A partial order on the set of prime knots with up to 11 crossings

TL;DR

This paper classifies the partial order of prime knots with up to 11 crossings based on the existence of surjective homomorphisms between their knot groups, using algebraic invariants to determine relations.

Contribution

It explicitly determines the partial order among prime knots up to 11 crossings, extending previous results with new computational methods.

Findings

Identified surjective homomorphisms for specific prime knots.

Used Alexander and twisted Alexander polynomials to prove non-existence.

Mapped the partial order structure for 801 prime knots.

Abstract

Let $K$ be a prime knot in $S^3$ and $G(K)=\pi_1(S^3-K)$ the knot group. We write $K_1 \geq K_2$ if there exists a surjective homomorphism from $G(K_1)$ onto $G(K_2)$. In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then $640,800$ should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial. This work is an extension of the result of \cite{KS1}.