The Jones slopes of a knot

TL;DR

This paper explores the relationship between the Jones polynomial of knots and geometric properties of their complements, introducing invariants called Jones slopes and Jones period, and verifying conjectures through computations on various knot classes.

Contribution

It introduces the Jones slopes and Jones period invariants for knots, formulates related conjectures, and verifies them for specific classes of knots, advancing understanding of knot invariants and geometry.

Findings

Jones slopes form a finite set of rational numbers.

Conjectures hold for alternating and torus knots.

Explicit computations support the proposed relations.

Abstract

The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, torus knots, the knots with at most 9 crossings, and the $(-2,3,n)$ pretzel knots.