Combinatorial Proofs of Properties of Double-Point Enhanced Grid Homology
Ollie Thakar
TL;DR
The paper provides a purely combinatorial construction of a skein exact sequence for double-point enhanced grid homology $GHL$ with integer coefficients, extending the base grid-homology framework to include a formal variable $v$ and sign assignments. It establishes invariance under grid moves (commutations, switches, stabilizations) via pentagon/hexagon-domain combinatorics and sign-augmented maps, and develops a skein exact sequence for links in the collapsed setting $cGHL$, along with explicit maps that realize the exactness. A central theme is comparing $GHL$ to the traditional knot Floer invariant via a spectral sequence from $GH^-(K)[v]$ to $GHL(K)$, and exploring proposals for invariants analogous to $\tau$ through endomorphisms like $\partial_1$ and its induced $\partial_{1*}$. The paper also reports on computations for alternating, quasi-alternating, and torus knots, providing evidence that $GHL(K) \cong GH^-(K)[v]$ in several key families, and discusses open questions about whether extra information exists beyond this relation and about potential filtered theories.
Abstract
We provide a purely combinatorial proof of a skein exact sequence obeyed by double-point enhanced grid homology. We also extend the theory to coefficients over $\mathbb{Z},$ and discuss alternatives to the Ozsváth-Szabó $τ$ invariant.