Date: 14 September 2012, 4:30pm
Location: Hempfield High School, Room 213
 Speaker: Neal Stoltzfus, Louisiana State University
   Title: Knots with Cyclic Symmetries and Knot Polynomial
   Constraints and Recursions

Abstract. (Joint work with J. Keller and M. Montee) For the
Alexander polynomial of a knot with a finite order cyclic symmetry,
Horst Seifert developed a relation constraining the Alexander polynomial
of such knots.

We develop similar constraints using the transfer method of generating
functions is applied to the ribbon graph rank polynomial. This
polynomial, denoted R(D;X,Y,Z), is due to  Bollob\'as, Riordan, Whitney
and Tutte. Given a sequence of ribbon graphs, D_n, constructed by
successive amalgamation of a fixed pattern ribbon graph, we prove by the
transfer method that the associated sequence of rank polynomials is
recursive:  that is, the polynomials R(D_n;X,Y,Z) satisfy a linear
recurrence relation with coefficients in Z[X,Y,Z].

We develop conditions for the Jones polynomial of links which admit a
periodic homeomorphism, by applying the above result and the work of
Dasbach et al showing that the Jones polynomial is a specialization of
the ribbon graph rank polynomial.