The Jones polynomial of an almost alternating link

TL;DR

This paper provides formulas for the Jones polynomial coefficients of almost alternating links, demonstrating their nontriviality and sign patterns, and discusses minimal crossing diagrams.

Contribution

It introduces explicit formulas for key Jones polynomial coefficients of almost alternating links and analyzes their properties, advancing understanding of their structure.

Findings

Jones polynomial of almost alternating links is nontrivial

First two or last two coefficients alternate in sign

Conditions for minimal crossing diagrams are identified

Abstract

A link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change. We give formulas for the first two and last two potential coefficients of the Jones polynomial of an almost alternating link. Using these formulas, we show that the Jones polynomial of an almost alternating link is nontrivial. We also show that either the first two or last two coefficients of the Jones polynomial of an almost alternating link alternate in sign. Finally, we describe conditions that ensure an almost alternating diagram has the fewest number of crossings among all almost alternating diagrams of the link.