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Category: 2016 Arctic Thunder
master category for 2016
Contact the Alaska Air Show Association today! If you'd like more information about the Alaska Air Show Association to participate in any of our events, please use our handy contact form below.
The Alaska Air Show Association supports aerospace educational activities for young people of all ages, as part of it's educational mission as a non-profit organization. In 2006, AASA established two scholarships for aviation majors at the University of Alaska Anchorage. Alaska residents who are full-time students enrolled in any aviation degree program at the University of Alaska Anchorage are eligible to apply for the Alaska Air Show Association Aviation Scholarship. The scholarship is administered by UAA https://www.uaa.alaska.edu/scholarships/air_show.cfm. Full-time students enrolled in the UAA professional piloting program are also eligible to receive an Alaska Air Show Association Flight Training Award. Flight Training Awards are administered directly by the UAA Flight Operations Department.
For children ages 10-12, AASA provides four scholarships to Anchorage area children to attend week long camp at the Challenger Learning Center in Kenai http://www.akchallenger.org/. The Challenger Learning Center inspires students to pursue education in science, technology, engineering and math by providing a series of fun, hands-on science experiences during the camper's week long stay.
AASA has also awarded up to two scholarships to participating teens at the Aviation Challenge Camp at the U.S. Space and Rocket Center located in Huntsville, Alabama http://www.spacecamp.com/.
For 2016, AASA is working with sponsors for matching UAA educational grants to broaden the number of scholarships young aviators.
2006 Update: Four children won trips to the week long science and technology adventure camp were drawn by lottery at the 2006 Arctic Thunder Air Show. They completed "Trainee Selection Cards" by visiting various static displays to learn about the aircraft and get their cards stamped. Two teens in the 13 – 18 year group that attended Arctic Thunder 2006 earned scholarships to Aviation Challenge. Three UAA students, Desiree McIntosh, Glenn Roose and Maria Terrazas each received $2000 AASA Aviation Scholarships for academic year 2007-2008.
2008 Update: In 2008, AASA contributed $12,000 to UAA to support aviation Scholarships. There were four $2,000 scholarships and a $4,000 flight training incentive award.
2010 Update: For 2010, AASA again contributed $12,000 to UAA to support aviation scholarships. In 2010, there were four $2,500 scholarship and $2,000 in flight training incentive awards.
2012 Update: In 2012, AASA supported 4 scholarships to the Challenger Learning Center, and $2,500 to UAA Aviation Technology program, supplemented with a $7,500 matching grant by the Historic Flight Foundation.
2014 Update: AASA provided $4,000 to UAA in support of the AASA Aviation Scholarship, and four scholarships to the Challenger Learning Center.
2016 Update: AASA intends to fully support four scholarships and flight training incentive awards to the UAA aviation technology program, as well as to provide scholarships to the Challenger Learning Center. Check back for further updates.
US Navy Blue Angels
Making their first appearance since Arctic Thunder 2010, the Association is thrilled to have the Blue Angels as the featured jet team for Arctic Thunder, 2016 at Joint Base Elmendorf Richardson. The Blue Angels showcase demonstrate the Unites States Navy and Marine Corps' commitment to professionalism, excellence and service through these world-class flight demonstrations and community outreach.
The Blue Angels are comprised of 16 officers fill the flying and support positions. Jet pilots have aircraft carrier qualifications and at least 1,250 tactical jet flight hours. Over 100 enlisted personnel support the demonstration squadron, with such specialties as airframes, avionics, aviation medicine, crew chiefs, logistics, among others.
The Blue Angels fly the Boeing F/A-18 Hornet and the Lockheed Martin C-130 Hercules.
Please visit their website at www.blueangels.navy.mil for more great information and exciting videos.
Gary Ward
Gary developed an interest in airplanes as an early teenager, soloed at age 16 in a Piper J-3 Cub and has been passionate about aviation ever since. After high school and a tour with the USAF as a jet engine mechanic, Gary graduated from the Georgia Institute of Technology with a degree in Aerospace Engineering. He worked as a design engineer on the Cessna Citation for Cessna for about a year before "switching hats" to work in a family operated lumber manufacturing business.
Gary started flying airshows in 1998 in a Pitts S2-B and then flew airshows in the Giles 202 from 1999 through 2005. In 2006 he became the first pilot to fly airshows in the awesome and exciting new MX2, an all carbon fiber monoplane powered by a 6 cylinder Lycoming engine producing 350 HP.
Gary puts the MX2 through one of the most exciting performances you will ever see. The action starts on take off as Gary pulls steeply up and goes into a spectacular take off maneuver! The entire sequence is jam packed with breathtaking gyrations that range from zero speed hovers to dives in excess of 250 mph!
MX-2 Specifications:
Engine Lycoming/LYCON IO540 350 HP+
Propeller 3 blade MT-Propeller
Empty Weight 1,307 lbs
Wing Span 24 ft
Cruise Speed 190 knots (218 mph)
Top speed level flight 210 knots (241 mph)
Max speed in dive 230 knots (264 mph)
Roll rate: Exceeds 400 degrees/second
Range 750 NM (862 miles)
Please visit Gary's website at http://www.garywardaerobatics.com/ for more cool information and videos.
Jason Newburg & his Pitts Viper
Jason Newburg is an extreme aerobatic pilot who loves sharing his high energy performances nationwide and overseas with air show audiences. Jason's first solo flight was at age 16 and he earned his pilot's license at 17. Jason is committed to providing aviation leadership to young people who share his passion in aviation and aerobatics. He has served as Vice President for the Embry-Riddle Sport Aviation Club for numerous years.
Viper Pitts S2S Specifications:
Engine 400 HP
Wing Span 20 ft 6 in
Top speed level flight 300 mph
Roll rate: 400 degrees/second
Range 600 miles
Learn more about Jason and Viper Airshows at www.viperairshows.com.
Marcus Paine
Following in the footsteps of former Alaskan aviator legends, Marcus brings the thrill of aerobatic flight to his hometown of Anchorage, where he has lived for more than 40 years. Raised on a homestead in Rabbit Creek, Marc has been a pilot for over 20 years and is the founder and Chief Flight Instructor of Unusual Attitudes LLC, a flight school that teaches Unusual Attitude Recovery, Stall/Spin Awareness and aerobatic flight. Marc is a skilled instructor adept at teaching pilots of all skill levels new ways of thinking about the principals of flight and mastering the airplane in every attitude. He is an FAA Safety Counselor, and a regular speaker on Stall/Spin Awareness at FAA Safety Seminars held throughout Alaska, and nationally at aviation events including Oshkosh and Sun N Fun. Marcus will be flying his 1941 Stearman for Arctic Thunder 2016.
Marcus was recently featured in Discovery Channel Canada's Airshow! Television series, which documented the risks, dangers and lives of air show professional performers. The show provided a unique, behind the scenes look at those who regularly risk it all in performances throughout North and South America.
Stearman Specifications:
Marcus over Columbia Glacier
Photographer Joe Prax
Wings of Freedom *
Wings of Freedom, founded by Mr. Chuck Miller, is dedicated to providing a collection of flyable warbirds that played a role in shaping Alaska's aviation heritage. Housed in a WW-II era hangar at Merrill Field, Wings of Freedom will be providing a T-6 and L-13 at Arctic Thunder 2016. Wings of Freedom had previously been the home of one of the world's last remaining Mitsubishi Zero, which flew in the 2012 Arcitc Thunder Air Show as well as in the 2013 Alaska Aviation Centennial Celebration.
The AT-6 Texan is a single-engine, advanced trainer aircraft used during WWII. The prototype first flew in 1935, and the first models were put into production in 1937. More than 17,000 were built. The aircraft being flow at Arctic thunder was built in 1943, and is in the Aleutian colors of aircraft flying there in WWII.
The L-13A was manufactured by Consolidated Vultee, and first flown in 1945. It was used for observation, liaison, and air ambulance duties. Like the L-2, man were later converted for civilian use in the Alaska bush.
Arctic Thunder 2016
Arctic Thunder 2016 Air Show
July 30 & 31, 2016
Joint Base Elmendorf-Richardson Anchorage, AK
Open to the public Admission Free
The 2016 Arctic Thunder Air Show and Open House is coming July 30 & 31.
Copyright 2013 by Patty Lafond.
The Association is dedicated to providing aviation inspiration to all generations of Alaskans.
Sponsorship Opportunities for Arctic Thunder 2016: Donate Online
New! Arctic Thunder 2016 News Reports & Other Media: Click Here
Arctic Thunder 2016 Performers: Click Here
Arctic Thunder 2016 Schedule: Click Here
Frequently Asked Questions: Click Here
Super Cub STOL Demonstration *
Super Cub STOL
The Super Cub is one of the most popular aircraft in all of Alaska and is well suited for the variety of conditions and rugged terrain encountered in rural and remote areas of the State. The plane is known for its incredible short take-off and landing capabilities, and is the star of the Valdez Fly-in, held in May. For Arctic Thunder 2016, we are privileged to have Mr. Bob Breeden and Mr. Hank Swan. Bob Breeden is the record holder, landing his Super Cub in a world record 20 feet, and a take-off distance of 24 feet. Hank flies an experimental hot rod Super Cub called "Got Rocks" and is making his second appearance at Arctic Thunder. They will be joined by a 1941 L-1 aircraft and the L-13 for a joint civilian and military short field take-off and landing.
Commemorative Air Force *
The Commemorative Air Force was founded in 1957 and chartered as a non-profit in 1961 to restore, preserve, operate and maintain WWII-era aircraft. The CAF also provides museums to protect and display aircraft, as well as to perpetuate their memory and heritage. The Alaska Wing is dedicated to preserving the military heritage of Alaskan aviation, and flies the bright yellow AT-6 Harvard Mk IV, the yellow and blue BT-13, and the L-2 Grasshopper. The Harvard MkIIB was built under license in Montreal for the Royal Canadian Air Force, the RAF and the USAAF, with a total of 2,557 built. The Harvard was known to fly to Annette Island in Alaska during WWII, where Canadian units were stationed, adding strength to the 11th Air Force. The BT-13 was first flown in 1939, and was the WWII-era basic trainer flown by most pilots. Pilots would graduate to the advanced trainer, the AT-6, prior to being assigned to their main fighter aircraft or bomber. The L-2 Grasshopper was manufactured by Taylorcraft beginning in 1941 and was used an observation aircraft during WW-II to spot enemy troops and supplies, as well as direct artillery fire. After WWII, many aircraft were converted to civilian use and flown around bush Alaska.


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Find out how to navigate the different parking and locations for Antrim, Grand Traverse and Leelanau County Courthouses.
Your Jury Board sends out a double sided juror qualification/personal history questionnaire every year to potential jurors. The names are randomly selected from the State of Michigan Driver's Licenses and Personal Identification cards.


