The slope conjecture for graph knots

TL;DR

This paper verifies the slope conjecture for graph knots, showing that the Jones slopes derived from colored Jones polynomials correspond to boundary slopes, thus confirming a key prediction in knot theory.

Contribution

The paper proves the slope conjecture specifically for graph knots, expanding the class of knots for which the conjecture is validated.

Findings

Confirmed the slope conjecture for all graph knots

Established the correspondence between Jones slopes and boundary slopes for these knots

Enhanced understanding of the relationship between quantum invariants and geometric properties

Abstract

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.