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On the Jones polynomial modulo primes

Valeriano Aiello, Sebastian Baader, Livio Ferretti

TL;DR

An upper bound on the density of Jones polynomials of knots modulo a prime number is derived within a sufficiently large degree range: $4/p^7$ .

Abstract

We derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$, within a sufficiently large degree range: $4/p^7$. As an application, we classify knot Jones polynomials modulo two of span up to eight.

On the Jones polynomial modulo primes

TL;DR

An upper bound on the density of Jones polynomials of knots modulo a prime number is derived within a sufficiently large degree range: .

Abstract

We derive an upper bound on the density of Jones polynomials of knots modulo a prime number , within a sufficiently large degree range: . As an application, we classify knot Jones polynomials modulo two of span up to eight.
Paper Structure (4 sections, 3 theorems, 15 equations, 1 table)

This paper contains 4 sections, 3 theorems, 15 equations, 1 table.

Table of Contents

  1. Introduction
  2. Listing potential Jones polynomials
  3. Jones polynomial modulo primes
  4. Jones polynomial modulo two

Key Result

Theorem 1

For all $a,b \in {\mathbb Z}$ with $b-a \geq 7$, the set of Laurent polynomials with coefficients in ${\mathbb F}_p={\mathbb Z}/p{\mathbb Z}$ within the degree range from $a$ to $b$, that are realised as Jones polynomials of knots, has density at most $4/p^7$.

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark