$C_n$-moves and the difference of Jones polynomials for links

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 ${\left(t-1\right)}^n\left(t^2+t+1\right)\left(t^2+1\right)$.

13 pages, 11 figures

References in corpus (1)

• Claspers and finite type invariants of links