Homological characterization of the unknot

Given a knot K in the 3-sphere, let QK be its fundamental quandle as introduced by Joyce. Its first homology group is easily seen to be . We prove that H2(QK)=0 if and only if K is trivial, and whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components.More detailed information can be derived from the conjugation quandle: let QKπ be the conjugacy class of a meridian in the knot group . We show that , where p is the number of prime summands in a connected sum decomposition of K.