The bigraded stabilization factor on small grids

The standard 2 × 2 grid has fully blocked Poincaré series W, and stabilization splitting an X-marking multiplies the series by W. So the 3 × 3 unknot grids obtained by stabilizing the standard 2 × 2 grid have series W ^ 2: the dependence of fully blocked homology on the grid size is visible in the bigrading.

Main results

• TauCeti.OddComponentGridDiagram.bigradedFullyBlockedHomologyPoincareSeries_twoByTwo_stabilizeX: the stabilizations of the standard 2 × 2 grid have Poincaré series W ^ 2.

References

Ozsváth--Stipsicz--Szabó, Grid Homology for Knots and Links, Section 5.2.

theorem TauCeti.OddComponentGridDiagram.bigradedFullyBlockedHomologyPoincareSeries_twoByTwo_stabilizeX (s : Fin 2) : (twoByTwo.stabilizeX s).bigradedFullyBlockedHomologyPoincareSeries = Bigraded.W ^ 2

The 3 × 3 unknot grids obtained by stabilizing the standard 2 × 2 grid have Poincaré series W ^ 2: their fully blocked homology is W ⊗ W.