Graduate Blog

DEPARTMENTAL PHD THESIS EXAM

Note: We are experimenting with a somewhat different examination protocol, 
in which the student will be given the option of giving a longer 
presentation. This necessitates changing the format of the departmental 
exam and the location of the SGS exam, and may lead to certain technical 
and administrative difficulties. Only 1-2 exams will be run in this 
format this summer and we may or may not allow this option in the future. 


PhD Candidate:  Zsuzsanna Dancso

PhD Advisor:  Dror Bar-Natan

Thesis Title:  A Universal Finite Type Invariant of Knotted
               Trivalent Graphs
	       http://www.math.toronto.edu/zsuzsi/research/thesis.pdf 


THURSDAY, APRIL 28, 2011:
1:45PM Goodies at lounge.
2:10PM Open lecture by Zsuzsanna Dancso at BA6183.
3:00PM Closed exam starts at BA6183, without a student presentation.


Thesis Abstract:

Knot theory is not generally considered an algebraic subject, due to the
fact that knots don't have much algebraic structure: there are a few
operations defined on them (such as connected sum and cabling), but these
don't nearly make the space of knots finitely generated. In this thesis,
following an idea of Dror Bar-Natan's, we develop an algebraic setting for
knot theory by considering the larger, richer space of knotted trivalent
graphs (KTGs), which includes knots and links. KTGs along with standard
operations defined on them form a finitely generated algebraic structure,
in which many topological knot properties are definable 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
homomorphicity properties of the resulting invariant, which turns out to
be homomorphic with respect to almost all of the KTG operations except for
edge unzip. Unfortunately, edge unzip is crucial for finite generation,
and we prove that in fact no universal finite type invariant of KTGs can
intertwine all the standard operations at once. To fix this, we 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 generated, includes knots,
and is closely related to Drinfel'd associators.

The thesis is based on two articles, one published and one pre-print, the
second one joint with Dror Bar-Natan.