Grid homology for singular links in lens space and a resolution cube

TL;DR

This paper develops a new combinatorial framework for studying singular links in lens spaces using grid homology, enabling the construction of a resolution cube for knot Floer homology and extending classical theories beyond $S^3$.

Contribution

It introduces grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology, generalizing previous work from $S^3$ to lens spaces.

Findings

Defined grid homologies for singular links in lens spaces

Constructed a resolution cube for knot Floer homology in lens spaces

Established equivalence with classical knot Floer homology over $bZ$

Abstract

In this paper, we define grid homologies for singular links in lens spaces and use them to construct a resolution cube for knot Floer homology of regular links in lens spaces. The results will first be proved over $\mathbb{Z}/2\mathbb{Z}$ and then over $\mathbb{Z}$ with the help of sign assignments. We will also identify the signed grid homology and classical knot Floer homology over $\mathbb{Z}$ for regular links in lens spaces, illustrating the fact that our resolution cube is genuinely one for knot Floer homology. The main advancement in the paper is that we give a complete description of singular knot theory in lens spaces which was only defined in $S^3$ previously and we construct a signed combinatorial resolution cube for knot Floer homology in lens spaces which may be powerful in relating $HFK^\circ$ to other link homology theories.