2019/02/27 by Traldi, Lorenzo
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1902.10603
Joyce showed that for a classical knot K, the involutory medial quandle IMQ(K) is isomorphic to the core quandle of the homology group H1(X2), where X2 is the cyclic double cover of \mathbb S 3, branched over K. It follows that |IMQ(K)| = | det K |. In the present paper, the extension of Joyce's result to classical links is discussed. Among other things, we show that for a classical link L of μ≥ 2 components, the order of the involutory medial quandle is bounded as follows: (μ| det L |)/(2) ≥ |IMQ(L)| ≥ \frac μ| det L | 2μ-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 H1(X2). That is, if L and L' are links with IMQ(L) ≅ IMQ(L'), then H1(X2) ≅ H1(X'2); but it is possible to have H1(X2) ≅ H1(X'2) and IMQ(L) \not ≅ IMQ(L'). In fact, it is possible to have X2 ≅ X'2 and IMQ(L) \not ≅ IMQ(L').