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Zeros of graph-counting polynomials and their accumulation sets

Thursday 24th January 2008 - 15:30 to 16:00
INI Seminar Room 1

We present zeros of several graph-counting polynomials, including flow polynomial F(G,q), reliability polynomial R(G,p) and Jones polynomial. For both flow and reliability polynomials, we consider, among others, square, triangular and honeycomb lattice strips with various fixed widths and arbitrarily great lengths with several different boundary conditions. We study the zeros of F(G,q) and R(G,p) in the complex q and p planes, respectively, and determine exactly the asymptotic accumulation sets of these zeros in the infinite-length limit. We calculate Jones polynomials for several families of alternating knots and links by computing the Tutte polynomials T(G,x,y) for the associated graphs G. For each of these families we determine the zeros of the Jones polynomial and the accumulation set in the limit of infinitely many crossings. A discussion will be given to the calculation of Jones polynomials for non-alternating links. We also calculate exactly the partition function Z(G,Q,v) of the Q-state Potts model for strip graphs with Q and temperature variable v restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two-dimensional lattices. In the infinite-length limit, we determine the continuous accumulation loci of the partition function zeros in the v and Q planes. General features of these loci are discussed and conjectures are given for properties applicable to arbitrarily large width. Finally, we present the zero distributions of the Q-state Potts model partition function for large Q, and show when these zeros take on approximately circular patterns in a complex plane.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons