[Paper Review] A note on knots with H(2)-unknotting number one

Yuanyuan Bao|Osaka City University (Osaka City University)|Sep 17, 2010

Geometric and Algebraic Topology9 references3 citations

TL;DR

This paper introduces a new obstruction to determining whether a knot has H(2)-unknotting number one using Heegaard Floer homology. By analyzing the correction term and Goeritz matrix of alternating knots, the authors derive a necessary condition involving a modular equation and inequality on the correction term map, which successfully proves that the pretzel knot P(13,4,11) has H(2)-unknotting number two, surpassing existing methods.

ABSTRACT

We give an obstruction to unknotting a knot by adding a twisted band, derived from Heegaard Floer homology.

Motivation & Objective

• To develop a new obstruction for determining whether a knot has H(2)-unknotting number one using Heegaard Floer homology.
• To extend Lickorish's classical obstruction based on the linking form to a stronger invariant using correction terms and Goeritz matrices.
• To provide a computable criterion that can distinguish knots with H(2)-unknotting number one from those with higher values.
• To demonstrate the effectiveness of the new obstruction by showing it works for the pretzel knot P(13,4,11), where previous methods fail.

Proposed method

• The method uses the Ozsváth-Szabó correction term d(Y,s) for the double branched cover Y=Σ(K) of an alternating knot K.
• It applies the correction term inequality from Heegaard Floer homology, which relates d(Y,s) to a quadratic form derived from the Goeritz matrix Q of a reduced alternating diagram.
• The correction term is compared to a map M_Q(α) defined via characteristic vectors of the Goeritz matrix, which computes a rational invariant modulo 2 and bounds the correction term.
• A necessary condition for u_2(K)=1 is formulated as a modular equation I_{φ,ε}(i) ≡ 0 mod 2 and I_{φ,ε}(i) ≤ 0 for all i in ℤ/|p|ℤ.
• The condition involves a group isomorphism φ: ℤ/|p|ℤ → G (where G is the group presented by Q) and a sign ε ∈ {+1, -1}.
• The method is applied to the pretzel knot P(13,4,11), where explicit computation of M_Q and the correction term map shows the obstruction is violated for all possible φ and ε, ruling out u_2(K)=1.

Experimental results

Research questions

• RQ1 Can Heegaard Floer homology provide a stronger obstruction than the linking form for determining whether a knot has H(2)-unknotting number one?
• RQ2 Does the correction term d(Y,s) of the double branched cover Σ(K) yield a computable invariant that can detect H(2)-unknotting number one in alternating knots?
• RQ3 For the pretzel knot P(13,4,11), is it possible to prove that its H(2)-unknotting number is greater than one using a method that fails for previous invariants?
• RQ4 Can the obstruction derived from the correction term and Goeritz matrix be used to rule out u_2(K)=1 when the linking form condition is satisfied?

Key findings

• The pretzel knot P(13,4,11) has H(2)-unknotting number two, as the obstruction derived from Heegaard Floer homology rules out u_2(K)=1.
• The obstruction fails for all possible isomorphisms φ: ℤ/239ℤ → G and signs ε ∈ {+1, -1}, as I_{φ,ε}(1) = 4 > 0, violating the necessary condition I_{φ,ε}(i) ≤ 0.
• The Goeritz matrix Q for P(13,4,11) is [[17, -4], [-4, 15]], with determinant 239, and G is isomorphic to ℤ/239ℤ.
• The correction term d(Σ(K), s) is bounded above by M_Q(α), and the modular condition d(Σ(K), s) ≡ M_Q(α) mod 2 is essential to the obstruction.
• The obstruction is stronger than Lickorish’s linking form criterion, as the linking form condition is satisfied for P(13,4,11), but the knot still has u_2(K) > 1.
• The method also shows that γ*(K) ≤ u_2(K) ≤ γ(K), but for P(13,4,11), γ(K) = 2 and γ*(K) is unknown, so this inequality alone cannot determine u_2(K), highlighting the need for the new obstruction.