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A whittled complex for the Khovanov homology of torus links

Carmen Caprau, Nicolle Gonzalez, Christine Ruey Shan Lee, Radmila Sazdanovic

TL;DR

This work tackles the combinatorial explosion in computing Khovanov homology for torus braids by introducing a Bar-Natan Gaussian-elimination-based whetting procedure that yields the whittled complex $\mathcal{FT}^k_n$. The authors prove that each enhanced Kauffman state in $\mathcal{FT}^k_n$ corresponds to either a TL-normal-form-shaped word or to a TL-move path to its Jones Normal Form, and they establish a concrete upper bound on the number of generators at fixed homological degree in terms of partition counts, TL-normal-form lengths, and Catalan numbers. A central technical achievement is proving the acyclicity of a graph built from distinguished Gaussian elimination isomorphisms, enabling a well-defined linear order to perform eliminations and produce a homotopy-equivalent simplified complex. The results offer a computationally tractable framework that aligns with the Gorsky–Oblomkov–Rasmussen conjecture’s generator structure, with potential to make differentials explicit and to enhance computations of the infinite torus-braid homology and related colored Jones categorifications. Overall, the paper provides a self-contained methodology to bound and reduce generators in Kh for torus braids, bridging TL-algebraic structure and Khovanov homology through a principled deflation to Jones Normal Form.

Abstract

We give an algorithm for reducing the number of generators of the Khovanov chain complex of the torus braid $ft^k_n = (σ_1σ_2\cdots σ_{n-1})^k$ on $n$ strands by applying Bar-Natan Gaussian elimination along a distinguished set of Gaussian elimination isomorphisms. We call the resulting complex $\mathcal{FT}^k_n$ a \emph{whittled complex} for the Khovanov homology of torus braids. Using this algorithm, we provide a bound for the number of generators at a fixed homological degree in our whittled complex.

A whittled complex for the Khovanov homology of torus links

TL;DR

This work tackles the combinatorial explosion in computing Khovanov homology for torus braids by introducing a Bar-Natan Gaussian-elimination-based whetting procedure that yields the whittled complex . The authors prove that each enhanced Kauffman state in corresponds to either a TL-normal-form-shaped word or to a TL-move path to its Jones Normal Form, and they establish a concrete upper bound on the number of generators at fixed homological degree in terms of partition counts, TL-normal-form lengths, and Catalan numbers. A central technical achievement is proving the acyclicity of a graph built from distinguished Gaussian elimination isomorphisms, enabling a well-defined linear order to perform eliminations and produce a homotopy-equivalent simplified complex. The results offer a computationally tractable framework that aligns with the Gorsky–Oblomkov–Rasmussen conjecture’s generator structure, with potential to make differentials explicit and to enhance computations of the infinite torus-braid homology and related colored Jones categorifications. Overall, the paper provides a self-contained methodology to bound and reduce generators in Kh for torus braids, bridging TL-algebraic structure and Khovanov homology through a principled deflation to Jones Normal Form.

Abstract

We give an algorithm for reducing the number of generators of the Khovanov chain complex of the torus braid on strands by applying Bar-Natan Gaussian elimination along a distinguished set of Gaussian elimination isomorphisms. We call the resulting complex a \emph{whittled complex} for the Khovanov homology of torus braids. Using this algorithm, we provide a bound for the number of generators at a fixed homological degree in our whittled complex.
Paper Structure (28 sections, 19 theorems, 60 equations, 19 figures)

This paper contains 28 sections, 19 theorems, 60 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Background
  5. Kauffman states and enhanced Kauffman states
  6. Khovanov homology and the Khovanov chain complex
  7. The Khovanov complex for tangles
  8. Abstract Gaussian elimination for chain complexes
  9. The Temperley-Lieb monoid and the Jones Normal Form
  10. Base case $n=2$:
  11. Induction step:
  12. Gaussian elimination
  13. Gaussian elimination isomorphisms
  14. Algorithm: choosing a distinguished Gaussian elimination isomorphism
  15. Whittling the complex via Gaussian elimination isomorphisms
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mathcal{FT}_n^k$ denote the whittled complex obtained from whittling the Khovanov chain complex $\mathrm{CKh}(ft_n^k)$, and let $(\sigma, \varepsilon)$ be an enhanced Kauffman state (Definition d.enhanced-K-state) on $ft_n^k$. Suppose $(\sigma, \varepsilon)$ appears in the whittled complex $\m where each $V_j = e_{j_1}e_{j_2}\cdots e_{n-1}$ is a subword consisting of consecutive elements in

Figures (19)

  • Figure 1: The $0$-and $1$-resolution of a Kauffman state.
  • Figure 2: Bar-Natan relations.
  • Figure 3: Delooping diagram from bar2007fast.
  • Figure 4: Merging and splitting a circle.
  • Figure 5: The braid generator $\sigma_i.$
  • ...and 14 more figures

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Kauffman state
  • Definition 2.2
  • Definition 2.3: Homological and Quantum grading Turner-5lectures
  • Lemma 2.4: Gaussian elimination bar2007fast
  • Lemma 2.5: Delooping
  • Lemma 2.6
  • Definition 2.7: $\mathrm{TL}$-moves
  • Remark 2.8
  • ...and 48 more