Knot quandles and infinite cyclic covering spaces

抄録

Let K be an n-dimensional knot (n ≥ 1), Q(K) the knot quandle of K, Z_q[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) → Z_q[t^±1]/J is isomorphic to Z_q[t^±1]/J ⊕ Hom_Z[t^±1] (H₁(C_∞(K)), Z_q[t^±1]/J) as Z[t^±1]-modules. Here, Hom_Z[t^±1](H₁(C_∞(K)), Z_q[t^±1]/J) denotes the set consisting of Z[t^±1]-homomorphisms H₁(C_∞(K)) → Z_q[t^±1]/J.