4-fold symmetric quandle invariants of 3-manifolds
The paper is entirely devoted to introduce 4-fold symmetric quandle invariants of a closed oriented 3-manifold \(M\), and to discuss some basic facts about them. Such invariants are defined starting from any realization of \(M\) as a 4-fold cover of \(S^3\) branched over a link \(L\). In fact, in terms of the monodromy of the covering map, \(M\) can be represented by an \(\mathcal S\)-coloring of a diagram \(D\) of \(L\), where \(\mathcal S\) is the set of the six transpositions in the symmetric group \(\mathfrak S_4\), with the quandle structure given by conjugation. Then, the invariance is based on the \(\mathcal S\)-colored Reidemeister moves and the on covering moves that relate any two \(\mathcal S\)-colored links diagrams representing diffeomorphic manifolds, according to [\textit{N. Apostolakis}, Algebr. Geom. Topol. 3, 117--145 (2003; Zbl 1014.57001)] and [\textit{I. Bobtcheva} and \textit{R. Piergallini}, ``Covering moves and Kirby calculus'', \url{arXiv:math/0407032}] More precisely, a symmetric pair \(X_\rho = (X,\rho)\), where \(X\) is a quandle and \(\rho: X \to X\) is an involution giving a symmetry, which makes \(X_\rho\) suitable for coloring non-oriented link diagrams, just like \(\mathcal S\). Then, a 4-fold symmetric quandle is a triple \((X,p_X,\rho)\), where \(X_\rho = (X,\rho)\) is a symmetric quandle and \(p_X: X \to \mathfrak S_4\) is a quandle epimorphism satisfying certain properties. For a closed oriented 3-manifold \(M\) and a link diagram \(D\) with an \(\mathcal S\)-coloring representing \(M\), the number of the \(X_\rho\)-colorings of \(D\) inducing that \(\mathcal S\)-coloring through \(p_X\) is a topological invariant of \(M\), being preserved by the above mentioned moves. The formal sum all those \(X_\rho\)-colorings of \(D\) gives another topological invariant of \(M\), which lives in the group ring \(\mathbb Z[\Pi_{2,\rho}^{4f}]\) associated to a certain quotient \(\Pi_{2,\rho}^{4f}(X)\) of the cobordism group \(\Pi_{2,\rho}(X)\) consisting of all \(X_\rho\)-colored link diagrams. This is named a 4-fold symmetric quandle homotopy invariant of \(M\). The author shows that 4-fold symmetric quandles can be represented in terms of cored groups, defined as pairs \(G_c = (G,c)\) where \(G\) is a group and \(c \in Z(G)\) is a central element of order 2. Then, he uses this representation to provide some estimation about the group \(\Pi_{2,\rho}^{4f}(X)\) and some specific computation of it. In particular, \(\Pi_{2,\rho}^{4f}(X)\) is shown to be a finite abelian group when \(X_{\rho}\) corresponds to a finite cored group. Moreover, the author introduces 4-fold symmetric quandle 2-cocycle invariants of \(M\) with coefficients in an abelian group \(A\), as potentially more computable reductions of the 4-fold symmetric quandle homotopy invariant of \(M\) associated to a given \(X_\rho\). These are \(\mathbb Z[A]\)-valued state sum invariants, defined terms of \(X_\rho\)-colorings of the link diagram and certain colorings of the complementary regions compatible to them.
scientific article; zbMATH DE number 5904609
scientific article