Hoste’s Conjecture and Roots of Link Polynomials

Abstract

In relation to a conjecture of Hoste on the roots of the Alexander polynomial of alternating knots, we prove that any root z of the Alexander polynomial of a 2-bridge (rational) knot or link satisfies $${| z^{1/2}-z^{-1/2} | < 2}$$ . We extend our result to properties of zeros for some Montesinos knots, and to an analogous statement about the skein polynomial. A similar estimate is derived for alternating 3-braid links.