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

Abstract

Let $K$ be a prime knot in $S^3$ and $G(K)={\pi }_1(S^3-K)$ the knot group. We write $K_1\ge K_2$ 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}.

26 pages

References in corpus (4)

• Epimorphisms between 2-bridge link groups
• Non-zero degree maps between $2n$-manifolds
• Twisted Alexander polynomials and a partial order on the set of prime knots
• Simon's conjecture for 2-bridge knots