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Grid homology for spatial graphs and a Künneth formula of connected sum

Hajime Kubota

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 Künneth formula for the knot Floer homology of connected sums in the framework of the grid homology.

Grid homology for spatial graphs and a Künneth formula of connected sum

Abstract

In this paper, we research the grid homology for spatial graphs with cut edges. We show that the grid homology for spatial graph is trivial if has sinks, sources, or cut edges. As an application, we give purely combinatorial proofs of some formulas including a Künneth formula for the knot Floer homology of connected sums in the framework of the grid homology.
Paper Structure (17 sections, 20 theorems, 49 equations, 15 figures)

This paper contains 17 sections, 20 theorems, 49 equations, 15 figures.

Table of Contents

  1. Introduction
  2. MOY graphs
  3. Main results
  4. Outline of the paper.
  5. grid homology for general MOY graphs
  6. The definition of the grid chain complex
  7. The invariance of HF
  8. The proof of Theorem \ref{['thm:cutedge']} (1)
  9. The preparations of Theorem \ref{['thm:cutedge']} (2)
  10. The proof of theorem \ref{['thm:cutedge']} (2)
  11. The structure of the grid chain complex
  12. The main idea of the proof
  13. The proof of Corollary \ref{['cor:wedge sum']}
  14. The proof of Theorem \ref{['cor:connected-sum-knot']}
  15. The proof of Theorem \ref{['thm:disjoint']}
  16. ...and 2 more sections

Key Result

Theorem 1.4

Let $(f,\omega)$ be an MOY graph.

Figures (15)

  • Figure 1: Spatial handcuff graphs. The rightmost one has no cut edge as a spatial graph.
  • Figure 2: MOY graphs in Definition \ref{['dfn: disjoint-connected-wedge-sum']}
  • Figure 3: Cyclic permutation and commutation$'$, gray lines are $\mathrm{LS}_1$ and $\mathrm{LS}_2$
  • Figure 6: Two graph grid diagrams representing $f$ and $f'$ respectively
  • Figure 7: The special grid-like diagram $C_n$
  • ...and 10 more figures

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Remark 2.1
  • ...and 47 more