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) = π 1 (S 3 - K) the knot group. We write K 1 ≥ 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.