This is a great investigation by the Washington Post. Trump lies and lies and lies.
I don't know if it's illegal. I do know it's not appropriate.
Great investigation by the New York Times.
It's too bad that all of the information on Trump's sleazy dealings, payoffs to women and tax avoidance wasn't known before the election.
This is the inside story of how Deutsche Bank officials – including 2 CEOs – were so eager to do business with @realDonaldTrump that they ignored years of internal warnings about the risks.
If true, @realDonaldTrump lied to Deutsche Bank to get a loan. That constitutes bank fraud.
At the end of 2017, Trump waived punishment for Deutsche Bank.
Jim Sciutto does an excellent job on TV but he was wrong to give the name of Trump's golf course in his tweet.
President Trump has arrived at the Trump Intl Gold Club. This is his 223rd day spent at a Trump property and his 172nd day spent at a Trump property golf course since becoming President.
Please don't mention the name of his property – don't give his business free publicity. Just say he's at his golf course and the location.
Just ask Donald Trump. In 2005, he collected $17 million in insurance money for non-existent damage to Mar-A-Lago after Hurricane Wilma.


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THRESHEReview
< Previous Issue Next Issue >
Bethel board approves strategic plan
New board members appointed, come from western U.S.
You can still sign up for the interterm trip to China
Join the Whites for dinner at the MEDA convention Nov. 5
16th-century martyr drama at museum Nov. 15
Worship and the Arts Symposium slated for Nov. 21
Guest composer at third annual honors band clinic Nov. 23
Donate to BC on Giving Tuesday!
Future Fall Fests
Bethel board approves strategic plan 2015-11-23T09:28:17-06:00 2015-11-23T09:28:17-06:00
The Bethel Board of Directors unanimously approved a strategic plan that has been in development since 2014 and is designed to carry the college through 2022.
"The board is appreciative of the hard work of the Strategic Planning Task Force and is excited about new initiatives on the horizon for the college," Board Chair Heather Esau Zerger '96 said.
Private colleges in particular are vulnerable in what Bethel President Perry D. White calls "treacherous times in higher education."
He cited "a tremendous increase in federal regulations, the call to make education more affordable while demanding easily assessed and sometimes rather shallow outcomes and, in Bethel's case, a state that may seek to remove tax-exempt status for educational institutions.
"This environment makes it necessary to do aggressive and innovative organizational planning, not only to survive but to thrive."
Work on the Bethel strategic plan began in 2014 with the selection of Credo Higher Education, a consulting firm that specializes in small, church-affiliated institutions.
Credo staff facilitated the process of collecting the ideas of stakeholder groups (faculty, staff, students, administrative team, alumni, Newton-area leaders and board), helped assess the collected data based on national trends and will continue to guide and monitor, through a software program, the first 14 months of plan implementation.
For more of this article, visit the Bethel College website.
New board members appointed, come from western U.S. 2015-11-23T09:29:31-06:00 2015-11-23T09:29:31-06:00
Among other actions at its fall meetings Oct. 8–9, the Board of Directors approved the appointments of two new members. Both are Bethel alumni who bring previous experience working at the college.
Lori Schmidt-Harrison '87, Pasadena, California, is filling an unexpired term representing the Alumni Association. Both she and Wynn Goering '77, Albuquerque, begin their terms Jan. 1, 2016.
Schmidt-Harrison is owner and president of Lime Twig Group, started in 2000, a marketing firm specializing in the area of higher education in the western United States.
A Hesston native, Schmidt-Harrison graduated from Bethel with a B.A. in speech and drama. She served as an admissions counselor, assistant director of admissions and director of admissions at Bethel between 1987 and 1992…
Goering, a native of Moundridge, has worked since 1997 at the University of New Mexico, where he is currently chief executive officer at UNM West in Rio Rancho, the UNM system's newest campus…
Goering graduated from Bethel with majors in English and music. He earned M.A. and Ph.D. degrees in English language and literature at the University of Chicago, including a four-year Danforth Fellowship during that time…
You can still sign up for the interterm trip to China 2015-11-23T09:30:56-06:00 2015-11-23T09:30:56-06:00
Bethel alumni and friends of the college can still sign up to join students and Chris Westover, director of instrumental music, on a trip to China in January 2016. It will be led by experienced tour guides Jim '57 and Shirley (Suderman) '57 Goering, North Newton.
The cross-cultural learning experience will focus on arts and music. Participants will visit Bejing, Xi'an, Shanghai and Suzhou, and will see the Central Academy of Drama, National Centre of the Performing Arts, Forbidden City, Tian'anmen Square, Great Wall, 798 Art Zone and Terra Cotta Warriors Museum.
The Goerings have visited China nearly every year since 1980, lived there from 1985–1989, have traveled widely throughout the country, spent 12 summers teaching English to Chinese middle school teachers and hosted 25 tours prior to this one.
The estimated cost, which includes airfare, transportation, lodging, lunches/dinners, fees and tickets, is $3,600 for Bethel students, faculty and staff, and $4,300 for non-students/community members.
For more information, contact Westover at 316-284-5264 or via e-mail.
Join the Whites for dinner at the MEDA convention Nov. 5 2015-11-23T09:31:29-06:00 2015-11-23T09:31:29-06:00
President Perry White and Dalene White invite Bethel alumni and friends to go out to dinner with them during the Mennonite Economic Development Associates (MEDA) annual convention at Richmond, Va. Those who live in the area, as well as convention participants, are welcome to join in.
The group will meet in the lobby of the Omni Richmond Hotel at 5 p.m. Friday, Nov. 6, and go from there, supporting the college by purchasing food and/or drinks on their own.
Fellowship with others and hear what's happening at BC!
16th-century martyr drama at museum Nov. 15 2015-11-23T09:32:23-06:00 2015-11-23T09:32:23-06:00
A new readers theater drama, "I Have Not Forsaken the Word of God," will be featured at Kauffman Museum's Sunday-Afternoon-at-the-Museum at 3 p.m. Nov. 15. The presentation is based on the confession of Jacques d'Auchy, whose betrayal and inquisition are recounted in the "Martyrs Mirror."
The Mirror of the Martyrs Trust commissioned Carrie and Gerald Mast of Bluffton (Ohio) University to write the script to celebrate the purchase of the long-lost etching plate depicting d'Auchy in prison, created by Jan Luyken for the 1685 edition of the "Martyrs Mirror."
The program is free and open to the public. For more information, contact Rachel Pannabecker via e-mail.
Worship and the Arts Symposium slated for Nov. 21 2015-11-23T09:33:58-06:00 2015-11-23T09:33:58-06:00
"Worship and Community: Cultivating Practices That Sustain Us" is the theme for the Worship and the Arts Symposium scheduled for Saturday, Nov. 21, on the Bethel campus.
During the day, keynote speakers Christine Pohl, associate provost and professor of social ethics at Asbury Theological Seminary, and Michael Driscoll, associate professor of liturgical studies and co-director of the master in sacred music program at the University of Notre Dame — along with local resource people — will lead plenary and breakout sessions exploring the intersections of worship and community and the practices that hold people of faith together.
The symposium also features a special exhibition in the Regier Art Gallery on "Beyond the Martys' Mirror: The Prints of Jan Luyken," co-curated by Rachel Epp Buller, associate professor of visual art and design, and Bethel senior Alexandra Shoup.
The evening worship service, "In Community We Sing," will feature Pohl and Driscoll, the Bethel Concert Choir, singers from area churches and congregational singing. The service is free and open to the public. An offering of non-perishable food and funds will be accepted for Harvest of Love.
Cost for the day-long event is $35 per individual for all or part of the symposium. A rate of $20 each applies when three or more individuals from the same congregation register as a group. The price includes Saturday lunch and supper (register no later than Nov. 16 to have the meals included).
The symposium is made possible by the Reimer-Boese Worship and the Arts Endowment. For more information, visit the Bethel College website.
Guest composer at third annual honors band clinic Nov. 23 2015-11-23T09:34:34-06:00 2015-11-23T09:34:38-06:00
The Bethel Wind Ensemble will present a concert at 7 p.m. Monday, Nov. 23, in Memorial Hall, as the culmination of the college's third annual Honors Band Clinic featuring guest composer David Maslanka, Missoula, Montana. The wind ensemble will perform his "Tears," "Letter to Martin" and "Give Us This Day" in addition to Leonard Bernstein's "Prelude, Fugue and Riffs."
Maslanka has written works for choir, wind ensemble, chamber music group and symphony orchestra. He has published nearly 100 pieces, including nine symphonies (seven of them for concert band), nine concerti, and a full Mass. He is best known for his works for wind ensemble.
Maslanka's musical style is rhythmically intense and complex, but also highly tonal and melodically-oriented. His pieces have been performed around the world.
The concert is free and open to the public.
Donate to BC on Giving Tuesday! 2015-11-23T09:36:14-06:00 2015-11-23T09:36:14-06:00
Dec. 1 is this year's worldwide Giving Tuesday. Bethel College encourages you to make your year-end gift early, on the Bethel College website.
According to the #GivingTuesday website, "#GivingTuesday is a global day of giving fueled by the power of social media and collaboration. Observed on the Tuesday following Thanksgiving (in the U.S.) and the widely recognized shopping events Black Friday and Cyber Monday, #GivingTuesday kicks off the charitable season, when many focus on their holiday and end-of-year giving.
"Since its inaugural year in 2012, #GivingTuesday has become a movement that celebrates and supports giving and philanthropy with events throughout the year and a growing catalog of resources."
Avoid the holiday rush and hassle. Donate to Bethel Dec. 1!
Future Fall Fests 2015-11-23T09:37:50-06:00 2015-11-23T09:38:02-06:00
Since 1971, Bethel College has held an annual Fall Festival — its own special version of homecoming — to celebrate education, diverse cultural traditions, special foods, faith, fine arts, sports, crafts, community and friendship.
Since fall 1987, the special weekend has begun with Taste of Newton downtown on a Thursday evening, continued with special events at the college on Friday, peaked with the Fall Fest fair and activities across campus on Saturday, and ended with worship and the final performance of a Fall Fest stage production on Sunday.
Upcoming dates for Saturday's annual Fall Festival fair on campus are:
with Taste of Newton downtown the Thursday before.
The dates are determined by the first home football game in October, as scheduled by the Kansas Collegiate Athletic Conference (available online).
© 2023 Bethel College • All Rights Reserved
THRESHEReview Contact: alumni@bethelks.edu

