Invariants of long knots

By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed groups, we illustrate the construction and the importance of consideration of long knots. Else, by using the restricted dual of algebras and Drinfeld's quantum double construction, we show that to any Hopf algebra $H$ with invertible antipode, one can associate a universal long knot invariant $Z_H(K)$ taking its values in the convolution algebra $((D(H))^o)^*$ of the restricted dual Hopf algebra $(D(H))^o$ of the quantum double $D(H)$ of $H$. That extends the known constructions of universal invariants previously considered mostly either in the case of finite dimensional Hopf algebras or by using some topological completions.

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