[Paper Review] Homologically thin, non-quasi-alternating links

Joshua Evan Greene|ArXiv.org|Jun 11, 2009

Geometric and Algebraic Topology23 references3 citations

TL;DR

This paper constructs the first known examples of homologically thin links that are not quasi-alternating, using the obstruction that their branched double covers do not bound a negative definite 4-manifold with non-torsion H₁. It completes the classification of quasi-alternating pretzel links and identifies 11n50 as the only 11-crossing knot that is homologically thin but not quasi-alternating.

ABSTRACT

We exhibit the first examples of links which are homologically thin but not quasi-alternating. To show that they are not quasi-alternating, we argue that none of their branched double-covers bounds a negative definite 4-manifold with non-torsion H_1. Using this method, we also complete the determination of the quasi-alternating pretzel links.

Motivation & Objective

• To resolve the open question of whether homologically thin links must be quasi-alternating.
• To provide a method to obstruct quasi-alternatingness using the topology of branched double covers.
• To complete the classification of quasi-alternating pretzel links.
• To identify 11n50 as a homologically thin, non-quasi-alternating link with determinant 25.
• To investigate the finiteness of quasi-alternating links with bounded determinant.

Proposed method

• Uses Donaldson's Theorem A to analyze the intersection form of negative definite 4-manifolds bounding branched double covers.
• Applies a cohomological lemma to detect when a 4-manifold with torsion-free H₁ cannot bound a negative definite manifold.
• Employs Khovanov and knot Floer homology computations to verify homological thinness of candidate links.
• Combines results from Champanerkar-Kofman and Widmer on Montesinos and pretzel links to extend classification.
• Uses Kanenobu's two-parameter knot family K(p,q) to construct infinite families of homologically thin knots with fixed invariants.
• Applies the long exact sequence in knot Floer homology to show invariance of HFK across the family K(n,3−n).

Experimental results

Research questions

• RQ1 Are there homologically thin links that are not quasi-alternating?
• RQ2 Can the branched double cover of a link obstruct it from being quasi-alternating?
• RQ3 Which pretzel links are quasi-alternating?
• RQ4 Are there infinitely many homologically thin knots with the same invariants but only finitely many quasi-alternating ones?
• RQ5 Does the set of quasi-alternating links of fixed determinant remain finite?

Key findings

• The paper constructs the first examples of links that are homologically thin (with respect to HFK, Kh, and odd Kh) but not quasi-alternating.
• The knot 11n50 is identified as the only 11-crossing knot that is homologically thin but not quasi-alternating.
• The branched double cover of 11n50 does not bound a negative definite 4-manifold with non-torsion H₁, which obstructs its quasi-alternatingness.
• The classification of quasi-alternating pretzel links is completed, with a full characterization in terms of parameters e, n, m, p_i, q_j.
• An infinite family of homologically thin knots K(n,3−n) exists with the same determinant (25) and homological invariants, but only K(1,2) is quasi-alternating.
• The paper provides evidence for Conjecture 3.1: there are only finitely many quasi-alternating links with a given determinant, as no such infinite family is known.