State Cycles, Quasipositive Modification, and Constructing H-thick Knots in Khovanov Homology

Abstract

We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee's map $\Phi$. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that specific families of such knots cannot be detected by Khovanov's thickness criteria. We also exhibit a sequence of prime links related by quasipositive modification whose width is increasing.

42 pages, color figures. Version 2 revisions: an error was corrected in Proposition 4.3, which requires a stronger hypothesis. This slightly widens the classification theorem of section 4, and has led to small revisions throughout. Theorem 4.7, which involved even all-1 state cycles, has been removed, as it has grown into a forthcoming paper

References in corpus (3)

• On the Khovanov and knot Floer homologies of quasi-alternating links
• Homology of torus links
• Khovanov homology and Rasmussen's s-invariants for pretzel knots

Cited by in corpus (1)

• Graphical methods establishing nontriviality of state cycle Khovanov homology classes