Apr

15

DEPARTMENTAL PHD THESIS EXAMNote: We are experimenting with a somewhat different examination protocol,in which the student will be given the option of giving a longerpresentation. This necessitates changing the format of the departmentalexam and the location of the SGS exam, and may lead to certain technicaland administrative difficulties. Only 1-2 exams will be run in thisPhD 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.format this summer and we may or may not allow this option in the future.

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