The Jones polynomial and the planar algebra of alternating links

TL;DR

This paper generalizes Thistlethwaite's result on the alternating nature of the Jones polynomial for non-split alternating links to the broader context of alternating tangles, using skein modules and planar algebra techniques.

Contribution

It introduces a new skein module-based alternating condition for tangles and proves its invariance under planar algebra compositions, extending the understanding of Jones polynomial properties.

Findings

Jones polynomial of tangles is valued in a skein module.

The alternating condition is preserved under planar algebra compositions.

For links, the condition equates to coefficient alternation in the Jones polynomial.

Abstract

It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles, and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial.