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

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

2009-06-22

arxiv.org/abs/0906.3943v1

Mathematics

Geometric Topology

26 pages

Scientific paper

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}.

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