The slope conjecture for Montesinos knots

Abstract

The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions.

Cancellations in the Degree of the Colored Jones Polynomial

“…See (3) for the definition of the quantum factorial. As a direct corollary, we recover the degree formula 1 from [7].…”

“…From (7), we see that for k 0 = k 1 + k 2 = t < n, there are parameters k = ( k0 , k1 , k2 ) with k0 = k1 + k2 = n such that the following inequality holds…”

“…for a positive integer j ≥ 0. When n = −1 + −2+w 1 +w 2 gcd(w 1 −1,w 2 −1) j , there is no cancellation and the degree of J P ,n corresponds to the maximum of δ(n, k) found in (7).…”

Jones diameter and crossing number of knots

“…The Slopes Conjecture [6], and its refinement [15], assert that the Jones slopes of a knot K are boundary slopes and that the cluster points of the linear terms {s 1 (n)} , {s * 1 (n)} contain information about the topology of incompressible surfaces that realize them. The conjectures have been proved for broad classes of knots including adequate and Montesinos knots [6,9,8], for some of their satellites including cables and Whitehead doubles [15,2], and they were shown to imply that the colored Jones polynomials detect torus knots and the figure-8 knot [13,14]. In this setting, Theorem 2.5 answers a question of Garoufalidis [6, Question 1].…”