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The Jones polynomial of periodic knots: an application of the
transfer method to the ribbon graph rank polynomial

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Neal Stoltzfus (LSU)

Abstract:
We develop a formula for the Jones polynomial of links left invariant by
an odd order periodic homeomorphism. Seifert had developed a formula for
the Alexander polynomial in this context.
The transfer method of generating functions is applied to the ribbon graph
rank polynomial $R(\mathbb{D};X,Y,Z)$ (due to Bollob\'as, Riordan,
Whitney and Tutte) of a sequence of ribbon graphs,$\mathbb{D}_{n}$
constructed by successive amalgamation of a fixed pattern ribbon graph. By
the transfer method this sequence of rank polynomials is shown to be
recursive: that is, polynomials $R(\mathbb{D}_{n};X,Y,Z)$ satisfy a
linear recurrence relation with coefficients in $\mathbb{Z}[X,Y,Z]$.
Finally, using the work of Dasbach et al showing that the Jones
polynomial is a specialization of the ribbon graph rank polynomial, we
give a characterization of the Jones polynomial of a link invariant under
a odd order periodic homeomorphism. This is joint work with Jordan Keller and MurphyKate Montee.