Hoste’s Conjecture and Roots of the Alexander Polynomial

Abstract

The Alexander polynomial remains one of the most fundamental invariants of knots and links in 3-space. Its topological understanding has led a long time ago to the insight of what (Laurent) polynomials occur as Alexander polynomials of arbitrary knots. 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. J. Hoste, based on computer verification, made the following conjecture about 15 years ago: If z is a complex root of the Alexander polynomial of an alternating knot, then \(\mathfrak {R}e\,z > -1\). We discuss some results toward this conjecture, about 2-bridge (rational) knots or links, 3-braid alternating links, and Montesinos knots.