redpajama_set_name: RedPajamaCommonCrawl


An international research team aims to protect human health and the environment by improving agricultural practices in four Asian countries.
As nations rapidly develop, agriculture intensifies, providing numerous economic benefits. But intensification can also damage the environment and pose serious health risks. Human health depends on the well-being of surrounding ecosystems. This has led to an area of study called "eco-health".
The Field Building Leadership Initiative, supported by IDRC, aims to address the associated health risks of intensive agricultural practices in Southeast Asia and China. Over the past five years, the international group has investigated specific problems and potential eco-health solutions in China, Indonesia, Thailand, and Vietnam. The scientists tested urine samples from farmers in Yuanmou, a key vegetable-growing region in China, where 28% of the samples contained traces of at least one pesticide, even though the region switched to "low toxic" pesticides in the early 2000s. The team distributed informational calendars and performed street theatre in villages to educate the farmers about how to reduce pesticides and use them safely.
Another team found an increase in the incidence of dengue and chikungunya around the rubber plantation areas in eastern Thailand, and the rubber workers had limited knowledge of these vector-borne diseases. Heavy metals and the bacteria E. coli and Salmonella were also found to be contaminating water in the area. Inappropriate disposal and handling of chemicals, including poor self-protection and sanitation, were observed. Apart from providing health education, the research team promoted the use of insecticide-impregnated screen jackets as one self-protection measure. A pilot vector-control intervention introduced super-sterile male mosquitoes into the environment to reduce vector populations and risk to the diseases they transmit.
In Vietnamese communes, the group tested biogas — fuel produced using human and animal waste — and found the wastewater resulting from its production exceeded standards for the bacteria E. coli and Salmonella, the parasite Giardia, and other harmful contaminants. Biogas wastewater used as fertilizer also put farmers at high risk of diarrhea. Villagers appreciated having a biogas expert speak to them about proper management, and some villages implemented new sanitation rules as a result of the project.
Almost 30% of Indonesia's milk production occurs in West Java. However, research in the West Java district of Pangalengan revealed that smallholder dairy farmers struggled with low milk yields and quality, and poor farm sanitation practices. The researchers developed ways to convert cow waste and earthworm feces into fertilizer. They also developed a food supplement for cows made of molasses, probiotics, turmeric, the leaves of star gooseberry and earthworms. These not only created a new source of income for farmers, they also proved effective in increasing farmers' yields.
While influencing decision-makers to adopt eco-health policies remains a challenge, the Field Building Leadership Initiative underscored that headway has been made through networking and engaging with local communities and other stakeholders. As a result of the Field Building Leadership Initiative, eco-health has been integrated into the curricula of four universities in Southeast Asia (Vietnam, Thailand, Indonesia) and China. More than 400 professionals and graduate students attended workshops across the region to learn about eco-health and leadership, building a new generation of eco-health professionals.
The original version of this article was published in the 2017 edition of Asia Research News.


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I'm already against the next war
By Antiwar.com from antiwardotcom link Jun 9, 2016
Shoot first and ask questions later.
It seems that the Land of the Free has retooled its foreign policy to mirror this B-movie dialogue. Perhaps this is an expected outcome after decades of standoff with the USSR and the ever escalating war technology. During the cold war, the visible antiwar movement lost its momentum. The U.S. was building the bombs but it wasn't dropping them on anyone. The war technology kept ramping up as a "deterrent" and we learned that, as treaties increasingly called on the U.S. to be the defender of allies and the police force in "problem" places in the world, the U.S. could stand on the neck of almost anyone and bring them into submission. This isn't right or righteous, this is tyranny.
You reap what you sow.
20 years ago, a small group of people recognized this growing post cold war pattern. Antiwar.com was born to collect and curate news and information that exposed this growing predilection for U.S. aggressive action in foreign land. Today we are harvesting the bitter fruit of this policy with increased terrorism, suitcase bombs, increased paranoia and burdensome security measures, loss of privacy, warrantless phone taps and other infringements of our rights here at home.
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Palau Now Is The Time To Dive These 5 Dive Sites Deeperblue
Palau, under the name palaos, was included in the malolos congress in 1898, the first revolutionary congress in the philippines, which wanted full independence from colonialists. palau, at the time, was part of the spanish east indies headquartered in the philippines. palau had one appointed member to the congress, becoming the only group of islands in the entire caroline islands granted high representation in a non colonial philippine congress. Palau, country in the western pacific ocean. it consists of some 340 coral and volcanic islands perched on the kyushu palau ridge. the palau (also spelled belau or pelew) archipelago lies in the southwest corner of micronesia, with guam 830 miles (1,330 km) to the northeast, new guinea 400 miles (650 km) to the south, and the philippines 550 miles (890 km) to the west, a huge barrier reef system, continuous on the west and broken on the east, encircles most of the …. Nov 23, 2020 · palau is a beautiful tropical paradise, and one of the true unspoiled destinations on earth. most of the 100 plus islands are small low lying coral islands, ringed by barrier reefs. the exception is the mountainous babulthuap which is volcanic in origin. the highest point is mount ngerchelchuus at an elevation of 715ft (215m). Aug 24, 2022 · the u.s. and palau cooperate on a broad range of issues, including strengthening regional security, promoting sustainable development and addressing climate change, and protecting fisheries and the environment. approximately 500 palauans serve as volunteers in the u.s. armed forces, and palau also has one of the highest levels of voting coincidence with the u.s. at the united nations. Jan 11, 2023 · palau voted against joining the federated states of micronesia in 1978 and adopted its own constitution in 1981, which stated that palau was a nuclear free country. in 1982, palau signed a compact of free association (cofa) with the us, which granted palau financial assistance and access to many us domestic programs in exchange for exclusive us military access and defense responsibilities.
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text:
\section{Introduction}
S. Baaj and G. Skandalis \cite{BS} and R.M. Kashaev \cite{Ka} found a striking fact that for the Heisenberg double $\mathcal{H}(H)$ of any finite dimensional Hopf algebra $H$, there exists a canonical element $T$ in $\mathcal{H}(H)^{\otimes 2}$ satisfying
the pentagon equation
\begin{align*}
T_{12}T_{13}T_{23}=T_{23}T_{12}.
\end{align*}
In \cite{Ka}, he also showed that the Drinfeld double $\mathcal{D}(H)$ of $H$ can be realized as a subalgebra of  $\mathcal{H}(H)^{\otimes 2}$, and observed that the universal $R$-matrix of $\mathcal{D}(H)$
can be represented as a combination of four copies of $T$, where  the quantum Yang-Baxter equation of the universal $R$-matrix follows from a sequence of the pentagon equation of $T$.
Using his results,  the second author \cite{S} reconstructed the universal quantum $\mathcal{D}(H)$ invariant of framed tangles by assigning a copy of the canonical element $T$ to each branched ideal tetrahedron of the tangle complements, and expected that this construction leads to invariants of pairs of a $3$-manifold and geometrical input. In the present paper, we show that this construction defines an invariant of closed oriented $3$-manifolds when the Hopf algebra $H$ is involutory, unimodular and counimodular.

In the formulation of our invariant, we use a diagrammatic representation of closed oriented $3$-manifolds introduced by R. Benedetti and C. Petronio  \cite{BP}.
Their diagrams, which are called closed normal o-graphs, are oriented virtual knot diagrams satisfying certain conditions.
They showed that homeomorphism classes of closed oriented $3$-manifolds are identified with equivalence classes of closed normal o-graphs up to certain moves. A crossing of a closed normal o-graph represents a branched ideal tetrahedron in the corresponding $3$-manifold, and the orientation of the strand specifies a way to extend these local branching structures to a global one.
Our invariant is obtained by assigning a copy of canonical element $T$ (or its inverse) to each crossing of closed normal o-graphs,
and by reading them along the strands.
The invariant takes values in the cyclic quotient $\mathcal{H}(H)/{[\mathcal{H}(H),\mathcal{H}(H)]}$,  which is isomorphic to the base field through the character of the Fock space representation.

The proof of the invariance will be performed by checking the invariance under each move for normal o-graphs.
There are two important types: the MP-moves and the  CP-move.
An MP-move represents a Pachner move equipped with a branching structure, which corresponds to a modified pentagon equation in the level of the invariant.
Here, the modification is related to the antipode of $H$, and we need to assume that the antipode is involutive.
Up to the MP-moves (and the 0-2 move),  a closed normal o-graph represents a closed oriented $3$-manifold with a combing.
The CP-move is a special move for branched triangulations, which changes the combing. The invariance under the CP-move will be shown using tensor networks, where we need to assume that $H$ is in addition both unimodular and counimodular.
Using tensor networks we will also show a connected sum formula of the invariant.

There are several invariants based on triangulations and involutory Hopf algebras \cites{TV, BW, Ku1}.
Our invariant uses only the canonical element and  the structure constants of Hopf algebras and does not use any representations, and thus the construction is rather simple.
Furthermore, the moves for closed normal o-graphs are local and finite, unlike  handle slide moves in the Kirby calculus for link surgery presentations of  $3$-manifolds, and each of these finite moves corresponds to an algebraic equation nicely, like the MP-moves and the pentagon equation. Thus the proof of the invariance is also rather easy to understand.

The main example of finite dimensional, involutory, unimodular and counimodular Hopf algebras is the group algebra $\mathbb{C}[G]$ of a finite group $G$. We show that in this case the invariant is same as the number of homomorphisms from the fundamental group of the $3$-manifold to $G$.
There are several other examples of Hopf algebras which would be interesting to be used. The restricted enveloping algebras of restricted Lie algebras \cite{J} are finite dimensional involutory Hopf algebras.
In \cite{MP}, S. Majid and A. Pachol classified Hopf algebras under dimension $\leq$ 4 over the field of characteristic $2$.
M. Kim \cite{Ki} also gave some examples of finite dimensional, involutory unimodular and counimodular  (commutative and cocommutative) Hopf algebras  which are not group algebras.

The rest of the paper is organized as follows. In Section \ref{sec:hd}, we discuss Hopf algebras and the Heisenberg double of them.
In Section \ref{sec:ce}, following \cite{BP}, we explain how to represent closed oriented $3$-manifolds in a combinatorial manner using closed normal o-graphs.
In Section \ref{sec:ri}, we explain the construction of our invariant $Z(M;\mathcal{H}(H))=Z(\Gamma ;\mathcal{H}(H))$ using the Heisenberg double  $\mathcal{H}(H)$  and a closed normal o-graph $\Gamma$ which represents a $3$-manifold $M$.
The proof of the invariance will be given in two sections.
In Section \ref{sec:mt}, we prove the invariance  under all moves except for the CP-move.
In Section \ref{sec:tnm}, we reformulate our invariant using tensor networks, and  by using them
we prove the  invariance  under the CP-move.
In Section \ref{sec:csf}, we show the connected sum formula, and study the case for the group algebra $\mathbb{C}[G]$ of a finite group $G$.

\begin{acknowledgment}
 We would like to thank S. Baseilhac, R. Benedetti, K. Hikami, M. Ishikawa, R.M. Kashaev, A. Kato, Y. Koda, T.T.Q. L$\hat{\mathrm{e}}$ for valuable discussions. This work is partially supported by
 JSPS KAKENHI Grant Number JP17K05243, JP19K14523, and by JST CREST Grant Number JPMJCR14D6.
\end{acknowledgment}

\section{Hopf algebra and Heisenberg double}
\label{sec:hd}
In this section, we quickly review the definition and some properties of the Heisenberg double of Hopf algebras.

