On knot Floer width and Turaev genus

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.Comment: 15 pages, 15 figure

Twisting quasi-alternating links

“…Finally, the Turaev genus plus two bounds the homological thickness for both Khovanov homology and knot Floer homology [16,5,13]. The Turaev genus is preserved after inserting a rational tangle as in Theorem 2.1.…”

Turaev genus, knot signature, and the knot homology concordance invariants

“…In [Low08], the second author interpreted the δ-grading in terms of information about the Tait graph of the knot diagram. The δ-grading corresponding to a spanning tree T is …”

“…Champanerkar, Kofman, and Stoltzfus [CKS07] use Theorem 4.1 to show that for any knot diagram, the difference between the maximum and minimum δ-gradings in the spanning tree complex for reduced Khovanov homology is equal to the genus of the Turaev surface of that diagram. Using a different approach, the second author [Low08] proved that for any knot diagram, the difference between the maximum and minimum δ-gradings in the spanning tree complex for knot Floer homology is equal to the genus of the Turaev surface of that diagram. In light of Proposition 3.3, the second author's result can be shown using the same approach as Champanerkar, Kofman, and Stoltzfus [CKS07], and both results can be encoded by the equation…”

The Turaev genus of an adequate knot

“…We do not know whether a mutant of an adequate link is adequate. We recall the Turaev surface associated to a connected link diagram (see [8,9] or [16]). First, we construct a cobordism Fig.…”

“…Note that Lowrance found an analogue for the homological width of the Floer homology of a knot [16].…”