On knot Floer width and Turaev genus

TL;DR

This paper establishes a bound relating the width of knot Floer homology to the Turaev genus of knots, providing new insights into the geometric and algebraic properties of knots.

Contribution

It introduces a bound connecting knot Floer homology width with Turaev genus and presents skein relations for these invariants.

Findings

Knot Floer homology width is bounded by Turaev genus plus one.

Skein relations are established for Turaev surface genus and Floer homology width.

The work links algebraic invariants with geometric properties of knots.

Abstract

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizontal distance between two such lines. Also, for each diagram $D$ of $K$ there is an associated Turaev surface, and the Turaev genus is the minimum genus of all Turaev surfaces for $K$. We show that the width of knot Floer homology is bounded by Turaev genus plus one. Skein relations for genus of the Turaev surface and width of a complex that generates knot Floer homology are given.