Some non-abelian covers of knots with non-trivial Alexander polynomial

Abstract

Let K be a tame knot embedded in S 3 . We address the problem of finding the minimal degree non-cyclic cover p : X → S 3 K. When K has non-trivial Alexander polynomial we construct finite non-abelian representations ρ : π1 S 3 K → G, and provide bounds for the order of G in terms of the crossing number of K which is an improvement on a result of Broaddus in this case. Using classical covering space theory along with the theory of Alexander stratifications we establish an upper and lower bound for the first betti number of the cover Xρ associated to the ker(ρ) of S 3 K, consequently showing that it can be arbitrarily large. We also demonstrate that Xρ contains non-peripheral homology for certain computable examples, which mirrors a famous result of Cooper, Long, and Reid when K is a knot with non-trivial Alexander polynomial.

Explicit special covers of alternating links

“…Assuming the generalized Riemann hypothesis, Kuperberg showed that for every non-trivial knot K, some non-abelian quotient of S 3 K is bounded in size by a function of the form exp(poly(c(K))), implying that unknot recognition is in the complexity class co-NP [23]. See also Broaddus [9] and Morris [30] for other constructions of non-abelian covers.…”