[Paper Review] Transverse Knots Distinguished by Knot Floer Homology

Lenhard Ng, Peter Ozsváth+1 more|arXiv (Cornell University)|Mar 15, 2007

Geometric and Algebraic Topology22 references4 citations

TL;DR

This paper introduces a new invariant, $φ(K)$, derived from knot Floer homology, to distinguish transverse knots with identical classical invariants—specifically, the same self-linking number. By analyzing the homology class of a canonical cycle in the grid diagram complex under transverse grid moves, the authors prove that certain pairs of transverse knots are not transversely isotopic, establishing new examples of transversely nonsimple knot types.

ABSTRACT

We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it.

Motivation & Objective

• To develop a nonclassical invariant for transverse knots using knot Floer homology to detect transverse isotopy classes beyond classical invariants.
• To resolve the question of whether transverse knots with identical self-linking numbers can be distinguished by stronger invariants.
• To provide the first examples of transversely nonsimple knot types using a Floer-theoretic invariant, rather than braid or convex surface theory.
• To explore the behavior of the invariant $φ(K)$ under transverse grid moves and its stability under stabilization.
• To investigate the potential for $φ(K)$ to detect multiple distinct transverse isotopy classes within a single topological knot type and self-linking number

Proposed method

• Construct a canonical cycle $Ø^+(G)$ in the knot Floer chain complex $χΧ(G)$ associated with a grid diagram $G$ of a transverse knot.
• Define the transverse invariant $φ(K)$ as the homology class of $Ø^+(G)$ in $χΤΤ(K)$, which is invariant under transverse grid moves.
• Use the fact that transverse grid moves induce isomorphisms on homology, ensuring $φ(K)$ is well-defined up to automorphism of the knot Floer homology group.
• Apply combinatorial knot Floer homology techniques to compute $φ(K)$ explicitly for specific grid diagrams, using quasi-isomorphisms under grid moves.
• Compare the image of $Ø^+(G)$ under sequences of transverse grid moves to detect non-homologous cycles, implying non-isotopy.
• Leverage the structure of knot Floer homology for two-bridge knots and the action of the mapping class group to predict the number of distinct $φ(K)$ images

Experimental results

Research questions

• RQ1 Can knot Floer homology detect transverse isotopy classes beyond the self-linking number for knots with the same classical invariants?

Key findings

• The invariant $φ(K)$, derived from the homology class of the canonical cycle $Ø^+(G)$ in the knot Floer complex, distinguishes transverse knots with identical self-linking numbers.
• For the transverse twist knots $E(1,5)^+$ and $E(2,4)^+$, the invariant $φ(K)$ detects non-isotopy, as the image of $Ø^+(G)$ under a sequence of transverse grid moves is not homologous to the canonical cycle in the target diagram.
• The authors provide evidence that for even $n$, the $⌈n/4⌉$ transverse twist knots $E(1,n-1)^+, E(3,n-3)^+, \dots, E(2⌈n/2⌉-1, 2⌌ n/2⌍+1)^+$ with self-linking number 1 are pairwise transversely nonisotopic.
• The conjecture is supported by the fact that the corresponding Legendrian knots are related by a satellite construction and that the knot Floer homology group has rank $n/2$, with a $ℤ/2ℤ$ action reducing the number of possible distinct images to $⌈n/4⌉$.
• The invariant $φ(K)$ is stable under positive stabilization and can detect non-isotopy even when classical invariants and some other invariants (e.g., $χΤΤ^{-}$) fail to distinguish the knots.
• The method provides a purely combinatorial framework to compute and compare $φ(K)$ using grid diagrams and their moves, avoiding reliance on pseudo-holomorphic curve counts