Combinatorial Proofs of Properties of Double-Point Enhanced Grid Homology

TL;DR

This paper presents a combinatorial proof of a skein exact sequence in double-point enhanced grid homology, extends the theory to integer coefficients, and explores alternatives to the Ozsváth-Szabó τ invariant.

Contribution

It offers a new combinatorial proof for key properties of double-point enhanced grid homology and extends the theory to broader coefficients and invariants.

Findings

Provided a combinatorial proof of the skein exact sequence.

Extended the homology theory to integer coefficients.

Discussed alternatives to the Ozsváth-Szabó τ invariant.

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 Ozsv\'ath-Szab\'o $\tau$ invariant.