Oriented Local Moves and Divisibility of the Jones Polynomial

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 $T\mapsto T^{\prime }$ is a replacement of $T$ with the $n$-tangle $T^{\prime }$ 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 $T\mapsto T^{\prime }$. 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.