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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.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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