Khovanov width and dealternation number of positive braid links

Abstract

We give asymptotically sharp upper bounds for the Khovanov width and the dealternation number of positive braid links, in terms of their crossing number. The same braid-theoretic technique, combined with Ozsváth, Stipsicz, and Szabó's Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six.

The Turaev genus of torus knots

“…The alternation number of a link was defined by Kawauchi [Kaw10]. An immediate consequence of the definition is that alt(L) ≤ dalt(L) for any link L. Feller, Pohlmann, and Zentner [FPZ15] computed the alternation number of torus knots on four or fewer strands, and Baader, Feller, Lewark, and Zentner [BFLZ16] gave bounds on the alternation and dealternating number of some families of torus links on six or fewer strands. Our arguments in Section 4 resemble those in [FPZ15].…”

Genus One Cobordisms Between Torus Knots

“…The modern Heegaard Floer concordance invariants ν + [HW16] and Υ [OSS17] lead to better bounds on cobordism distance depending on the braid indices [BCG17,FK17]. And for small braid indices, these invariants allow to compute the cobordism distance completely [Fel16,BFLZ16].…”