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

2009/06/22 by Keiichi Horie, Teruaki Kitano, Horie, Keiichi +5
Mathematics · #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M25

paper · pdf · doi:10.48550/arxiv.0906.3943

26 pages

arxiv created 2009/06/22 · arxiv updated 2009/12/01

Abstract

Let K be a prime knot in S³ and G(K)=π₁(S³-K) the knot group. We write K₁ ≥ K₂ if there exists a surjective homomorphism from G(K₁) onto G(K₂). 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 \citeKS1.

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