# Invariants of long knots

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Rinat Kashaev

**31/7/19**Published in : arXiv:1908.00118

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, we illustrate the construction and the importance of consideration of long knots by introducing racks associated with pointed groups. 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 in the case of finite dimensional Hopf algebras.