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Khovanov Homology for Tangles in Connected Sums

Alan Du

Abstract

Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting the 3-manifold along a separating sphere, we construct type D and type A structures that are invariants of tangles in the two halves following the work of Roberts. Gluing the type D and type A structures along the common boundary recovers the Khovanov homology of the link.

Khovanov Homology for Tangles in Connected Sums

Abstract

Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting the 3-manifold along a separating sphere, we construct type D and type A structures that are invariants of tangles in the two halves following the work of Roberts. Gluing the type D and type A structures along the common boundary recovers the Khovanov homology of the link.
Paper Structure (20 sections, 25 theorems, 60 equations, 22 figures)

This paper contains 20 sections, 25 theorems, 60 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Links in Connected Sums of Interval Bundles
  3. Tangles and Resolutions
  4. Resolutions
  5. 1-1 bifurcations in pairs of bridges
  6. Cleaved Links Algebra
  7. The vertices of Gamma_n
  8. The edges of Gamma_n
  9. The cleaver category
  10. A differential on BGamma_n
  11. The APS Chain Complex
  12. Types of Bridges
  13. Existence of Squares
  14. The APS Differential
  15. Type D Structure
  16. ...and 5 more sections

Key Result

Theorem 1.1

The chain homotopy type of the $A_\infty$ pairing $\llangle {\mathcal{L}}\rrangle=\llangle {\mathcal{T}}_1\rrbracket\boxtimes\llbracket {\mathcal{T}}_2\rrangle$ equipped with the box tensor product differential $\partial^\boxtimes$ is an invariant of the link ${\mathcal{L}}={\mathcal{T}}_1\natural {

Figures (22)

  • Figure 1: $\Sigma$ for $M=\#^3{\mathbb{R}} P^3$, where we consider ${\mathbb{R}} P^3$ minus a point as the twisted interval bundle over ${\mathbb{R}} P^2$, which we draw as a disk quotiented by the antipodal map on the boundary. The disks marked $A$ and $B$ are cut out and glued together by a reflection.
  • Figure 2: Reidemeister I, II, and III moves
  • Figure 3: Finger and Mirror moves: the two moves take the left diagram to the right diagram on the same row. In each diagram, the two black circles are $\partial D_i^\pm$ and are identified by a reflection across the line $y=-x$.
  • Figure 4: A handleslide of the blue arc below and above $\partial D_i$.
  • Figure 5: Positive and Negative crossings
  • ...and 17 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Lemma 3.6
  • ...and 57 more