A slope conjecture for links

TL;DR

This paper extends the slope conjecture, which relates the colored Jones polynomial to boundary slopes, from knots to links, and confirms its validity for alternating, adequate, and torus links.

Contribution

It generalizes the slope conjecture to links and proves it for a broad class of links including alternating, adequate, and torus links.

Findings

Proved the generalized slope conjecture for all alternating links.

Verified the conjecture for adequate links.

Confirmed the conjecture for torus links.

Abstract

The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating and more generally adequate links. We also verify the conjecture for torus links.