Roots of the Alexander polynomial and Hoste's conjecture

Alexander Stoimenow (Keimyung University)

Monday 23rd January, 2012 15:00-16:00 416


The Alexander polynomial remains one of the most fundamental invariants of knots and links in 3-space. It topological understanding has led a long time ago to the insight what (Laurent) polynomials occur as Alexander polynomial of an arbitrary knot. Ironically, the question to characterize the Alexander polynomials of alternating knots turns out to be far more difficult, even although in general alternating knots are much better understood. Hoste, based on computer verification, made the following conjecture about 10 years ago: If z is a complex root of the Alexander polynomial of an alternating knot, then Re z > -1. We discuss some results toward this conjecture, about 2-bridge (rational) knots or links, 3-braid alternating links, and Montesinos knots.

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