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An Extension of Khovanov Homology to Immersed Surface Cobordisms

Scott Carter, Benjamin Cooper, Mikhail Khovanov, Vyacheslav Krushkal

TL;DR

This work extends Khovanov homology to oriented surface cobordisms with double point singularities by associating maps to double points via Hopf-link Khovanov homology and extending Carter–Saito movie moves to the singular setting. It constructs chain maps $A$ and $B$ with distinct $(t,q)$-degrees, establishing a functorial extension to a 2-functor $\tilde{\kappa}'$ on immersed surface cobordisms, defined up to an overall sign. The authors formulate retinal charts and decorated charts to encode cross-sections of immersed surfaces in $\mathbb{R}^4$ and prove a movie-move theorem for immersed surfaces, including new moves MM16–MM19 involving nodes. Together, these results broaden the applicability of Khovanov-type invariants in 4-dimensional topology by incorporating singular surface cobordisms into the functorial framework.

Abstract

We show that an oriented surface in $\mathbb{R}^4$ containing double point singularities induces a map between the Khovanov homology groups of its boundary links in a functorial way. As part of this work, the movie moves of Carter and Saito are extended to surfaces with double points.

An Extension of Khovanov Homology to Immersed Surface Cobordisms

TL;DR

This work extends Khovanov homology to oriented surface cobordisms with double point singularities by associating maps to double points via Hopf-link Khovanov homology and extending Carter–Saito movie moves to the singular setting. It constructs chain maps and with distinct -degrees, establishing a functorial extension to a 2-functor on immersed surface cobordisms, defined up to an overall sign. The authors formulate retinal charts and decorated charts to encode cross-sections of immersed surfaces in and prove a movie-move theorem for immersed surfaces, including new moves MM16–MM19 involving nodes. Together, these results broaden the applicability of Khovanov-type invariants in 4-dimensional topology by incorporating singular surface cobordisms into the functorial framework.

Abstract

We show that an oriented surface in containing double point singularities induces a map between the Khovanov homology groups of its boundary links in a functorial way. As part of this work, the movie moves of Carter and Saito are extended to surfaces with double points.
Paper Structure (14 sections, 9 theorems, 28 equations, 13 figures)

This paper contains 14 sections, 9 theorems, 28 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Review of Khovanov homology
  3. Khovanov Homology
  4. Extension of Khovanov homology to double points
  5. Chain maps associated with double points
  6. Related work in the literature
  7. Checking the new movie moves
  8. Movie moves for immersed surface cobordisms
  9. Encoding the cross-sections of a surface
  10. Definition and properties of charts
  11. The edges
  12. Edge decorations
  13. The vertices
  14. The movie move theorem

Key Result

Theorem 1

The maps induced on the Khovanov homology of oriented links are well-defined, up to an overall sign, on isotopy classes of surface cobordisms with double points. These maps give rise to a functor from the category of surface cobordisms with double points between oriented links in $\mathbb R^3$ to th

Figures (13)

  • Figure 1: The definition of the chain maps $A$, $B$ assigned to double points
  • Figure 2: The grid shows the homology of the Hopf links $H_\pm$ from Ex. \ref{['ex-dual']}. As in Prop. \ref{['prop-abhopf']} the generators labelled $A$ and $B$ correspond to the maps $A$ and $B$ under the duality isomorphism. A hollow circle is a $\mathbb{Z}$-summand which is not in the image of the operation $X$ (determined by the Frobenius algebra). A filled circle is in the image of $X$. Multiplication by $X$ at a component of a link has $q$-degree $-2$.
  • Figure 3: Types of crossings and critical points (optima) of links, and the corresponding bookmark labels
  • Figure 4: Connected sum of $4_1\# (-4_1)$
  • Figure 13: MM16: passing a node over a type-II move.
  • ...and 8 more figures

Theorems & Definitions (23)

  • Theorem
  • Theorem
  • Theorem
  • Proposition 2.1
  • Lemma 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 13 more