Mathematics Colloquia and Seminars

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We will discuss the idea of creating knot invariants by converting a knot (or link or braid) into a linear map between vector spaces. This is done by defining linear maps corresponding to crossings, "births", and "deaths", and then decomposing a knot into a composition of these pieces. We will then show how this naturally leads to the notion of a Hopf algebra (which is an algebra with extra structures that let it act on tensor products of modules, duals of modules, and a trivial module) where the above mentioned vector spaces correspond to representations of the Hopf algebra.

Prerequisite knowledge: 1) the definition of a ring and a module/representation of a ring. 2) Some vague impression of what a knot is.