\subsection{Hopf algebra}\label{Hopf}
A \textit {Hopf algebra} $H$ over a field $\mathbb{K}$ is a vector space equipped with five linear maps
\begin{align*}
&M\colon\thinspace H\otimes H \to H, \quad
1\colon\thinspace \mathbb{K} \to H, \quad
\Delta\colon\thinspace H  \to H\otimes H, \quad
\epsilon\colon\thinspace H \to \mathbb{K}, \quad
S\colon\thinspace H \to H,
\end{align*}
called multiplication, unit, comultiplication, counit and antipode respectively, satisfying the standard axioms of Hopf algebras.
When the antipode is involutive, i.e., $S^2 = \mathrm{id}_H$, we call $H$  \textit{involutory}.
Throughout the paper, $H$ will denote a finite dimensional Hopf algebra and $H^*$ will denote the \textit {dual Hopf algebra} of $H$.
We will also use the Sweedler notation $\Delta(x)=x_{(1)}\otimes x_{(2)}$ for $x\in H$.

Recall that a \textit {right integral}
of a Hopf algebra $H$ is an element $\mu_{R} \in H^*$ satisfying $\mu_{R}\cdot f=\mu_{R}f(1)$ for every $f\in H^*$.
A \textit {left integral} is defined similarly.
Since $H$ is finite dimensional, a left (resp. right) integral of the dual Hopf algebra $H^*$ is an element of $H$ and is called a \textit {left (resp. right) cointegral} of $H$.
It is well known that for a finite dimensional Hopf algebra, an integral always exists and is unique up to scalar multiplication.
We say $H$ is \textit {unimodular} when the left cointegrals are also right cointegrals, and \textit {counimodular} when the left integrals are also the right integrals.
For more details on Hopf algebras and its integrals, see \cites{R, Ku1}.

\subsection{Heisenberg double}
We use the left action of $H$ on $H^*$ defined by $(a\rightharpoonup f)(x):=f(xa)$, for $a, x\in H$ and $f\in H^*$.
The \textit{Heisenberg double}
\begin{align*}
\mathcal{H}(H)=H^{*}\otimes H
\end{align*}
of a Hopf algebra $H$ is a $\mathbb{K}$-algebra with unit $\epsilon \otimes 1$ and multiplication given by
\begin{gather*}
(f\otimes a)\cdot (g\otimes b)=f\cdot (a_{(1)}\rightharpoonup g)\otimes a_{(2)}b,
\end{gather*}
for $a,b\in H$ and $f,g\in H^*$.

Let $\{e_i\}$ be the basis of $H$ and $\{e^i\}$ its dual basis.
Then the canonical element is given by
\begin{align}
T=\sum_i(\epsilon \otimes e_i)\otimes (e^i\otimes 1) \quad \in \mathcal{H}(H)^{\otimes 2} \label{eq:can}
\end{align}
and its inverse by
\begin{align}
\overline{T}=\sum_i(\epsilon \otimes S(e_i))\otimes (e^i\otimes 1) \quad \in \mathcal{H}(H)^{\otimes 2}. \label{eq:can inv}
\end{align}

In the case of the Drinfeld double $\mathcal{D}(H)$ of $H$, the canonical element satisfies the quantum Yang-Baxter equation, which in turn produces invariants of links and 3-manifolds \cites{H, KR}. One important feature of the Heisenberg double which plays a central role in our construction of invariants is that the canonical element satisfies the pentagon equation.
\begin{prop}(\cites{BS, Ka})\label{pentagon}
The pentagon equation
\begin{align}\label{eq:pentagon equation}
T_{12}T_{13}T_{23}=T_{23}T_{12}
\end{align}
holds in $\mathcal{H}(H)^{\otimes 3}$.
\end{prop}
\begin{proof}
Note that
\begin{align*}
T_{12}T_{13}T_{23}&=\sum\limits_{i,j,k}(\epsilon\otimes e_i)(\epsilon\otimes e_j)\otimes(e^i\otimes 1)(\epsilon\otimes e_k)\otimes(e^j\otimes 1)(e^k\otimes 1)
\\
&=\sum\limits_{i,j,k}(\epsilon\otimes e_i e_j)\otimes(e^i\otimes e_k)\otimes(e^j e^k\otimes 1)\quad \in \epsilon\otimes (H\otimes H^*)^{\otimes 2}\otimes 1,
\\
T_{23}T_{12}&=\sum\limits_{i,j}(\epsilon\otimes e_j)\otimes(\epsilon\otimes e_i)(e^j\otimes 1)\otimes(e^i\otimes 1)\\
&=\sum\limits_{i,j}(\epsilon\otimes e_j)\otimes(e_{i(1)}\rightharpoonup e^j\otimes e_{i(2)})\otimes(e^i\otimes 1) \quad  \in \epsilon\otimes (H\otimes H^*)^{\otimes 2}\otimes 1.
\end{align*}
Let us identify $(H\otimes H^*)^{\otimes 2}$ with $\text{End}(H\otimes H)$ through the map
\begin{align*}
& \iota\colon x\otimes f\otimes y\otimes g\mapsto(a\otimes b\mapsto f(a)x\otimes g(b)y),
\end{align*}
for $x,y,a,b \in H$ and $f,g\in H^*$.
Then, after identifying between $\epsilon\otimes (H\otimes H^*)^{\otimes 2}\otimes 1$ and $(H\otimes H^*)^{\otimes 2}$, we can see that the both elements $T_{12}T_{13}T_{23}$ and $T_{23}T_{12}$ are sent by $ \iota $ to
the same element as follows.
\begin{align*}
\iota \left(T_{12}T_{13}T_{23}\right)(a\otimes b)&=e^i(a)e_ie_j\otimes e^je^k(b)e_k
\\
&=ae_j\otimes e^j(b_{(1)})e^k(b_{(2)})e_k
\\
&=ab_{(1)}\otimes b_{(2)},
\\
 \iota (T_{23}T_{12}) (a\otimes b)
&= e_j (e_{i(1)}\rightharpoonup e^j)(a)\otimes e^i(b)e_{i(2)}
\\
&=e_je^j(ae_{i(1)})\otimes e^i(b)e_{i(2)}
\\
&=ae_{i(1)}\otimes e^i(b)e_{i(2)}
\\
&=ab_{(1)}\otimes b_{(2)}.
\end{align*}
Thus we have the assertion.
\end{proof}

The Heisenberg double $\mathcal{H}(H)$ has a canonical module $F(H^{*})=H^*$, which we call the \textit{Fock space}, with the  action $\phi:\mathcal{H}(H)\to\mathrm{End}(H^*)$ given by
\begin{align}
\phi(f\otimes a)(g)=(f\otimes a)\triangleright g:=f\cdot (a\rightharpoonup g) \label{eq:fock}
\end{align}
for $(f\otimes a) \in \mathcal{H}(H)$ and $g\in H^*$.
Let $\raisebox{2pt}{$\chi$}_{Fock}$ be the character associated to the Fock space.
For a $\mathbb{K}$-algebra $A$, let $[A,A]$ be the subspace spanned by $\{ xy-yx \mid x, y \in A\}$ over $\mathbb{K}$.
We are interested in the quotient space ${\mathcal{H}(H)}/{[\mathcal{H}(H),\mathcal{H}(H)]}$ since this is the space where the invariant takes values in.

\begin{prop}\label{prop:K}
The character of the Fock space
\begin{align*}
\raisebox{2pt}{$\chi$}_{Fock}:\bigslant{\mathcal{H}(H)}{[\mathcal{H}(H),\mathcal{H}(H)]}\to \mathbb{K}
\end{align*}
is an isomorphism between vector spaces.
\end{prop}

\begin{proof}
In  \cite{L}*{Proposition 6.1} it was shown that
for $F \in \mathrm{End}(H^*)$, the element
\begin{align*}
\sum_{i,j}F(e^i)e^j\otimes S^{-1}(e_j)e_i \in \mathcal{H}(H)
\end{align*}
is in the preimage of $F$ by $\phi$, i.e.,
for any $g \in H^*$ we have,
\begin{align*}
\sum_{i,j}\phi(F(e^i)e^j\otimes S^{-1}(e_j)e_i)(g) &= \sum_{i,j}F(e^i)e^j\cdot(S^{-1}(e_j)e_i\rightharpoonup g)\\
&= \sum_{i,j}g_{(3)}(e_i)\cdot F(e^i)\cdot g_{(2)}(S^{-1}(e_j))e^j\cdot g_{(1)}\\
&= F(g_{(3)})\cdot S^{-1}(g_{(2)})\cdot g_{(1)}\\
&=F(g_{(2)})\epsilon(g_{(1)})\\
&= F(g).\\
\end{align*}
Since dim$\mathcal{H}(H)=\text{dim}\text{End}(H^*)$, it follows that $\phi$ is bijective, hence $\phi$ is an algebra isomorphism.

Note that $\text{End}(H^*)$ is a matrix algebra, thus the canonical trace
\begin{align*}
\text{tr}\colon\thinspace\bigslant{\text{End}(H^*)}{[\text{End}(H^*),\text{End}(H^*)]}\to \mathbb{K}
\end{align*}
is also an isomorphism. Thus we have the assertion.
\end{proof}

When the antipode $S$ is involutive, $\raisebox{2pt}{$\chi$}_{Fock}$ can be given in terms of integrals.
Let $\mu_{R} \in H^*$ and $e_{L} \in H$ be a right integral and a left cointegral satisfying $\mu_{R}(e_{L})=1$.
\begin{prop}\label{prop:fock is int}
For an involutory Hopf algebra $H$, we have
\begin{align*}
\raisebox{2pt}{$\chi$}_{Fock}(f\otimes a)&=f(e_L)\mu_R(a)
\end{align*}
for $f\otimes a\in\mathcal{H}(H)$.
\end{prop}
\begin{proof}
For $F \in \mathrm{End}(H^*)$, the trace map is given (cf. \cite{R}*{Chap. 10}) by
\begin{align*}
\mathrm{tr}(F)=\langle e_L,F(\mu_{R(2)})S(\mu_{R(1)})\rangle.
\end{align*}
Thus we have
\begin{align*}
\raisebox{2pt}{$\chi$}_{Fock}(f\otimes a)&=\langle e_L, f(a\rightharpoonup\mu_{R(2)})S(\mu_{R(1)})\rangle\\
&=\langle e_L, \mu_{R(3)}(a)f\cdot\mu_{R(2)}\cdot S(\mu_{R(1)})\rangle\\
&=f(e_L)\mu_R(a).\\
\end{align*}
The third equality follows from the fact that $x_{(2)}S(x_{(1)})={\epsilon}(x)1$ for an involutory Hopf algebra.
\end{proof}

