Topology Seminar: Zsuzsanna Dancso (Toronto)

In this talk we present an algebraic context for knot theory. Knotted trivalent graphs (KTGs) along with standard operations defined on them form a finitely presented algebraic structure which includes knots, and in which many topological knot properties are defineable using simple formulas. Thus, a homomorphic invariant of KTGs provides an algebraic way to study knots. We present a construction for such an invariant: the starting point is extending the Kontsevich integral of knots to KTGs. This was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki in the late 90's using the theory of associators. We present an elementary construction building on Kontsevich's original definition, and discuss the homomorphic properties of the invariant, which, as it turns out, intertwines all the standard KTG operations except for one, called the edge unzip. We prove that in fact no universal finite type invariant of KTGs can intertwine all the standard operations at once, and present an alternative construction of the space of KTGs on which a homomorphic universal finite type invariant exists. This space retains all the good properties of the original KTGs: it is finitely presented, includes knots, and is closely related to Drinfel'd associators. Partly joint work with Dror Bar-Natan.