[Paper Review] Computations of Heegaard-Floer knot homology

John A. Baldwin, W. D. Gillam|ArXiv.org|Oct 5, 2006

Geometric and Algebraic Topology9 references13 citations

TL;DR

This paper presents a combinatorial algorithm for computing Heegaard-Floer knot homology (HFK) of knots in S³ using grid diagrams and arc presentations, enabling the first systematic computation of HFK for all knots with up to 12 crossings. The method leverages a filtered chain complex over Z₂ with explicit boundary maps defined by rectangle counts, and computes the τ invariant for knots through 11 crossings, revealing that s = 2τ holds for all computed knots despite the invariants differing in general.

ABSTRACT

Using a combinatorial approach described in a recent paper of Manolescu, Ozsváth, and Sarkar we compute the Heegaard-Floer knot homology of all knots with at most 12 crossings as well as the $τ$ invariant for knots through 11 crossings. We review the basic construction of \cite{MOS}, giving two examples that can be worked out by hand, and explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, closely examining the Kinoshita-Teraska knot $KT_{2,1}$ and its Conway mutant.

Motivation & Objective

• To develop an algorithmic method for computing Heegaard-Floer knot homology (HFK) for knots in S³, which previously lacked a systematic computational approach.
• To extend the combinatorial framework of Manolescu, Ozsváth, and Sarkar to compute HFK and the τ invariant for all knots with up to 12 crossings.
• To analyze the spectral sequence of the filtered complex and determine the behavior of differentials (d₁, d₂) in the E₁ and E₂ terms.
• To investigate the relationship between the Rasmussen s-invariant and the τ invariant via explicit computations.
• To study the knot Floer homology of the Kinoshita-Terada and Conway mutant pair (KT2,1 and C2,1), which have trivial Alexander polynomials but different genera.

Proposed method

• The method uses arc presentations of knots to construct a genus-one multi-pointed Heegaard diagram, which gives rise to a combinatorial filtered chain complex over Z₂.
• Generators of the complex are permutations of {1, ..., n}, corresponding to intersection points of α and β curves on a torus, with Maslov and Alexander gradings assigned via geometric winding and square-counting formulas.
• The boundary map d is defined combinatorially: d(σ) includes σ′ if they differ in exactly two positions and exactly one of the four rectangles formed by their differing coordinates contains no white dot or point from the knot graph.
• The filtered chain homotopy type of the complex computes HFK(K), and successive quotients yield the homology groups HFKj(Y, K, i).
• The algorithm computes the E₁ term of the spectral sequence from the filtered complex, then determines E₂ via d₁ and d₂ differentials, with the τ invariant derived from the Maslov grading of the top non-trivial homology class.
• The implementation uses a C++ program and the gridlink tool to generate and process grid diagrams for knots with up to 12 crossings.

Experimental results

Research questions

• RQ1 Can the Heegaard-Floer knot homology of all knots with up to 12 crossings be computed algorithmically using a combinatorial framework?
• RQ2 What is the behavior of the spectral sequence (E₁, E₂, d₁, d₂) for knots with small crossing number?
• RQ3 Does the Rasmussen s-invariant equal 2τ for all knots in the computed set, despite their general non-equality?
• RQ4 How do the knot Floer homologies of mutant pairs like KT2,1 and C2,1 differ, especially given their identical Alexander polynomials?
• RQ5 What is the relationship between arc-index and knot complexity, particularly for non-alternating knots?

Key findings

• The paper computes the full knot Floer homology polynomial for all 12-crossing knots and the τ invariant for all 11-crossing knots using a combinatorial algorithm.
• For 11n42 (KT2,1), the HFK polynomial is (q⁻² + q⁻¹)t⁻² + 4(q⁻¹ + 1)t⁻¹ + 7 + 6q + 4(q + q²)t + (q² + q³)t², and for its Conway mutant 11n34 (C2,1), it is (q⁻³ + q⁻²)t⁻³ + 3(q⁻² + q⁻¹)t⁻² + 3(q⁻¹ + 1)t⁻¹ + 3 + 2q + 3(q + q²)t + 3(q² + q³)t² + (q³ + q⁴)t³.
• The τ invariant was computed for all knots through 11 crossings, and in every case, the Rasmussen s-invariant was found to equal 2τ, supporting a conjecture despite general non-equality.
• The spectral sequence for 10₁₅₄ shows non-trivial d₁ and d₂ differentials, with E₁ and E₂ terms indicating complex behavior not captured by shape alone.
• The arc-index of non-alternating knots tends to be lower than that of alternating knots, with 10₁₂₄ having arc-index 8, while all alternating knots with ≤10 crossings have arc-index c(K) + 2.
• The program implementation is publicly available at http://www.math.columbia.edu/~wgillam/hfk, enabling reproducibility and extension of the results.