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 ').

Title	Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected)
Creator	Traldi, Lorenzo
Publisher	Journal of Knot Theory and Its Ramifications
Academic Department	Mathematics
Division	Natural Sciences
Organization	Lafayette College
Date Issued	December 2020
Date Available	2022-05-17
Type	Article
Language	English
Keyword	Alexander module
	branched double cover
	determinant
	link coloring
	involutory medial quandle
Bibliographic Citation	Traldi, Lorenzo. (2020 Dec) "Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected)." Journal of Knot Theory and Its Ramifications 29 (14): 2050093.
Standard Identifier	DOI 10.1142/S0218216520500935
Standard Identifier	Handle 10385/9019s4038
Permalink	http://hdl.handle.net/10385/9019s4038
Rights Statement	In Copyright
Rights Holders	World Scientific Publishing Company