$C_{n}$-moves and the difference of Jones polynomials for links

TL;DR

This paper investigates how the Jones polynomial differences behave under $C_{n}$-moves, establishing divisibility properties and providing explicit examples of knots with specific polynomial differences.

Contribution

It proves divisibility conditions for Jones polynomial differences under $C_{n}$-moves and constructs explicit knot pairs demonstrating these properties.

Findings

Difference of Jones polynomials divisible by specific factors under $C_{n}$-equivalence

Existence of knot pairs with polynomial difference exactly equal to the divisibility factor

Provides new insights into the relationship between $C_{n}$-moves and Jones polynomial variations

Abstract

The Jones polynomial $V_{L}(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_{n}$-equivalent is divisible by $\left(t-1\right)^{n}\left(t^{2}+t+1\right)\left(t^{2}+1\right)$, and (2) there exists a pair of two oriented knots which are $C_{n}$-equivalent such that the difference of the Jones polynomials for them equals $\left(t-1\right)^{n}\left(t^{2}+t+1\right)\left(t^{2}+1\right)$.