4-fold symmetric quandle invariants of 3-manifolds
type: Item
instance of: scholarly article
title: 4-fold symmetric quandle invariants of 3-manifolds
zbMATH Open document ID: 1230.57015
author: Takefumi Nosaka
published in: Algebraic \& Geometric Topology
publication date: 2011-06-07T00:00:00Z
review text: 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 \(\Pi_{2,\rho}(X)\) …
reviewed by: Riccardo Piergallini
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 5904609
zbMATH Keywords: 3-manifold
zbMATH Keywords: symmetric quandle
zbMATH Keywords: quandle cocycle invariant
zbMATH Keywords: colored link diagram
zbMATH Keywords: branched covering
zbMATH Keywords: covering moves
zbMATH Keywords: rack space
MaRDI profile type: Publication
full work available at URL: agt.2011.11.1601
OpenAlex ID: W2060319620
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DOI: 10.2140/AGT.2011.11.1601
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description: scientific article; zbMATH DE number 5904609