For two prime knots \(K_1, K_2\) in \(S^3\), we write \(K_1\geq K_2\) if there exists a surjective homomorphism from the knot group of \(K_1\) to that of \(K_2\). This relation is a partial order on the set of prime knots and it is an open problem to determine whether there exists a surjective homomorphism between the knot groups of two given knots.
A useful criterion is given by the Alexander polynomial and more generally by the twisted Alexander polynomial. Using these criteria, two of the authors completely determined, in a previous paper, this partial order for Rolfsen’s knot table which contains 249 prime knots up to 10 crossings.
In this paper, the authors determine this partial order on the set of prime knots with up to 11 crossings which contains 801 prime knots. For this purpose, they either construct explicit surjective homomorphisms or prove the non-existence of such homomorphism by checking the divisibility of twisted Alexander polynomials and using 2-dimensional unimodular representations of knot groups over finite prime fields.