\section{Closed normal o-graph}
\label{sec:ce}
In order to define an invariant, we first recall the method introduced in \cite{BP} by R. Benedetti and C. Petronio to represent closed oriented 3-manifolds in a combinatorial manner.
This method is based on the theory of branched spines, which are the dual of branched ideal triangulations.

\begin{figure}[b]
  \centering
  \includegraphics{type_+_vertex.pdf}
  \hspace{10mm}
  \includegraphics{type_-_vertex.pdf}
  \caption{}
  \label{fig:vertex}
\end{figure}

\begin{defi}[\cite{BP}]\textbf{}
\label{def:bp}
A \textit{closed normal o-graph} is an oriented virtual knot diagram, i.e.,
a finite connected $4$-valent graph $\Gamma$ immersed in $\mathbb{R}^2$ with the following conditions:
\begin{spacing}{0.5}
\end{spacing}
\begin{description}
\setlength{\itemsep}{1mm}
\setlength{\parskip}{1mm}
\item[N1] At each vertex, a sign $+$ or $-$ is indicated, which is represented by the over-under notation as in Figure \ref{fig:vertex},
\item[N2] Each edge has an orientation such that it matches among two edges which are opposite to each other at a vertex,
\item[C1] If one removes the vertices and joins the edges which are opposite to each other, the result is a unique oriented circuit,
\end{description}
satisfying the following additional conditions:
\begin{description}
\setlength{\itemsep}{1mm}
\setlength{\parskip}{1mm}
\item[C2] The trivalent graph obtained from $\Gamma$ by the rules of \cite{BP}, page 7. Figure 1.2. is connected,
\item[C3] Consider the disjoint union of oriented circuits obtained from $\Gamma$ by the rules of \cite{BP}, page 7. Figure 1.3. Then the number of these circuits is exactly one more than the number of vertices of $\Gamma$.
\end{description}
\end{defi}

Let $\mathcal{G}$ be the set of closed normal o-graphs and $\mathcal{M}$ the set of oriented closed 3-manifolds up to orientation preserving homeomorphisms.
Given a closed normal o-graph $\Gamma\in\mathcal{G}$, one can canonically construct a $3$-manifold $\Phi(\Gamma) \in\mathcal{M}$ as follows.
We fix an orientation of $\mathbb{R}^3$ and place a closed normal o-graph on $\mathbb{R}^2\subset \mathbb{R}^3$.
Then we replace each of its vertices with a tetrahedron (with the orientation given by $\mathbb{R}^3$), and glue the faces of the ideal tetrahedra. The way to glue the faces of ideal tetrahedra is specified by the order of vertices of ideal tetrahedra defined as in Figure \ref{fig:reconstruction map}, i.e.,
we glue faces by the orientation reversing map which preserves the order of vertices.
Conditions on closed normal o-graphs ensure that the result is an ideally triangulated 3-manifolds with $S^2$ boundary.
Then, after we cap the boundary, the result defines an element $\Phi(\Gamma) \in\mathcal{M}$.
Here, for the geometrical meaning of the order of the vertices of ideal tetrahedra, see Remark \ref{branch}, where the meaning of the technical conditions \textbf{C1} \textbf{C2} and \textbf{C3} are also explained.
We denote the construction map obtained in the above way by $\Phi\colon\thinspace \mathcal{G} \to \mathcal{M}$.

\begin{figure}
    \centering
    \includegraphics{Bp-diagram_and_ideal-triangulation.pdf}
    \caption{}
    \label{fig:reconstruction map}
\end{figure}

The map $\Phi$ is not one to one, and in order to make it so, we need the following local moves of the diagrams.
\begin{description}
\setlength{\itemsep}{1mm}
\setlength{\parskip}{1mm}
\item[1]Planer isotopy of the diagram and the \textit{Reidemeister type moves} described in Figure \ref{fig:RM}.
\item[2]The \textit{0-2 move} and the \textit{MP-moves} described in Figure \ref{fig:02} and Figure \ref{fig:MP}, respectively.
\item[3]The \textit{CP-move} in Figure \ref{fig:CP}.
\end{description}

\begin{figure}
  \centering
  \begin{minipage}{0.62\columnwidth}
    \centering
    \includegraphics{Reidemeister_type_moves.pdf}
    \caption{The Reidemeister type moves.}
    \label{fig:RM}
  \end{minipage}

  \begin{minipage}{0.37\columnwidth}
    \centering
    \includegraphics{0-2move.pdf}
    \caption{The 0-2 move.}
    \label{fig:02}
  \end{minipage}
\end{figure}

\begin{figure}
  \centering
  \includegraphics{MP-move.pdf}
  \caption{The MP-moves. The orientation of each non-oriented edge in the figure is arbitrarily if it matches before and after the move.}
  \label{fig:MP}
\end{figure}

\begin{figure}
  \centering
  \includegraphics{Pmove.pdf}
  \caption{The CP-move.}
  \label{fig:CP}
\end{figure}

All of the above local moves preserve the axioms of closed normal o-graphs.
We say that two closed normal o-graphs are equivalent if one can be obtained from the other by planer isotopy and finite sequence of moves defined above.
Let us denote this equivalence relation by $\sim $.
The following was proved in \cite{BP}.

\begin{prop}
The map
\begin{align}
\Phi\colon\thinspace \bigslant{\mathcal{G}}{\sim}\to \mathcal{M}
\end{align}
is well-defined and one to one.
\label{thm:BP}
\end{prop}

\begin{ex}
The closed normal o-graph of the lens space $L(p,1)$, $p\geq 1$, is given by the following graph with $p$ vertices.
\begin{figure}[H]
  \centering
  \includegraphics{example_of_lens_space.pdf}
  \label{lens_space}
\end{figure}
\end{ex}

\begin{rem}\label{branch}
We briefly remark on the geometrical meaning of the representation of $3$-manifolds by closed normal o-graphs; see \cite{BP} for more details.
The order of vertices of ideal tetrahedra as in Figure \ref{fig:reconstruction map} specifies a branching structure for the ideal triangulation, which gives a combing, i.e., a non-vanishing vector field up to homotopy, to the underlying $3$-manifold.
The technical conditions \textbf{C1}, \textbf{C2} and \textbf{C3} ensure that the 3-manifold corresponding to a closed normal o-graph has a $S^2$ boundary with a nice branching structure, where the associated combing can be extended canonically to the closed $3$-manifold after we cap $B^3$ on the boundary.
In this case, a closed normal o-graph up to the 0-2 move and the MP-moves represents a 3-manifolds with a combing.
The CP-move in Figure \ref{fig:CP} changes the combing while preserving the underline 3-manifold, thus one gets the complete representation of $\mathcal{M}$ as in Proposition \ref{thm:BP}.
Here, the CP-move is an interpretation of the Pontryagin surgery in terms of branched standard spine; see \cite{BP}*{Chap. 6} for details.
\end{rem}

\section{Invariant}
\label{sec:ri}
\begin{figure}
  \centering
  \begin{minipage}{0.61\columnwidth}
    \centering
    \includegraphics{colored_curve_move.pdf}
    \caption{}
    \label{fig:crm}
  \end{minipage}
    \begin{minipage}{0.37\columnwidth}
    \centering
    \includegraphics{beads_slide.pdf}
    \caption{}
    \label{fig:bs}
  \end{minipage}
\end{figure}

\begin{figure}
  \centering
  \begin{minipage}{0.5\columnwidth}
    \centering
    \includegraphics{beads_slide_2.pdf}
    \caption{}
    \label{fig:bs2}
  \end{minipage}
\end{figure}

In this section,  we define a scalar $Z(\Gamma;\mathcal{H}(H))$ for a closed normal o-graph $\Gamma$.
In the following section, we show that this scalar is an invariant of  closed oriented 3-manifolds  when the Hopf algebra is involutory, unimodular and counimodular.

Let $A$ be a $\mathbb{K}$-algebra. \textit{$A$-decorated diagram} is an oriented closed curve immersed in $\mathbb{R}^2$ with the finite number of dots each of which is labeled with an element of $A$.
These dots are called beads.
We shall consider the $A$-decorated diagram up to planer isotopy and moves in Figure \ref{fig:crm} and \ref{fig:bs}.
We also allow the beads to slide along the curve.

We define the scalar $Z(\Gamma;\mathcal{H}(H))$ as follows.
Recall the definition of the canonical element $T$ and its inverse $\overline{T}$ given in (\ref{eq:can}) and (\ref{eq:can inv}).
Using the Sweedler notation, we write the canonical element as $T=T_1\otimes T_2\in\mathcal{H}(H)^{\otimes 2}$ and its inverse as $\overline{T}=\overline{T}_1\otimes \overline{T}_2\in\mathcal{H}(H)^{\otimes 2}$.
Given a closed normal o-graph, we replace its vertices with the following diagram to get a $\mathcal{H}(H)$-colored diagram $C_{\Gamma}$.
\begin{figure}[H]
  \centering
  \includegraphics{vertex_beads_+.pdf}
  \hspace{8mm}
  \includegraphics{vertex_beads_-.pdf}
  \caption{How to associate beads to vertices.}
  \label{fig:beads associated to vertices}
\end{figure}
\noindent
Since closed normal o-graph satisfies axiom \textbf{C1} in the definition \ref{def:bp}, we can perform the moves in Figure $\ref{fig:crm}$, $\ref{fig:bs}$ and slide beads on $C_{\Gamma}$ to get a closed circle with a bead $J_{\Gamma}$ in $\mathcal{H}(H)$.
Because one can permute the beads as in Figure \ref{fig:bs2}, $J_{\Gamma}$ is well-defined in the quotient space $\mathcal{H}(H)/[\mathcal{H}(H),\mathcal{H}(H)]$, which can be identified with $\mathbb{K}$ by Proposition \ref{prop:K}.

\begin{defi}
 $Z(\Gamma ;\mathcal{H}(H)):=\raisebox{2pt}{$\chi$}_{Fock}(J_{\Gamma})$.
\end{defi}

\begin{ex}The invariant of $S^3=L(1,1)$:

\begin{figure}[H]
  \centering
  \includegraphics{example_of_invariant_S3.pdf}
\end{figure}
\noindent
Thus we have  $Z(S^3;\mathcal{H}(H))=\raisebox{2pt}{$\chi$}_{Fock}(T_2T_1)=\sum_i\raisebox{2pt}{$\chi$}_{Fock}(e^i\otimes e_i)$. By Proposition \ref{prop:fock is int}, we have
\begin{align*}
Z(S^3;\mathcal{H}(H))&=e^i(e_L)\mu_R(e_i)\\
&=\mu_R(e^i(e_L)e_i)\\
&=\mu_R(e_L)\\&=1.
\end{align*}
\end{ex}

\begin{ex}The invariant of $\mathbb{R}P^3=L(2,1)$:

\begin{figure}[H]
  \centering
  \includegraphics{example_of_invariant_proj.pdf}
\end{figure}
\noindent
Thus we have  $Z(\mathbb{R}P^3;\mathcal{H}(H))=\raisebox{2pt}{$\chi$}_{Fock}(T_2T_1T^{\prime}_1T^{\prime}_2)=\sum\limits_{i,j}\raisebox{2pt}{$\chi$}_{Fock}(e^i\cdot(e_{i(1)}e_{j(1)}\rightharpoonup e^j)\otimes e_{i(2)}e_{j(2)})=\mathrm{Tr}(S)$, where $S$ is the antipode.
\end{ex}

