Knot quandles and infinite cyclic covering spaces

Abstract

Let K be an n-dimensional knot (n ≥ 1), Q(K) the knot quandle of K, Zq[t±1]/J an Alexander quandle, and C∞(K) the infinite cyclic covering space of Sn+2$\backslash$K. We show that the set consisting of homomorphisms Q(K) → Zq[t±1]/J is isomorphic to Zq[t±1]/J ⊕ HomZ[t±1] (H1(C∞(K)), Zq[t±1]/J) as Z[t±1]-modules. Here, HomZ[t±1](H1(C∞(K)), Zq[t±1]/J) denotes the set consisting of Z[t±1]-homomorphisms H1(C∞(K)) → Zq[t±1]/J.