Why is density of water maximum at 4 degree celsius? Why is density of water maximum at 4 degree celsius?

We were having a discussion to this question in class and I could gather the following points. 

1) $H_2O$ exists in a cage like structure in ice form, due to extensive hydrogen bonding.

2) density=mass/ volume : We are not changing the mass here, therefore increase in volume will directly correspond to decrease in density and vice versa.

2) This cage like structure can trap gases and result in increase in volume. Therefore, forming of this cage like structure will result in increase in volume and breaking of this structure will result in decrease in volume. 

3) At 273K , this cage like structure of ice exists as shown. On heating, we find that molecules gain kinetic energy and this cage like structure breaks. Due to this breaking, the volume decreases. At 277 K or (4 degree celsius), it so happens that the entire cage like structure is broken which results in decrease in volume resulting in maximum density. 
Therefore, density of water is maximum at this temperature. 

4) At this stage, the cage has been entirely broken. Now, the molecules form this cage like structure again due to which volume again starts to increase as a result of which density decreases. 

Here's what I couldn't understand:


  1) I cannot understand why trapping of gases results in increase of
  volume. The gases are trapped in the free space inside the cage.
  Whether that free space is occupied by gases or whether it remains
  unoccupied shouldn't result in increase of volume. Then why it's said
  that trapping of gases inside the cage like structure formed by ice
  results in increase in volume?
  
  2) I am not sure I heard the 4th point right because it clearly
  doesn't make sense. If at 277K, the cage like structure broke, why it
  will reform on further heating. On heating, we are giving the
  molecules more kinetic energy.They should roam about and not get tied
  down to a cage like structure. Why that cage like structure is being
  formed again after 4 degree celsius?
The answer appears to be fairly complicated, but fundamentally it results from a competition between two different options for water molecules interacting. 

One place to start would be The structure of water; from ambient to deeply supercooled, Lars G.M. Pettersson and Anders Nilsson, J. Non-Crystalline Solids 407 399-417 (2015) where they discuss the 'inhomogeneous structure hypothesis'. At its heart they define high density liquid (HDL) and low density liquid (LDL) (not connected to cholesterol!) structures. LDL is tetrahedral, HDL is more distorted. LDL is directional H-bonded, while HDL is more van der Waals influenced. LDL minimized enthaly, HDL maximizes entropy. These are all consistent with a variety of experimental and simulation work given in the paper. 

A recent paper applying DFT to the problem, Interstitial Voids and Resultant Density of Liquid Water: A First-Principles Molecular Dynamics Study, Sohag Biswas et al., ACS Omega 3 2010-2017 (2018), suggests a similar general picture. Again, the tradeoff of hydrogen vs van der Waals bonding appears to be important. Different exchange/correlation terms provide somewhat different answers. One suggested reason for the HDL structure is that, by incorporating an 'interstitial' water molecule into the LDL, one disrupts the directional hydrogen bonds and enhances the breaking and making of them, blurring out the structure (while being higher density).

So, it isn't a simple one-phase picture, but still seems to be a tradeoff of entropy and enthalpy of different configurations in the water.