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Growth of Salem numbers and Mahler measures of fibered knots and links

Eriko Hironaka �iFlorida State Univ.�j

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The Mahler measure of a fibered knot or link is the product of expanding eigenvalues of the monodromy matrix. A fibered knot or link is Salem if the monodromy matrix has exactly one root with norm larger than 1. McMullen's work implies that for arborescent links associated to star graphs, the Alexander polynomial has at most one root outside the unit circle. The size of the corresponding Salem number is ordering preserving with respect to graph inclusion, and the smallest Salem number comes from the (2,3,7)-star graph. Increasing the branch of a star graph corresponds to doing a Hopf plumbing on the corresponding link, or doing a half twist along a well-chosen pair of strands. Silver and Williams showed that the Mahler measure of links converges under iterated twisting. In this talk we discuss the effect of iterated Hopf plumbing on the Mahler measure of fibered links, and discuss criteria for Salem numbers to occur.

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