\section{Main theorem}
\label{sec:mt}
Recall from Section \ref{Hopf} that a Hopf algebra $H$ is called unimodular if the left cointegrals are also the right cointegrals, and counimodular if the left integrals are also the right integrals.
We prove the following main theorem.
\begin{thm} \label{mt}
Let $H$ be a finite dimensional involutory unimodular counimodular Hopf algebra over $\mathbb{K}$, and $\Gamma$ a closed normal o-graph of a closed oriented 3-manifold $M$.
Then, $Z(\Gamma,\mathcal{H}(H))$ is an invariant of $M$.
\end{thm}

According to Proposition \ref{thm:BP}, in order to prove that $Z(\Gamma,\mathcal{H}(H))$ is an invariant, we need to show that $Z(\Gamma,\mathcal{H}(H))$ is an invariant under planer isotopy and the local moves (the Reidemeister type moves, the 0-2 move, the MP-moves, and the CP-move) of closed normal o-graph described in Figures \ref{fig:RM} to \ref{fig:CP}.
In \cite{S}, it was essentially proved that the value  $Z(\Gamma,\mathcal{H}(H))$ is an invariant under planer isotopy, the Reidemeister type moves, the 0-2 move, and the MP-moves. Here the Reidemeister type moves are nothing but the \textit{symmetry moves} in \cite{S}, the 0-2 move is a special case of the \textit{colored (0,2) moves},  and the MP-moves are obtained by the \textit{colored Pachner (2,3) moves} and the colored (0,2) moves. Even so, since the frameworks are slightly different, we give another proof in this section. The invariance under the CP-move was not observed in \cite{S}, and we give the proof in the following section, where we use a framework of tensor networks.

\begin{proof} [Proof of Theorem \ref{mt}]
We prove that $Z(\Gamma,\mathcal{H}(H))$ is an invariant under planer isotopy and the Reidemeister type moves, the 0-2 move, and the MP-moves. The proof of the invariance under the CP-moves will be shown in Section \ref{sec:tnm}.
\\
\newline
\textbf{Invariance under planer isotopy and the Reidemeister type moves}
It is obvious from the construction that $Z(\Gamma,\mathcal{H}(H))$ is an invariant under planer isotopy and the Reidemeister type moves described in the Figure \ref{fig:RM}.
\\
\newline
\textbf{Invariance under the 0-2 move.}
Let us calculate the local tensor associated to the right hand side of Figure \ref{fig:02}.
\begin{figure}[H]
  \centering
  \includegraphics{0-2_move_invariance.pdf}
\end{figure}
\noindent
Thus, we need to show that $T_2\overline{T}_2\overline{T}_1T_1=\epsilon \otimes 1$.
Let $T=\sum\limits_{i}(\epsilon \otimes e_i)\otimes (e^i \otimes 1)$ and $\overline{T}=\sum\limits_{j}(\epsilon \otimes S(e_j))\otimes (e^j \otimes 1)$.
Then,
\begin{align}
T_2\overline{T}_2\overline{T}_1T_1 &= \sum\limits_{i,j}(e^i \otimes 1)(e^j \otimes 1)(\epsilon \otimes S(e_j))(\epsilon \otimes e_i) \\
&=\sum\limits_{i,j}e^ie^j \otimes S(e_j)e_i \label{eq:0-2}
\end{align}
Identifying $H^*\otimes H$ with $\mathrm{End}(H)$, (\ref{eq:0-2}) becomes $x \mapsto S(x_{(2)})x_{(1)}$.
Since $S$ is assumed to be involutive, we have $S(x_{(2)})x_{(1)}=\epsilon(x)1$ .
Thus we have the assertion $T_2\overline{T}_2\overline{T}_1T_1=\epsilon \otimes 1$.
\\
\newline
\textbf{Invariance under the MP moves.}
There are 16 MP moves. Let us write all the calculations down.
\begin{figure}[H]
  \centering
  \includegraphics{M11_and_M21.pdf}
\end{figure}
\begin{figure}[H]
  \centering
  \includegraphics{M12_and_M22.pdf}
\end{figure}
\begin{figure}[H]
  \centering
  \includegraphics{M13_and_M23.pdf}
\end{figure}
\begin{figure}[H]
  \centering
  \includegraphics{M14_and_M24.pdf}
\end{figure}
\begin{figure}[H]
  \centering
  \includegraphics{M31_and_M41.pdf}
\end{figure}
\begin{figure}[H]
  \centering
  \includegraphics{M32_and_M42.pdf}
\end{figure}
\begin{figure}[H]
  \centering
  \includegraphics{M33_and_M43.pdf}
\end{figure}
\begin{figure}[H]
  \centering
  \includegraphics{M34_and_M44.pdf}
\end{figure}
Namely, we need to check the following 16 equalities:
\setlength{\itemsep}{2mm}
\setlength{\parskip}{2mm}
\begin{align*}
\textbf{MP1.1 :}& \quad T_{23}T_{13} = \overline{T}_{21}T_{13}T_{21} &
\textbf{MP2.1 :}& \quad \overline{T}_{32}\overline{T}_{31} = T_{12}\overline{T}_{31}\overline{T}_{12}\\
\textbf{MP1.2 :}& \quad \overline{T}_{13}\overline{T}_{23} = \overline{T}_{21}\overline{T}_{13}T_{21} &
\textbf{MP2.2 :}& \quad T_{31}T_{32} = T_{12}T_{31}\overline{T}_{12}\\
\textbf{MP1.3 :}& \quad \overline{T}_{23}T_{13} = T_2^{\prime}T_1\overline{T}_2 \otimes \overline{T}_1T_1^{\prime}\otimes T_2 &
\textbf{MP2.3 :}& \quad T_{32}\overline{T}_{31} = \overline{T}_1^{\prime}\overline{T}_2T_1\otimes T_2\overline{T}_2^{\prime}\otimes \overline{T}_1\\
\textbf{MP1.4 :}& \quad \overline{T}_{13}T_{23} = T_2\overline{T}_1^{\prime}\overline{T}_2 \otimes \overline{T}_1T_1\otimes \overline{T}_2^{\prime} &
\textbf{MP2.4 :}& \quad T_{31}\overline{T}_{32} = \overline{T}_1T_2^{\prime}T_1\otimes T_2\overline{T}_2\otimes T_1^{\prime}\\
\\
\textbf{MP3.1 :}& \quad T_{23}\overline{T}_{31} = \overline{T}_{31}T_{23}T_{21} &
\textbf{MP4.1 :}& \quad T_{31}\overline{T}_{23} = \overline{T}_{21}\overline{T}_{23}T_{31}\\
\textbf{MP3.2 :}& \quad \overline{T}_{23}\overline{T}_{31} = \overline{T}_{31}\overline{T}_{21}\overline{T}_{23} &
\textbf{MP4.2 :}& \quad T_{31}T_{23} = T_{23}T_{21}T_{31}\\
\textbf{MP3.3 :}& \quad T_{23}T_{31} = \overline{T}_2T_2^{\prime}\otimes T_1\overline{T}_1\otimes T_1^{\prime}T_2 &
\textbf{MP4.3 :}& \quad \overline{T}_{31}\overline{T}_{23} = \overline{T}_2T_2\otimes T_1\overline{T}_1^{\prime}\otimes \overline{T}_2^{\prime}\overline{T}_1\\
\textbf{MP3.4 :}& \quad \overline{T}_{23}\overline{T}_{31} = T_{21}T_{31}\overline{T}_{23} &
\textbf{MP4.4 :}& \quad \overline{T}_{31}T_{23} = T_{23}\overline{T}_{31}\overline{T}_{21}
\end{align*}

Let us see that each equation above is equivalent to the pentagon equation (\ref{eq:pentagon equation}).
Define the map $\tau_{\mathcal{H}}:\mathcal{H}(H)^{\otimes 2}\to\mathcal{H}(H)^{\otimes 2}$ by $\tau_{\mathcal{H}}(x\otimes y)=y\otimes x$ for $x, y\in\mathcal{H}(H)$.
Then, for example, if we multiply $T_{21}$ on \textbf{MP1.1} from left and then apply $\tau_{\mathcal{H}}\otimes \text{id}$, one sees that the result is exactly the pentagon equation.
Similarly, we can reduce \textbf{MP1.2},\textbf{MP2.1},\textbf{MP2.2},\textbf{MP3.1},\textbf{MP3.2},\textbf{MP3.4},\textbf{MP4.1},\textbf{MP4.2}, and \textbf{MP4.4} to the pentagon equation.

Let us see the other 6 equalities.
Define the map $\mathcal{S}:\mathcal{H}(H)\to \mathcal{H}(H)$ by $\mathcal{S}(f\otimes a)=S(f)\otimes S(x)$, where $S$ is the antipode of the Hopf algebra $H$.
Then, for example, we can transform \textbf{MP1.3} into \textbf{MP1.1} by applying $\text{id}\otimes\mathcal{S}\otimes\text{id}$ to the both sides  as follows.
\begin{align*}
(\text{id}\otimes\mathcal{S}\otimes\text{id})(\overline{T}_{23}T_{13}) &= (\text{id}\otimes\mathcal{S}\otimes\text{id})(T_2^{\prime}T_1\overline{T}_2 \otimes \overline{T}_1T_1^{\prime}\otimes T_2)\\
\Leftrightarrow\quad\sum\limits_{i,j}(\epsilon\otimes e_j)\otimes(\epsilon\otimes S^2(e_i))\otimes(e^ie^j\otimes 1)&=\sum\limits_{i,j,k}(e^j(e_{(1)}\rightharpoonup e^k)\otimes e_{(2)})\otimes (\epsilon\otimes S(e_j)S^2(e_k))\otimes(e^i\otimes 1)\\
\Leftrightarrow\quad T_{23}T_{13} &= \overline{T}_{21}T_{13}T_{21},
\end{align*}
where we used the involutivity of $S$ for the last equivalence. Thus \textbf{MP1.3} is also equivalent to pentagon equation. We can show that
\textbf{MP1.4}, \textbf{MP2.3}, \textbf{MP2.4}, \textbf{MP3.3}, \textbf{MP4.3} are also equivalent to the pentagon equation in similar manners.

