Cyclotomic expansions for double twist knots with an odd number of half-twists

TL;DR

This paper derives a cyclotomic expansion formula for the colored Jones polynomial of double twist knots with an odd number of half-twists, using skein theory, addressing a question from prior research.

Contribution

It provides the first explicit cyclotomic expansion formula for this class of knots, expanding understanding of their quantum invariants.

Findings

Derived explicit cyclotomic expansion formula

Applied Kauffman bracket skein theory for computation

Addresses a previously open question

Abstract

In this note, we compute the cyclotomic expansion formula for colored Jones polynomial of double twist knots with an odd number of half-twists $\mathcal{K}_{p,\frac{s}{2}}$ by using the Kauffman bracket skein theory. It answers a question proposed by Lovejoy and Osburn in 2019.