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Computations of HOMFLY homology

Keita Nakagane, Taketo Sano

TL;DR

The paper develops and implements an algorithm to compute the reduced HOMFLY homology $\mathcal{H}(L)$ for knots, using an edge-ring presentation, $q$-grading slicing, and variable-exclusion techniques to render the computation finite and feasible. It reports results for all prime knots up to 11 crossings, distinguishing KR-thick from KR-thin knots (96 KR-thick up to 11 crossings, including a unique alternating example), and provides the kr-calc software and data. The work demonstrates that HOMFLY homology is a strictly stronger invariant than the HOMFLY polynomial by exhibiting knot pairs with identical polynomials but different homologies, and discusses spectral-sequence-based constraints to connect to $sl(N)$-homology for efficiency. Overall, it advances computational knot homology by delivering a practical method and a substantial data set that supports deeper structural insights and future refinements.

Abstract

Khovanov--Rozansky's HOMFLY homology is determined for all prime knots with up to 11 crossings by direct computations.

Computations of HOMFLY homology

TL;DR

The paper develops and implements an algorithm to compute the reduced HOMFLY homology for knots, using an edge-ring presentation, -grading slicing, and variable-exclusion techniques to render the computation finite and feasible. It reports results for all prime knots up to 11 crossings, distinguishing KR-thick from KR-thin knots (96 KR-thick up to 11 crossings, including a unique alternating example), and provides the kr-calc software and data. The work demonstrates that HOMFLY homology is a strictly stronger invariant than the HOMFLY polynomial by exhibiting knot pairs with identical polynomials but different homologies, and discusses spectral-sequence-based constraints to connect to -homology for efficiency. Overall, it advances computational knot homology by delivering a practical method and a substantial data set that supports deeper structural insights and future refinements.

Abstract

Khovanov--Rozansky's HOMFLY homology is determined for all prime knots with up to 11 crossings by direct computations.

Paper Structure

This paper contains 9 sections, 14 theorems, 30 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithm
  4. Free variables of the edge ring
  5. Slicing $C(D)$ by the $q$-grading
  6. Exclusion of variables
  7. Computations
  8. Observations
  9. Computation Results

Key Result

Theorem 1

Up to $11$ crossings, the following $96$ are the only KR-thick prime knots, all of which are supported in two $\Delta$-gradings. (The actual results are listed in sec:results.)

Figures (3)

  • Figure 1: The definition of $C(D_c)$
  • Figure 2: Generators $x_c$ and $y_c$ of $R(D)$.
  • Figure 3: Partial computation result for $11a_{280}$.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2: KR08Ras15
  • Definition 2.1
  • Remark 2.2
  • Theorem 3: KR08Ras15
  • Proposition 2.3: Kho07Wu08
  • Proposition 2.4: DGROR20gorsky2021:tautological
  • Proposition 2.5: EMAK
  • Proposition 2.6
  • proof
  • ...and 10 more