A Refined Jones Polynomial for Symmetric Unions
نویسندگان
چکیده
Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. We develop a twovariable refinement WD(s,t) of the Jones polynomial that is invariant under symmetric Reidemeister moves. If D is a symmetric union diagram, representing a ribbon knot K, then the polynomial WD(s,t) nicely reflects their topological properties. 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 f (t) · f (t−1) reminiscent of the Alexander polynomial of ribbon knots.
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