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

TL;DR

This work extends grid homology to singular links in lens spaces $L(p,q)$ and constructs a purely combinatorial resolution cube for knot Floer homology in that setting. It builds three grid-based chain complexes, establishes two gradings, and proves invariance under grid moves, then derives skein exact sequences and a resolution cube leading to a spectral sequence to $HFK^-$; the theory is subsequently lifted to integral coefficients via sign assignments and related to the canonical orientation framework. By leveraging universal covers and sign conventions, the authors connect the combinatorial grid model to the classical knot Floer theory and show an oriented, sign-sensitive version recovers the standard oriented HFK in $S^3$ and lens spaces. The results provide a robust, computable combinatorial toolkit for singular knots in lens spaces and lay groundwork for relating $HFK^ullet$ to other link homology theories. Overall, the paper delivers a complete combinatorial framework for singular knot theory in lens spaces, including a signed resolution cube that recovers classical knot Floer homology and offers avenues for cross-theory comparisons.

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.

Figures (23)

• Figure 1: An example of grid diagram
• Figure 2: Combined diagram for resolution of crossing.
• Figure 3: Examples and a non-example of special trivalent graph. (a) orientation at each thick edge; (b) an example of special trivalent graph; (c) non-orientable example
• Figure 4: A standard disk with its triangulation.
• Figure 5: A transverse disk that separates incoming and outgoing edges

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3