A refined Jones polynomial for symmetric unions

Abstract

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams $D$ we develop a two-variable refinement $W_{D}(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $t$ are associated to the two types of crossings, respectively on and off the symmetry axis. From sample calculations we deduce that a ribbon knot can have essentially distinct symmetric union presentations even if the partial knots are the same. If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the polynomial $W_{D}(s,t)$ nicely reflects the geometric properties of $K$. In particular it elucidates the connection between the Jones polynomials of $K$ and its partial knots $K_{\pm}$: we obtain $W_{D}(t,t) = V_{K}(t)$ and $W_{D}(-1,t) = V_{K_{-}}(t) \cdot V_{K_{+}}(t)$, which has the form of a symmetric product $f(t) \cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots.