A REFINED JONES POLYNOMIAL FOR SYMMETRIC UNIONS

作者：Eisermann Michael^*; Lamm Christoph

来源：Osaka Journal of Mathematics, 2011, 48(2): 333-370.

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 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 W(D)(t, t) = V(K)(t) and W(D)(-1, t) = V(K-)(t).V(K+)(t), which has the form of a symmetric product f(t).f(t(-1)) reminiscent of the Alexander polynomial of ribbon knots.

• 出版日期2011-6