Since the pentagon equation holds in the Heisenberg double $\mathcal{H}(H)$, we conclude that $Z$ is an invariant under the MP-moves.
\\
\newline
\textbf{Invariance under the CP-move.}
This will be proved in Proposition \ref{prop:CP} using tensor network.
\end{proof}

\begin{rem}
The assumption of unimodularity and counimodularity of $H$ will be used only for invariance of the CP-move.
Involutivity was used for the 0-2 move and the MP-moves \textbf{MP1.3}, \textbf{MP1.4}, \textbf{MP2.3}, \textbf{MP2.4}, \textbf{MP3.3} and \textbf{MP4.3}.
The other 10 equalities for the MP-moves do not need any restriction on Hopf algebras to hold.
\end{rem}
\begin{rem}
In \cite{S}*{Theorem 5.1}, there is an error; even for an involutory Hopf algebra, $J$ is not an invariant under the colored Pachner (2,3) move in \cite{S}*{Figure 16}, which cannot be obtained by rotating the allowed one. This excluded move corresponds to the MP-moves with some strands reversed, which are not actually the MP-moves.
\end{rem}

\section{Tensor network approach}
\label{sec:tnm}
In this section, we give a quick review of tensor networks which enable graphical calculus of tensors and linear maps. Then we reformulate the invariant using tensor networks, and prove  the invariance under the CP-move.
\subsection{Tensor network}
\label{ssec:tn}
A \textit{tensor network} over a vector space $V$ is an oriented graph which represents a tensor labeled by the set of open edges, where each incoming (resp. outgoing) edge labels $V^*$ (resp. $V$).
For example, the diagram in Figure \ref{fig:T and F} presents an $(m,n)$ tensor $T\in  (V^*)^{\otimes\mathcal{I}}\otimes V^{\otimes \mathcal{O}}=\mathrm{Hom}(V^{\otimes\mathcal{I}}, V^{\otimes\mathcal{O}})$, where $\mathcal{I}=\{i_1,\ldots, i_m\}$ is the set of incoming edges and $\mathcal{O}=\{o_1,\ldots, o_n\}$ is the set of outgoing edges.

\begin{figure}[ht]
  \centering
  \includegraphics{tensor_T.pdf}

  \caption{}
  \label{fig:T and F}
\end{figure}

One important feature of tensor networks, which makes this notion practical, is the contraction of tensors.
Given two tensor networks $T$ and $S$, one gets a new tensor network by connecting an outgoing edge $o$ of $T$ and an incoming edge $i$ of $S$ (see Figure \ref{fig:contruction}), which represents the tensor  obtained from $T\otimes S$ by contracting $V_o$ and $(V^*)_i$.
\begin{figure}[ht]
  \centering
  \includegraphics{contraction_of_tensors.pdf}
  \caption{}
  \label{fig:contruction}
\end{figure}

For example, the left diagram in Figure \ref{fig:f and g} represents the composition $g\circ f$ of two maps $f, g\colon\thinspace V\to V$ and the right diagram represents the trace $\sum_{i}f^i_i = \text{tr}(f)
\in \mathbb{K}$ of a map $f\colon\thinspace V\to V$.

\begin{figure}[H]
  \centering
  \includegraphics{f_and_g.pdf}
  \hspace{10mm}
  \includegraphics{trace_F.pdf}
  \caption{}
    \label{fig:f and g}
\end{figure}

The axioms of Hopf algebras $(H, M, 1, \Delta, \epsilon, S)$ are represented as follows;
\begin{figure}[H]
  \centering
  \includegraphics{axiom_of_M.pdf}
\end{figure}
\noindent Here, we assume that the incoming edges (resp. outgoing edges) are counted in counter-clockwise
 (resp. clockwise) order from the edge subscripted by $1$, which labels $V^{\otimes m}$ (resp. $(V^*)^{\otimes m}$) in order from the left.
In obvious cases, the subscript $1$ will be omitted.

\subsection{Reformulation of invariant}
\label{sec:IH}
We assume $H=(H, M, 1, \Delta, \epsilon, S)$ to be a finite dimensional involutory unimodular counimodular Hopf algebra.

An \textit{o-tangle} is an oriented virtual tangle diagram in $[0,1]^2$ such that each boundary point is on the bottom  $[0,1]\times \{0\}$ or on the top $[0,1]\times \{1\}$.
For finite sequences $\varepsilon,\varepsilon'$ of $\pm$, an $(\varepsilon,\varepsilon')$ o-tangle is an o-tangle having boundary points on the bottom and top compatible to $\varepsilon $ and $\varepsilon'$, respectively, where compatible means if an edge is oriented upwards (resp. downwards) then it is connected to $+$ (resp.  $-$).

Let $\mathcal{T}_{o}$ be the \textit{category of o-tangles}, where objects are finite sequences of $\pm$ including the empty sequence $\emptyset$, and morphisms $\mathcal{T}_{o}(\varepsilon,\varepsilon')$ from  $\varepsilon$ to $\varepsilon'$ are isotopy classes of $(\varepsilon,\varepsilon')$ o-tangles.
As usual, $\mathcal{T}_{o}$ is a strict monoidal category with the unit object $\emptyset$, and the composition $\Gamma' \circ \Gamma$ of $(\varepsilon,\varepsilon')$ o-tangle $\Gamma$ and  $(\varepsilon',\varepsilon'')$ o-tangle $\Gamma'$ is obtained by connecting the $\varepsilon'$ type boundary points on the top of $\Gamma$ to these on the bottom of $\Gamma'$.
We can construct a monoidal functor $Z(\,*\,;H)$ from the category of o-tangle $\mathcal{T}_{o}$ to the category of finite dimensional vector spaces $\text{Vect}_{\mathbb{K}}$ as follows.

For the object $+$ (resp. $-$), we set $Z(+;H):=H^*$ (resp. $Z(-;H):=H$).
For a sequence $\varepsilon=(\varepsilon_1, \ldots,\varepsilon_n)$ in ${\pm}$, let $H^{\varepsilon}$ denote $H^{\varepsilon_1}\otimes \cdots \otimes  H^{\varepsilon_n}$ with $H^+=H^*$ and $H^-=H$.
To a given $(\varepsilon,\varepsilon')$ o-tangle $\Gamma$, we associate a tensor network over $H$, which represents a linear map;
we replace each positive (resp. negative) crossing of $\Gamma$ with the tensor network as the left (resp. right) picture below, and then we obtain $Z(\Gamma;H)\in \mathrm{Hom}(H^{\varepsilon},H^{\varepsilon'})$ by connecting the boundary points of the above tensor networks following the strands of $\Gamma$. Note that the corresponding strands of $Z(\Gamma;H)$ are oriented in the opposite direction to these of $\Gamma$, and the linear map $Z(\Gamma;H)$ sends a tensor $T\in H^{\varepsilon}$ to the tensor $Z(\Gamma;H)(T)$ which is obtained by concatenating $T$ to the bottom of $Z(\Gamma;H)$.

\begin{figure}[H]
  \centering
  \includegraphics{vertex_tn_-.pdf}
\end{figure}

Note that a maximum point plays a role of evaluation map, and a minimum point plays a role of coevaluation map as below.

\begin{figure}[H]
    \centering
    \includegraphics{cap_cup_vect.pdf}
\end{figure}

Since a closed normal o-graph $\Gamma$ is an $(\emptyset,\emptyset)$ o-tangle, it is sent to an endomorphism $Z(\Gamma;H)$ of $\mathbb{K}$, which is presented by a scalar in $\mathbb{K}$. By abusing the notation we also denote the scalar by $Z(\Gamma;H)\in \mathbb{K}$.

\begin{ex}
For a closed normal o-graph  $\Gamma$ for $S^3$, which is $(\emptyset,\emptyset)$ o-tangle as the left picture below, the resulting tensor network $Z(\Gamma;H)$ is the right picture below.
\begin{figure}[H]
  \centering
  \includegraphics{example_tn.pdf}
\end{figure}
Thus we have $Z(\Gamma;H)=\text{tr}(M\circ\tau\circ\Delta)\in\mathbb{K}$, where $\tau(x\otimes y)=y\otimes x$.
\end{ex}
Let $\Gamma$ be a closed normal o-graph and $Z(\Gamma;\mathcal{H}(H))$ the invariant defined in Section \ref{sec:ri}.
\begin{prop}\label{prop:same invariants}
$Z(\Gamma;\mathcal{H}(H))=Z(\Gamma;H)$.
\end{prop}
\begin{proof}
First, let us consider an oriented strand with a bead $f\otimes a \in \mathcal{H}(H)$ as in the left picture below.
We will think of this strand with a bead as the action of $f\otimes a$ on the Fock space $F(H^*)$, which was given by $\phi(f\otimes a)\colon\thinspace g\mapsto f(a\rightharpoonup g)$ for $g \in H^*$, where $f(a\rightharpoonup g)\colon\thinspace x \mapsto f(x_{(1)})g(x_{(2)}a)$ for $x\in H$.
Graphically this map $\phi(f\otimes a)\in \mathrm{Hom}(H^*,H^*)$ can be represented by the tensor network in the right picture below.
\begin{figure}[H]
    \centering
    \includegraphics{strand_action.pdf}
\end{figure}
\noindent
If there are multiple beads, we replace each bead with the above tensor network. Since each map is an action, the result is well-defined under the move of Figure \ref{fig:bs}.

Then, let us consider the oriented closed curve with a bead $f\otimes a\in\mathcal{H}(H)$.
Recall that $\raisebox{2pt}{$\chi$}_{Fock}(f\otimes a)$ is the trace of the linear map defined by the action of $f\otimes a$.
As remarked in Section \ref{ssec:tn}, in terms of tensor network, taking a trace is just connecting the incoming edge with the outgoing edge.
Thus, we have the following.
\begin{figure}[H]
    \centering
    \includegraphics{Fock_action_in_tn.pdf}
\end{figure}

Finally, let $\Gamma$ be a closed normal o-graph.
Replacing its vertices as in Figure \ref{fig:beads associated to vertices} and sliding beads, we get a single bead $J_{\Gamma}\in\mathcal{H}(H)$, and the invariant $Z(\Gamma, \mathcal{H}(H))$ was defined as $\raisebox{2pt}{$\chi$}_{Fock}(J_{\Gamma})$.
Here, before the sliding process, we replace each bead with a corresponding tensor network as above, and compare the result to $Z(\Gamma;H)$.
Since the beads only appear at the vertices of $\Gamma$, we just need to look at the associated tensor network for these beads:
\begin{figure}[H]
    \centering
    \includegraphics{vertices_action_on_Fock_3.pdf}
\end{figure}
\noindent These are the same tensor networks associated with vertices in the definition of $Z(\Gamma;H)$.
Thus we have the assertion.
\end{proof}

