### A homomorphic universal finite type invariant of knotted trivalent graphs

- Series
- Geometry Topology Seminar
- Time
- Monday, November 29, 2010 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 269
- Speaker
- Zsuzsanna Dancso – University of Toronto

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
places knot theory in an algebraic context. In this talk we construct 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.)