Monomial reduction of knot polynomials

TL;DR

This paper constructs knots with skein polynomials that reduce trivially modulo primes and explores the relationship between polynomial evaluations and knot properties, including classification of certain polynomial forms.

Contribution

It introduces a method to find knots with skein polynomials that have trivial modular reductions and classifies specific polynomial forms modulo 2.

Findings

Existence of knots with skein polynomial evaluations that are trivial modulo p

Realization of all polynomials with bounded a-span by knots with bounded braid index

Complete classification of polynomials P_K(a,1) mod 2 with a-span ≤ 10

Abstract

For all natural numbers $N$ and prime numbers $p$, we find a knot $K$ whose skein polynomial $P_K(a,z)$ evaluated at $z=N$ has trivial reduction modulo $p$. An interesting consequence of our construction is that all polynomials $P_K(a,N)$ (mod~$p$) with bounded $a$-span are realised by knots with bounded braid index. As an application, we classify all polynomials of the form $P_K(a,1)$ (mod $2$) with $a$-span $\leq 10$.