A Jones slopes characterization of adequate knots

TL;DR

This paper characterizes adequate knots using the degree of their colored Jones polynomial and relates it to Jones slopes and essential surfaces, assuming the Strong Slope conjecture, extending results for alternating knots.

Contribution

It provides a new characterization of adequate knots via Jones polynomial degrees and Jones slopes, linking quantum invariants with geometric topology.

Findings

Characterization of adequate knots through Jones polynomial degrees

Reformulation of the characterization assuming the Strong Slope conjecture

Extension of results to alternating knots using recent work

Abstract

We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of "Jones slopes" of knots and the essential surfaces that realize the slopes .For alternating knots the reformulated characterization follows by recent work of J. Greene and J. Howie.