Grid homology for spatial graphs and a K\"{u}nneth formula of connected sum

TL;DR

This paper explores grid homology for spatial graphs with cut edges, demonstrating triviality in certain cases and providing combinatorial proofs for formulas like the K"unneth formula in knot Floer homology.

Contribution

It introduces new results on the triviality of grid homology for spatial graphs with sinks, sources, or cut edges, and offers combinatorial proofs of key formulas in knot Floer homology.

Findings

Grid homology is trivial for spatial graphs with sinks, sources, or cut edges.

Provides combinatorial proofs of the K"unneth formula for knot Floer homology.

Establishes connections between spatial graph properties and homological invariants.

Abstract

In this paper, we research the grid homology for spatial graphs with cut edges. We show that the grid homology for spatial graph $f$ is trivial if $f$ has sinks, sources, or cut edges. As an application, we give purely combinatorial proofs of some formulas including a K\"{u}nneth formula for the knot Floer homology of connected sums in the framework of the grid homology.