A note on Grid Homology in lens spaces: $\mathbb{Z}$ coefficients and computations

TL;DR

This paper provides a combinatorial proof of sign refined grid homology in lens spaces, confirms the differential squares to zero over integers, and offers computational tools and evidence regarding torsion in these homology groups.

Contribution

It introduces a combinatorial proof for the existence of sign refined grid homology in lens spaces and a Sage program for computations, with empirical analysis of torsion.

Findings

Proof that ^2=0 over  coefficients

Sage program for computing b6b6GH in lens spaces

Empirical evidence suggests absence of torsion in these groups

Abstract

We present a combinatorial proof for the existence of the sign refined grid homology in lens spaces, and a self contained proof that $\partial_{\mathbb{Z}}^2 = 0$. We also present a Sage program that computes $\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence of torsion in these groups.