A Geometric Characterization of the Upper Bound for the Span of the Jones Polynomial

Abstract

Let D be a link diagram with n crossings, sA and sB its extreme states and |sAD| (resp. |sBD|) the number of simple closed curves that appear when smoothing D according to sA (resp. sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing 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 |sAD| + |sBD| = 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 [14]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al [3], 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.