The Slope Conjecture for a Family of Montesinos Knots

TL;DR

This paper verifies the Slope Conjecture and the Strong Slope Conjecture for a specific family of Montesinos knots, establishing a connection between the degree of their colored Jones polynomial and boundary slopes.

Contribution

It provides the first verification of the Slope Conjecture for this class of Montesinos knots with particular parameter restrictions.

Findings

Confirmed the Slope Conjecture for the specified Montesinos knots.

Confirmed the Strong Slope Conjecture for the same class.

Established a link between colored Jones polynomial degree and boundary slopes.

Abstract

The Slope Conjecture relates the degree of the colored Jones polynomial to the boundary slopes of a knot. We verify the Slope Conjecture and the Strong Slope Conjecture for Montesinos knots $M(\frac{1}{r},\frac{1}{s-\frac{1}{u}},\frac{1}{t} )$ with $r,u,t$ odd, $s$ even and $u\leq-1$, $r<-1<1<s,t$.