Slopes and colored Jones polynomials of adequate knots

TL;DR

This paper verifies Garoufalidis's conjecture linking boundary slopes and colored Jones polynomials for adequate knots, expanding the class of knots for which this relationship is confirmed.

Contribution

It proves the conjecture for adequate knots, a broad class that includes many important knot types, thus advancing understanding of the boundary slope and polynomial degree relationship.

Findings

Confirmed the conjecture for adequate knots

Extended the class of knots where the conjecture holds

Provided evidence supporting the conjecture's validity

Abstract

Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.