# Normal and Jones surfaces of knots

## TL;DR

This paper introduces an algorithm to determine if a knot satisfies the Strong Slope Conjecture based on the colored Jones polynomial, and explores the relationship between Jones periods and Jones surfaces with supporting evidence.

## Contribution

It presents a novel normal surface algorithm for the Strong Slope Conjecture and investigates the connection between Jones periods and Jones surfaces, including experimental validation.

## Key findings

- Algorithm effectively decides the Strong Slope Conjecture for knots with known Jones polynomial degree.
- Established a relation between Jones period and the number of sheets of Jones surfaces.
- Provided numerical evidence supporting a conjectured relation between Jones period and Jones surfaces.

## Abstract

We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open questions. We establish a relation between the Jones period of a knot and the number of sheets of the surfaces that satisfy the Strong Slope Conjecture (Jones surfaces). We also present numerical and experimental evidence supporting a stronger such relation which we state as an open question.