##
**A functor-valued extension of knot quandles.**
*(English)*
Zbl 1283.57012

A quandle is a set \(Q\) with a binary operation which satisfies certain axioms and a pointed quandle is a pair \((Q, h)\) consisting of a quandle \(Q\) and one of its elements \(h\). In the paper under review, for each oriented knot \(K\) the author constructs a quandle invariant functor \(I_K\) in three different ways (algebraic, combinatorial, and geometric method), which is a functor from the category of pointed quandles to the category of quandles. As a consequence, for each pointed quandle \((Q, h)\) a quandle-valued invariant of a knot \(I_K(Q, h)\) is obtained. The classical knot quandle appears as the quandle valued invariant which corresponds to the trivial \(1\)-quandle and so this functor-valued invariant of a knot is an extension of the knot quandle. The author also constructs an extension of quandle cocycle invariants by using these quandle-valued invariants of knots, and studies their properties.

Reviewer: Sang Youl Lee (Pusan)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

PDF
BibTeX
XML
Cite

\textit{T. Ito}, J. Math. Soc. Japan 64, No. 4, 1147--1168 (2012; Zbl 1283.57012)

### References:

[1] | N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras, Adv. Math., 178 (2003), 177-243. · Zbl 1032.16028 |

[2] | J. S. Birman, Braids, Links, and Mapping Class Groups, Ann. of Math. Stud., 82 , Princeton University Press, 1974. |

[3] | J. Crisp and L. Paris, Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups, Pacific J. Math., 221 (2005), 1-27. · Zbl 1147.20033 |

[4] | J. Carter, M. Elhamdadi, M. Graña and M. Saito, Cocycle knot invariants from quandle modules and generalized quandle homology, Osaka J. Math., 42 (2005), 499-541. · Zbl 1089.57008 |

[5] | J. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc., 355 (2003), 3947-3989. · Zbl 1028.57003 |

[6] | M. Eisermann, Homological characterization of the unknot, J. Pure. Appl. Algebra, 177 (2003), 131-157. · Zbl 1013.57002 |

[7] | D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure. Appl. Algebra, 23 (1982), 37-65. · Zbl 0474.57003 |

[8] | S. V. Matveev, Distributive groupoids in knot theory (Russian), Mat. Sb. (N.S.), 119 (1982), 78-88. · Zbl 0523.57006 |

[9] | M. Wada, Group invariants of links, Topology, 31 (1992), 399-406. · Zbl 0758.57008 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.