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 we develop a two-variable refinement WD(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 WD(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: we obtain WD(t,t) = VK(t) and WD(-1,t) = VK-(t) · VK+(t), which has the form of a symmetric product f(t) · f(t-1) reminiscent of the Alexander polynomial of ribbon knots.