The Strong Slope Conjecture and torus knots

TL;DR

This paper shows that the strong slope conjecture implies the colored Jones polynomial uniquely identifies torus knots, and that adequate knots with the same polynomial degrees as a torus knot are specifically (2,q)-torus knots.

Contribution

It establishes a connection between the strong slope conjecture and the detection of torus knots by the colored Jones polynomial, providing new characterizations.

Findings

The strong slope conjecture implies the colored Jones polynomial detects all torus knots.

Adequate knots with the same polynomial degrees as a torus knot are (2,q)-torus knots.

The degree of the colored Jones polynomial can distinguish certain classes of knots.

Abstract

We observe that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a $(2,q)$-torus knot.