Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new computational tool for Khovanov cobordism maps

Zsombor Fehér

TL;DR

The paper introduces a Python tool to compute Khovanov cobordism maps $\mathrm{Kh}(\Sigma)$ for cobordisms between links, filling a gap in computational capabilities for this invariant. It formulates the maps via movie moves (births, deaths, saddles, Reidemeister moves) and enforces crossing-order and mirror dualities to ensure robust, invariant results, including a Khovanov–Jacobsson class when ending in the empty set. The authors apply the software to two main problems: (i) determining cobordism maps for all incompressible Seifert surfaces of prime knots up to $10$ crossings and extracting gcd data from the induced maps, and (ii) distinguishing ribbon disks arising from knot symmetries via their cobordism data, with extensive data and code provided. The work demonstrates the practical utility of Khovanov cobordism maps for distinguishing smooth surface isotopy classes and symmetry-related disks, and provides publicly available tools for researchers to reproduce and extend these analyses.

Abstract

We describe a Python module that we developed to calculate cobordism maps induced on Khovanov homology. As applications of our program, we compute these maps for all incompressible Seifert surfaces for prime knots up to 10 crossings, and distinguish many ribbon disks arising from symmetries of the boundary knots.

A new computational tool for Khovanov cobordism maps

TL;DR

The paper introduces a Python tool to compute Khovanov cobordism maps for cobordisms between links, filling a gap in computational capabilities for this invariant. It formulates the maps via movie moves (births, deaths, saddles, Reidemeister moves) and enforces crossing-order and mirror dualities to ensure robust, invariant results, including a Khovanov–Jacobsson class when ending in the empty set. The authors apply the software to two main problems: (i) determining cobordism maps for all incompressible Seifert surfaces of prime knots up to crossings and extracting gcd data from the induced maps, and (ii) distinguishing ribbon disks arising from knot symmetries via their cobordism data, with extensive data and code provided. The work demonstrates the practical utility of Khovanov cobordism maps for distinguishing smooth surface isotopy classes and symmetry-related disks, and provides publicly available tools for researchers to reproduce and extend these analyses.

Abstract

We describe a Python module that we developed to calculate cobordism maps induced on Khovanov homology. As applications of our program, we compute these maps for all incompressible Seifert surfaces for prime knots up to 10 crossings, and distinguish many ribbon disks arising from symmetries of the boundary knots.

Paper Structure

This paper contains 10 sections, 2 theorems, 10 equations, 1 figure, 6 tables.

Table of Contents

  1. Background
  2. Khovanov homology of links
  3. Crossing order change
  4. Khovanov homology of cobordisms
  5. Mirroring
  6. The computer program
  7. Classes and methods
  8. Applications
  9. Seifert surfaces up to 10 crossings
  10. Ribbon disks from symmetries

Key Result

Theorem 1

Let $\Sigma\subseteq S^3\times[0,1]$ be a cobordism from $L_0$ to $L_1$. Then the map $\mathrm{Kh}(\Sigma)\colon \mathrm{Kh}(L_0)\to\mathrm{Kh}(L_1)$ is invariant up to multiplication by $\pm1$ under any diffeomorphism of $S^3\times[0,1]$ that restricts to the identity on the boundary.

Figures (1)

  • Figure 1: Band diagrams for ribbon disks obtained by symmetry. Solid lines of different colours represent non-isotopic disks distinguished by Khovanov homology.

Theorems & Definitions (3)

  • Theorem 1: Lipshitz-Sarkar*Proposition 3.7
  • Proposition 2
  • proof