Khovanov width and dealternation number of positive braid links

TL;DR

This paper establishes tight upper bounds for the Khovanov width and dealternation number of positive braid links based on crossing number, and uses braid theory and the Upsilon invariant to compute exact cobordism distances between certain torus knots.

Contribution

It provides asymptotically sharp bounds for key invariants of positive braid links and applies braid-theoretic techniques with the Upsilon invariant to determine precise cobordism distances.

Findings

Sharp upper bounds for Khovanov width and dealternation number.

Exact cobordism distance between specific torus knots.

Application of braid theory and Upsilon invariant in knot invariants.

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 Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon invariant, allows us to determine the exact cobordism distance between torus knots with braid index two and six.