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Computing colored Khovanov homology

Karim Ritter von Merkl

TL;DR

The paper addresses the problem of unifying multiple finite-dimensional categorifications of colored Jones polynomials by proving that eight proposed constructions of colored Khovanov homology are isomorphic over fields of characteristic zero. It leverages a cable-based framework to show that the two-variable Poincaré polynomials of colored homology satisfy the same linear relations as colored Jones polynomials, enabling computation of $\mathrm{Kh}_{\mathrm{Sym}^n}(L)$ from cable homologies and supporting an inverse Chebyshev-Polynomials-based translation. A practical outcome is a data-rich online database of colored superpolynomials and a conjectural closed formula for the Poincaré series of the skein lasagna module of $\overline{\mathbb{CP}^2}$, highlighting deep connections to 4-manifold invariants. The work also clarifies how various variants (Inv, Coinv, $H^*_+$, $H^*_-$, Ker, Coker, Im, Coim) relate, enabling robust cross-verification of colored-link invariants and informing skein-lasagna-type constructions in topology and physics.

Abstract

We compare eight versions of finite-dimensional categorifications of the colored Jones polynomial and show that they yield isomorphic results over a field of characteristic zero. As an application, we verify a physics-motivated conjectural formula for colored superpolynomials based on Poincaré polynomials of the Khovanov homology of cables. We also obtain a conjectural closed formula for the Poincaré series of the skein lasagna module of $\overline{\mathbb{CP}^2}$. Accompanying this note is an online database of colored superpolynomials.

Computing colored Khovanov homology

TL;DR

The paper addresses the problem of unifying multiple finite-dimensional categorifications of colored Jones polynomials by proving that eight proposed constructions of colored Khovanov homology are isomorphic over fields of characteristic zero. It leverages a cable-based framework to show that the two-variable Poincaré polynomials of colored homology satisfy the same linear relations as colored Jones polynomials, enabling computation of from cable homologies and supporting an inverse Chebyshev-Polynomials-based translation. A practical outcome is a data-rich online database of colored superpolynomials and a conjectural closed formula for the Poincaré series of the skein lasagna module of , highlighting deep connections to 4-manifold invariants. The work also clarifies how various variants (Inv, Coinv, , , Ker, Coker, Im, Coim) relate, enabling robust cross-verification of colored-link invariants and informing skein-lasagna-type constructions in topology and physics.

Abstract

We compare eight versions of finite-dimensional categorifications of the colored Jones polynomial and show that they yield isomorphic results over a field of characteristic zero. As an application, we verify a physics-motivated conjectural formula for colored superpolynomials based on Poincaré polynomials of the Khovanov homology of cables. We also obtain a conjectural closed formula for the Poincaré series of the skein lasagna module of . Accompanying this note is an online database of colored superpolynomials.
Paper Structure (8 sections, 4 theorems, 18 equations, 2 figures)

This paper contains 8 sections, 4 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Colored superpolynomials from superpolynomials of cables
  3. Versions of colored Khovanov homology
  4. Preliminaries
  5. The Temperley--Lieb category
  6. Variations of colored Khovanov homology
  7. Conventions
  8. Defining variations of colored Khovanov homology

Key Result

Theorem 1.1

Let $K$ be a framed oriented knot in $B^3$ and $n\in \mathbb{N}_0$. Then the Poincaré polynomial of the finite-dimensional colored Khovanov homology $\mathop{\mathrm{Kh}}\nolimits_{\mathop{\mathrm{Sym}}\nolimits^{n}}(K)$ can be computed from the Poincaré polynomials of (ordinary) Khovanov homologies

Figures (2)

  • Figure 1: The complex $\overline{C}_4$ over the additive closure of $\Gamma_4$. Red arrows indicate morphisms that carry the sign $-1$. On the right, we see the poset structure of $\Gamma_4$ embedded in the three dimensional cube highlighting the complement in gray. Quotienting by the gray part and totalizing (in either order) yields an isomorphic complex.
  • Figure 2:

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • Remark 3
  • Definition 1
  • Remark 4
  • Example 1
  • Lemma 1
  • ...and 2 more