On the Jones polynomial modulo primes

TL;DR

This paper establishes an upper bound on the density of Jones polynomials of knots modulo a prime p and classifies certain Jones polynomials modulo two, advancing understanding of their distribution.

Contribution

It provides a new upper bound on the density of Jones polynomials modulo primes and classifies Jones polynomials modulo two for spans up to eight.

Findings

Upper bound on density: 4/p^7 within large degree range

Classification of Jones polynomials modulo two for span up to eight

Enhanced understanding of Jones polynomial distribution modulo primes

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.