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Instantons and odd Khovanov homology

Christopher W. Scaduto

Abstract

We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route, we provide a new proof of Floer's surgery exact triangle for instanton homology using metric stretching maps, and generalize the exact triangle to a link surgeries spectral sequence. Finally, we relate framed instanton homology to Floer's instanton homology for admissible bundles.

Instantons and odd Khovanov homology

Abstract

We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route, we provide a new proof of Floer's surgery exact triangle for instanton homology using metric stretching maps, and generalize the exact triangle to a link surgeries spectral sequence. Finally, we relate framed instanton homology to Floer's instanton homology for admissible bundles.

Paper Structure

This paper contains 53 sections, 28 theorems, 388 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Background & Motivation
  3. Outline & Acknowledgments
  4. The Surgery Story
  5. Bundles in the Exact Triangle
  6. Dehn Surgery with Bundles
  7. The Surgery Bundle $\mathbb{Y}_i$
  8. The Surgery Cobordism $\mathbb{X}_{ij}$
  9. The Bundle $\mathbb{S}_i$
  10. The Bundle $\mathbb{T}$
  11. An Involution of $\mathbb{E}$
  12. Geometric Representatives
  13. Instanton Homology
  14. Instanton Groups
  15. Maps from Cobordisms
  16. ...and 38 more sections

Key Result

Theorem 1.1

Given an oriented link $L$ in $S^3$, there is a spectral sequence whose second page is $\overline {\text{Kh}'}(L)$ that converges to $I^\#(\overline{\Sigma(L)})$. Each page of the spectral sequence comes equipped with a $\mathbb{Z}/4$-grading, which on $\overline {\text{Kh}'}(L)$ is given by where $\sigma$ and $\nu$ are the signature and nullity of $L$, respectively, and the induced $\mathbb{Z}/4

Figures (19)

  • Figure 1: Local surgery diagrams. The slanted line in each case is the knot $K$. Each row represents a possible construction for a surgery triad.
  • Figure 2: From the diagram $\text{D}$ to a resolution diagram $\text{D}_v$.
  • Figure 3: On the left, we want to color the regions of the diagram so that at each crossing exactly one of the four regions is colored. On the right, we go from an oriented diagram to a disjoint union of oriented circles.
  • Figure 4: The two hypersurfaces $Y_1$ and $S_1$ in the interior of $X_{02}$. The 3-sphere $S_1$ separates off a copy of $-\mathbb{C}\mathbb{P}^2$ minus a 4-ball.
  • Figure 5: The intersections of the five hypersurfaces in the interior of $X_{03}$. The $S^1\times S^2$ hypersurface $T$ divides $X_{03}$ into two pieces, $V$ and $E$. This picture first appeared in kmos.
  • ...and 14 more figures

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 2.1: Floer
  • Theorem 2.2
  • Theorem 2.3
  • ...and 30 more