin{align} \label{eq:khovanov-and-jones-reduced}
        Kh = q^{-a} \left ( (q + q^{-1})
        \left (1 + \frac{1}{2} \left ( 1 -(-1)^a \right) (t q^2)^{b-a} \right)
        + \left ( q^{-1} + t q^2 \cdot q \right ) Kh'(t q^2)
        \right )
        \\ \notag
        J = q^{-a} \left ( (q + q^{-1})
        \left (1 + \frac{1}{2} \left ( 1 -(-1)^a \right) (- q^2)^{b-a} \right)
        + \left ( q^{-1} - q^2 \cdot q \right ) J'(- q^2)
        \right ),
    \end{align}
    where polynomials $Kh'$ and $J'$ are actually equal, so if one considers
    $J'$ as a function of $q$ one can restore $Kh'$ by substituting $q \rightarrow q \sqrt{-t}$.

    Now, for the Jones polynomial we can obtain the following evolution
    on the whole parameter plane (using, for instance, RT-formalism)
    \begin{align}
      J_{\substack{a = 2l \hfill \\ b = 2k + 1}} & =
      \left(\begin{array}{cc} (1)^b & (- q^2)^b \end{array}\right)
      N
      \left(\begin{array}{c} q^{-a} \\ (- q^3)^{-a} \end{array}\right) \\ \notag
      N & =
      \left(
      \begin{array}{cc}
        \frac{\left(q^2 + 1 + q^{-2}\right)}{\left(q+q^{-1}\right)} &
        \frac{1}{q+q^{-1}} \\
        -\frac{\left(q^2 + 1 + q^{-2}\right)}{\left(q+q^{-1}\right)} &
        \frac{\left(q^2 + 1 + q^{-2}\right)}{\left(q+q^{-1}\right)}
        \\
      \end{array}
      \right)
    \end{align}

    Using formula \eqref{eq:khovanov-and-jones-reduced} to first express
    $J'$ through $J$, then $Kh'$ through $J'$ and finally obtain $Kh$
    through $a$ and $b$ one can straightforwardly check that, indeed,
    $Kh$ depends on $a$ and $b$ as in formulas \eqref{eq:evolution-formula} and \eqref{eq:evolution-matrices}.
    This completes the proof of the theorem.
  }

{\section{Conclusion}
  {In this paper we've built on the results of \cite{AnoM} by considering what
    happens to the evolution method for Khovanov polynomial for the simplest
    non-torus family of knots (see Section~\ref{sec:fig-eight-like-knots}),
    that in particular includes two-strand and twist knots.}

  {For this two-parametric family we found a peculiar chamber structure (see Section~\ref{sec:phase-diagram}),
    which generalizes ``mirror anomaly'' observed in \cite{AnoM}.
    We also found an unexpected third eigenvalue of the (hypothetical) fundamental $R$-matrix,
    which drops out in the limit $(t \rightarrow -1)$.}

  { At the heart of the proof of our evolution formulas is the theorem~\cite{\fixme{???}}
    that relates Khovanov polynomial for any alternating knot to its Jones polynomial.
    While computer experiments clearly show that evolution method extends beyond alternating knots,
    new ideas are required to extend the proof, even to the case of 2-strand braid insertion into \textit{arbitrary} knot.
  }

  {It is very interesting, what is the relevant generalization of the Reshetikhin-Turaev formalism.
    It should be capable of describing the observed chamber structure.
    The abrupt jumps of the evolution coefficients that occur when one goes between the chambers
    may hint that another deformation (in addition to $q$- and $t$-deformations) is required to
    smoothen these jumps out and embed the problem into the framework of the usual linear algebra.
    This indication is in accordance with recent developments of the representation theory of DIM algebra \cite{ZenDIM},
    where one more deformation parameter also begs to be introduced.}

  {At the moment it is not clear where exactly does the simple evolution break down.
    There is only a vague idea that it breaks when ``something interesting'' happens to the planar diagram
    -- either it goes from being disconnected to being connected, or it untwists in an unusual manner
    (see Section~\ref{sec:phase-diagram}).
  }

  {Study of more complicated families of knots is needed to clarify the situation.
    To perform such a study one needs better computer programs that allow to specify families of knots
    more conveniently
    (manually working out through all the odd-even cases for parameters and figuring orientations of strands is a bit tedious).
    We continue to work in this direction.}
}

{

redpajama_set_name: RedPajamaArXiv