A geometric characterization of the upper bound for the span of the Jones polynomial

TL;DR

This paper provides a geometric formula for the span of the Jones polynomial in k-almost alternating link diagrams, linking topological properties to polynomial bounds and introducing a new perspective on Turaev genus.

Contribution

It introduces a geometric characterization of the span of the Jones polynomial for k-almost alternating diagrams, generalizing previous bounds and relating Turaev genus to dealternator crossings.

Findings

Derived a formula for |s_AD|+|s_BD| in k-almost alternating diagrams.

Provided a geometric proof of the Jones polynomial span upper bound for dealternator connected diagrams.

Established that Turaev genus equals the number of dealternator crossings in dealternator connected diagrams.

Abstract

Let D be a link diagram with n crossings, s_A and s_B its extreme states and |s_AD| (resp. |s_BD|) the number of simple closed curves that appear when smoothing D according to s_A (resp. s_B). We give a general formula for the sum |s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |s_AD|+|s_BD|=n+2-2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu in 1997. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al in 1992, is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.