#### Volume 11, issue 3 (2011)

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$4$–fold symmetric quandle invariants of $3$–manifolds

### Takefumi Nosaka

Algebraic & Geometric Topology 11 (2011) 1601–1648
##### Abstract

The paper introduces $4$–fold symmetric quandles and $4$–fold symmetric quandle homotopy invariants of $3$–manifolds. We classify $4$–fold symmetric quandles and investigate their properties. When the quandle is finite, we explicitly determine a presentation of its inner automorphism group. We calculate the container of the $4$–fold symmetric quandle homotopy invariant. We also discuss symmetric quandle cocycle invariants and coloring polynomials of $4$–fold symmetric quandles.

##### Keywords
quandle, symmetric quandle, quandle cocycle invariant, the rack space, link, $3$–manifold, branched covering
##### Mathematical Subject Classification 2010
Primary: 57M12, 57M25, 57M27, 57N70, 58K65
Secondary: 55Q52, 22A30, 11E57, 55R40, 05E15
##### Publication
Revised: 22 March 2011
Accepted: 24 March 2011
Published: 3 June 2011
##### Authors
 Takefumi Nosaka Research Institute for Mathematical Sciences Kyoto University Sakyo-ku Kyoto 606-8502 Japan