Osaka Journal of Mathematics

A refined Jones polynomial for symmetric unions

Michael Eisermann and Christoph Lamm

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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.

Article information

Source
Osaka J. Math. Volume 48, Number 2 (2011), 333-370.

Dates
First available in Project Euclid: 6 September 2011

Permanent link to this document
http://projecteuclid.org/euclid.ojm/1315318344

Mathematical Reviews number (MathSciNet)
MR2831977

Zentralblatt MATH identifier
06005130

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Citation

Eisermann, Michael; Lamm, Christoph. A refined Jones polynomial for symmetric unions. Osaka J. Math. 48 (2011), no. 2, 333--370. http://projecteuclid.org/euclid.ojm/1315318344.


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