The purpose of this talk is to understand the algebraic structures on the spaces of long knots and string links. Thanks to the Schubert's theorem, we already know that the isotopy classes of longs knots give rise to a commutative monoid freely generated by the isotopy classes of prime knots. Thereafter, R.Budney has been able to extend this result at the level of the topological space of long knots using the little cubes operad.

During this presentation, we will show similar statements for the space of string links with two strands. The isotopy classes of the latter one is a non-commutative monoid, the center of which consists of the pure braid groups and three copies of the isotopy classes of long knots. We will use colored operad in order to describe the space of string 2-links in terms of free algebra. This result is a part of a project (in collaboration with E.Batelier and D.Kosanovic), the main purpose of which is to compare the Vassiliev's invariants of finite type and the Goodwillie-Weiss' functor calculus.

(Collaboration with Etienne Batelier)

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