Oriented Local Moves and Divisibility of the Jones Polynomial

TL;DR

This paper develops a framework to analyze how local oriented moves on virtual links affect their Jones polynomials, deriving divisibility conditions and applying them to classical knot equivalence.

Contribution

It introduces a general decomposition of the Jones polynomial for virtual links and establishes divisibility conditions under local moves, including the $ riangle$-move.

Findings

Divisibility conditions for differences in Jones polynomials under local moves

A necessary condition for classical knots to be $S$-equivalent

Application to broad classes of local moves including $ riangle$-move

Abstract

For any virtual link $L = S \cup T$ that may be decomposed into a pair of oriented $n$-tangles $S$ and $T$, an oriented local move of type $T \mapsto T'$ is a replacement of $T$ with the $n$-tangle $T'$ in a way that preserves the orientation of $L$. After developing a general decomposition for the Jones polynomial of the virtual link $L = S \cup T$ in terms of various (modified) closures of $T$, we analyze the Jones polynomials of virtual links $L_1,L_2$ that differ via a local move of type $T \mapsto T'$. Succinct divisibility conditions on $V(L_1)-V(L_2)$ are derived for broad classes of local moves that include the $\Delta$-move and the double-$\Delta$-move as special cases. As a consequence of our divisibility result for the double-$\Delta$-move, we introduce a necessary condition for any pair of classical knots to be $S$-equivalent.