Multivariate Alexander Quandles, Ii. The Involutory Medial Quandle Of A Link (Corrected)

Joyce showed that for a classical knot K, the involutory medial quandle IMQ(K) is isomorphic to the core quandle of the homology group H-1(X-2), where X-2 is the cyclic double cover of S-3, branched over K. It follows that |IMQ(K)| = |det K|. In this paper, the extension of Joyce's result to classical links is discussed. Among other things, we show that for a classical link L of mu >= 2 components, the order of the involutory medial quandle is bounded as follows:mu|det L|/2 >=|IMQ(L)|>= mu|det L|/2(mu-1) .In particular, IMQ(L) is infinite if and only if det L = 0. We also show that in general, IMQ(L) is a strictly stronger invariant than H-1(X-2). That is, if L and L ' are links with IMQ(L) congruent to IMQ(L '), then H-1(X-2) congruent to H-1(X-2 '); but it is possible to have H-1(X-2) congruent to H-1(X-2 ') and IMQ(L)not congruent to IMQ(L '). In fact, it is possible to have X-2 congruent to X-2 ' and IMQ(L)not congruent to IMQ(L ').