# The Jones polynomial and boundary slopes of alternating knots

## TL;DR

This paper establishes a direct relationship between boundary slopes of essential spanning surfaces and the Jones polynomial for alternating knots, providing explicit formulas involving the knot signature and polynomial degrees.

## Contribution

It introduces a formula linking boundary slopes of alternating knots to the Jones polynomial and signature, extending to Montesinos knots.

## Key findings

- Boundary slopes are given by signature plus twice the Jones polynomial degrees.
- For alternating Montesinos knots, these boundary slopes are minimal and maximal.
- The results unify boundary slope calculations with polynomial invariants.

## Abstract

We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is given by the signature plus twice the maximum degree of the Jones polynomial. For alternating Montesinos knots, these are the minimal and maximal boundary slopes.