\subsection{Invariance under CP-move}
We prove the invariance of $Z(\Gamma;H)$ under the CP-move.  Recall from  Section \ref{sec:hd} for the definition of integrals.
Let $e_R\in H$ and $\mu_L\in H^*$ be a left cointegral and a left integral satisfying $\mu_L(e_L)=1$.
In terms of tensor networks, we have
\begin{figure}[H]
    \centering
    \includegraphics{integral_in_tensornetwork.pdf}
\end{figure}

\begin{lem}The following equality holds in any Hopf algebra.
\begin{figure}[H]
    \centering
    \includegraphics{integral_lemma.pdf}
\end{figure}
\label{lem:integral}
\end{lem}

\begin{proof}
See \cite{Ku2}, Lemma 3.3.
\end{proof}
Set $e_R:=S(e_L)$. Since $S$ is an anti-algebra map, $e_R$ is a (non-zero) right cointegral.

\begin{lem}\label{lem:integral is cyclic}
For a finite dimensional involutory counimodular Hopf algebra, we have
\begin{align*}
\Delta^{\mathrm{op}}(e_R) &= \Delta(e_R),
\end{align*}
where $\Delta^{\mathrm{op}}(x):=x_{(2)}\otimes x_{(1)}$.
\end{lem}

\begin{proof}
See \cite{R}, Theorem 10.5.4.
\end{proof}

\begin{lem}\label{tw}
Let $H$ be an involutory Hopf algebra. We have the following.
\begin{figure}[H]
    \centering
    \includegraphics{twist_lemma.pdf}
\end{figure}
\end{lem}

\begin{proof}
Replacing the vertex of the o-tangle with the corresponding tensor network we have
\begin{figure}[H]
    \centering
    \includegraphics{twist_lemma_proof.pdf}
\end{figure}
Using Lemma \ref{lem:integral} we get
\begin{figure}[H]
    \centering
    \includegraphics{twist_lemma_proof_2.pdf}
\end{figure}
\end{proof}

\begin{prop}
\label{prop:CP}
Let $H$ be a finite dimensional involutory unimodular counimodular Hopf algebra.
Then $Z(\Gamma;H)$ is an invariant under the CP-move in Figure \ref{fig:CP}.
\end{prop}

\begin{proof}
First, we evaluate the left hand side of the CP-move.
Using Lemma \ref{tw}, the twists in the closed normal o-graph can be replaced by integrals,  and thus we have
\begin{figure}[H]
    \centering
    \includegraphics{CP_proof.pdf}
\end{figure}
\noindent
Next, we evaluate a part of the right hand side of the CP-move.
\begin{figure}[H]
    \centering
    \includegraphics{CP_proof_2.pdf}
\end{figure}
\noindent
We used the definition of left integral $\mu_L$ for the equality above.
From Lemma \ref{lem:integral is cyclic}, $e_R$ is cyclic. Thus,
\begin{figure}[H]
    \centering
    \includegraphics{CP_proof_3.pdf}
\end{figure}
\noindent
The second equality follows from coassociativity of the comultiplication, and the last equality follows from  the property of the antipode.
Thus right hand side of the CP-move becomes
\begin{figure}[H]
    \centering
    \includegraphics{CP_proof_4.pdf}
\end{figure}
\noindent
Finally we need only to show that the following equality holds for involutory unimodular counimodular Hopf algebra.
\begin{figure}[H]
    \centering
    \includegraphics{integral_eq_for_CP.pdf}
\end{figure}
\noindent
Since $H$ is unimodular, $e_R=ke_L$ for some non-zero $k \in \mathbb{K}$.
Since $\mu_L(e_L)$ is $1$, $\mu_L(e_R)=k$ and the left hand side of the above tensor network is the same as right hand side times $k^2$.
Applying $S^2$ to $e_L$ and using the fact that $S$ is involutive, we see that $k^2=1$, and the above equality follows.
\end{proof}

\section{Properties}
\label{sec:csf}
We continue to assume that $H$ is a finite dimensional involutory unimodular counimodular Hopf algebra over $\mathbb{K}$.
For a  closed oriented $3$-manifold $M$ and a closed normal o-graph $\Gamma$ representing $M$, set  $Z(M;H):=Z(\Gamma;H)$.

\subsection{Connected sum formula}
For  closed oriented $3$-manifolds $M,N$, let $M\# N$ be the connected sum of them.
\begin{prop}
$Z(M\# N;H)=Z(M;H)Z(N;H)$.
\end{prop}

\begin{proof}
Let $\Gamma_M$  and $\Gamma_N$ be closed normal o-graphs representing $M$ and $N$ respectively.
Let $\Gamma_M\# \Gamma_N$ be the connected sum of closed normal o-graphs defined in Figure \ref{fig:connected sum}.

\begin{figure}[ht]
    \centering
    \includegraphics{connected_sum.pdf}
    \caption{}
    \label{fig:connected sum}
\end{figure}

In \cite{Ko}, Y. Koda showed that for two closed normal o-graphs $\Gamma_M$ and $\Gamma_N$, the $3$-manifold represented by $\Gamma_M\# \Gamma_N$ is $M\# N$. We show the assertion by comparing the tensor networks for the closed normal o-graphs in the left and right hand sides of the Figure \ref{fig:connected sum}.
Note that
\begin{figure}[H]
    \centering
    \includegraphics{connected_sum_proof.pdf}
\end{figure}
Thus we have
\begin{figure}[H]
    \centering
    \includegraphics{connected_sum_proof_2.pdf}
\end{figure}
which implies the assertion.
\end{proof}

\subsection{Group algebra}
We show that the invariant $Z(M;\mathbb{C}[G])$ with the group algebra $\mathbb{C}[G]$ of a finite group $G$ counts the number $|\mathrm{Hom}(\pi_1M,G)|$ of group homomorphisms from the fundamental group $\pi_1 M$ of $M$ to $G$.
The proof essentially follows the line of [KR].

The algebra $\mathbb{C}[G]$ has a canonical basis given by $\{ g \}_{g\in G}$ and the Hopf algebra structure given by $\Delta(g)=g\otimes g$, $\epsilon(g)=1$, and $S(g)=g^{-1}$.
Note that $\mathbb{C}[G]$ is involutory and also unimodular and counimodular.
The dual group algebra $\mathbb{C}(G):=\mathbb{C}[G]^*$ has the dual basis $\{ \delta_g \}_{g\in G}$, and the dual Hopf algebra structure is given as follows.
\begin{center}
    $\delta_{g}\cdot\delta_{h} = \delta _{g,h}\delta_{g} \qquad 1_{\mathbb{C}(G)} = \sum\limits_{g\in G}\delta_{g}$
\end{center}
\begin{center}
    $\Delta(\delta_{g}) = \sum\limits_{hk=g}\delta_{h}\otimes \delta_{k} \qquad {\epsilon}(\delta_g) = \delta_g(e)$
\end{center}
\begin{center}
    $S(\delta_g) = \delta_{g^{-1}}$
\end{center}
where  $\delta_{g, h}\in \{ 0,1 \}$ is $1$ if $g=h$ and $0$ otherwise, and $e$ is the unit of $G$.

The left action of $x\in G\subset\mathbb{C}[G]$ on $\delta_{g}\in\mathbb{C}(G)$ is given by
\begin{align*}
x\rightharpoonup \delta_g = \delta_{gx^{-1}}.
\end{align*}

\begin{prop}
For a closed oriented $3$-manifold $M$ and a finite group $G$, we have $Z(M;\mathbb{C}[G])=|\mathrm{Hom}(\pi_1M,G)|$.
\end{prop}

\begin{proof}
Let $\Gamma$ be a closed normal o-graph representing $M$.
In \cites{Ko, Ko3}, Y. Koda gave an explicit formula for the fundamental group $\pi_1 M$  in terms of closed normal o-graph $\Gamma$.  Let $E$ be the set of all edges of $\Gamma$. We consider the group generated by $E$ and consider the relation set $R$ consisting of  $g=l$, $hg=k$ for the edges  $g, l, h, k$ around each vertex as below.
Then the resulting group $\langle E\,|\,R\rangle$ is isomorphic to the fundamental group $\pi_1M$ of $M$,  and the number $|\mathrm{Hom}(\pi_1M,G)|$ is equal to the number of edge colorings  $c\colon\thinspace E \to G$ such that $c(R)$ holds in $G$.
\begin{figure}[H]
    \centering
    \includegraphics{fundamental_group.pdf}
\end{figure}

We show that the invariant $Z(M;\mathbb{C}[G])$ indeed counts such edge colorings.
As we explained in Section \ref{sec:IH}, each vertex of closed normal o-graph can be treated as a linear map between $\mathbb{C}(G)^{\otimes 2}$:
\begin{figure}[H]
    \centering
    \includegraphics{homeo_counting_proof.pdf}
\end{figure}
\noindent Here we have,
\begin{align*}
T_1\triangleright\delta_h\otimes T_2\triangleright\delta_g &= \sum\limits_{x\in G}({\epsilon}\otimes x)\triangleright\delta_h\otimes (\delta_x\otimes e)\triangleright\delta_g \\
&= \sum\limits_{x\in G}x\rightharpoonup\delta_h\otimes\delta_x\cdot\delta_g\\
&= \sum\limits_{x\in G}\delta_{hx^{-1}}\otimes \delta_{x,g}\delta_g\\
&= \delta_{hg^{-1}}\otimes \delta_g\\
\end{align*}
and
\begin{align*}
\overline{T}_2\triangleright\delta_g\otimes\overline{T}_1\triangleright\delta_h &= \sum\limits_{x\in G}(\delta_x\otimes e)\triangleright\delta_g\otimes ({\epsilon}\otimes x^{-1})\triangleright\delta_h\\
&= \sum\limits_{x\in G}\delta_x\cdot\delta_g\otimes x^{-1}\rightharpoonup\delta_h\\
&= \sum\limits_{x\in G}\delta_{x,g}\delta_g\otimes\delta_{hx}\\
&= \delta_{g}\otimes\delta_{hg}.\\
\end{align*}
We draw these maps as follows.
\begin{figure}[H]
    \centering
    \includegraphics{homeo_counting_proof_2.pdf}
\end{figure}
\noindent Note that the subscripts of $\delta$ give nothing but an edge coloring (around the vertices) as desired. Furthermore, to connect those vertices by strands means that we insert maximum and minimum points among them. Recall from Section \ref{sec:IH} that maximum and minimum points correspond to evaluations and coevaluations, respectively. In the present case they are the maps shown below, which means after all we sum up all edge colorings.
\begin{figure}[H]
    \centering
    \includegraphics{cap_cup.pdf}
\end{figure}
\end{proof}

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\end{document}

redpajama_set_name: RedPajamaArXiv


text:
from django import forms
from app.models import *
from django.contrib.auth.models import User
from django.core.validators import email_re

def is_valid_email(email):
    return True if email_re.match(email) else False

class AddFriendForm(forms.Form):
    friend = forms.EmailField()

    def is_valid(self, request):
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            return True
        return False

    def save(self, request):
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        friend, _       = Friends.objects.get_or_create(creator = request.user, friend = a)
        friend.save()
        print 'dsdadas'

redpajama_set_name: RedPajamaGithub