[Paper Review] Knot Floer homology of (1,1)-knots

Hiroshi Goda, Hiroshi Matsuda+1 more|ArXiv.org|Nov 6, 2003

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper presents a combinatorial algorithm for computing knot Floer homology with ℤ-coefficients for (1,1)-knots in S³, leveraging genus 2 Heegaard splittings and explicit boundary operator calculations. The method enables computation of the homology for non-alternating (1,1)-knots with ten crossings and pretzel knots P(−2,m,n), leading to exact determination of their unknotting numbers and 4-genera, with τ(P(−2,m,n)) = −g where g = (m+n)/2.

ABSTRACT

We present a combinatorial method for a calculation of knot Floer homology with Z-coefficient of (1,1)-knots, and then demonstrate it for non-alternating (1,1)-knots with ten crossings and the pretzel knots of type (-2,m,n). Our calculations determine the unknotting numbers and 4-genera of the pretzel knots of this type.

Motivation & Objective

• To develop a systematic combinatorial method for computing knot Floer homology with ℤ-coefficients for (1,1)-knots.
• To extend existing Floer homology computations beyond alternating knots and low-crossing knots to include non-alternating (1,1)-knots.
• To compute the knot Floer homology for pretzel knots of type P(−2,m,n) and derive topological invariants such as 4-genus and unknotting number.
• To provide explicit formulas for the boundary operator in the CFK^∞ complex of (1,1)-knots using a genus 2 Heegaard splitting.

Proposed method

• Construct a genus 2 Heegaard splitting W_α ∪ W_β for the complement of a (1,1)-knot K in S³, with K contained in W_β.
• Define meridian discs D_α^1, D_α^2 for W_α and D_β^1, D_β^2 for W_β, with D_β^1 intersecting K transversely once.
• Label generators of the CFK^∞ complex using decorated points y₁, y₂, y₃, y₄ and x_{p,q} indexed by integer gradings and topological data.
• Explicitly compute the sign and structure of the differential ∂ on the CFK^∞ complex using lemmas that define ∂ on each generator type.
• Use the resulting complex to compute the knot Floer homology ĤFK(S³, K, i) and extract invariants such as τ and 4-genus.
• Verify that the class of ĤF(S³) ≅ ℤ is represented by [ỹ₄; 0, −g], confirming τ(P(−2,m,n)) = −g.

Experimental results

Research questions

• RQ1 Can a combinatorial method be developed to compute knot Floer homology with ℤ-coefficients for all (1,1)-knots?
• RQ2 What is the structure of the CFK^∞ complex for non-alternating (1,1)-knots with ten crossings?
• RQ3 How can the knot Floer homology of pretzel knots P(−2,m,n) be computed explicitly using this method?
• RQ4 What are the 4-genus and unknotting number of pretzel knots P(−2,m,n) as determined by their knot Floer homology?
• RQ5 Does the invariant τ(P(−2,m,n)) equal −g, where g = (m+n)/2?

Key findings

• The knot Floer homology ĤFK(S³, K, i) is computed explicitly for non-alternating (1,1)-knots with ten crossings, including two with identical homology (Example 4.4).
• For pretzel knots P(−2,m,n), the CFK^∞ complex is generated by specific generators indexed by i ∈ ℤ and topological parameters m, n, with explicit boundary operators defined by Lemmas 5.6–5.10.
• The differential ∂ on the CFK^∞ complex is fully determined by sign-annotated relations, allowing exact computation of homology groups.
• The invariant τ(P(−2,m,n)) is computed as −g, where g = (m+n)/2, confirming the conjectured value for this family.
• The 4-genus and unknotting number of P(−2,m,n) are determined as g and g respectively, based on the knot Floer homology structure.
• The class of ĤF(S³) ≅ ℤ is represented by the generator [ỹ₄; 0, −g], confirming the absolute grading and the value of τ.