Jun

01

*Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.*

Wednesday, June 20 2018

11:10 a.m.

BA6183

PhD Candidate: Huan Vo

Supervisor: Dror Bar-Natan

Thesis title: Alexander Invariants of Tangles via Expansions

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Abstract:

In this thesis we describe a method to extend the Alexander polynomial to tangles. It is based on a

technology known as expansions, which is inspired by the Taylor expansion and the Kontsevich integral.

Our main object of study is the space of w-tangles, which contains usual tangles, but has a much simpler

expansion. To study w-tangles, we introduce an algebraic structure called meta-monoids. An expansion

of w-tangles together with a particular Lie algebra, namely the non-abelian two-dimensional Lie algebra,

gives us a meta-monoid called Γ-calculus that recovers the Alexander polynomial. Using the language

of Γ-calculus, we rederive certain important properties of the Alexander polynomial, most notably the

Fox-Milnor condition on the Alexander polynomials of ribbon knots [Lic97, FM66]. We argue that our

proof has some potential for generalization which may help tackle the slice-ribbon conjecture. In a sense

this thesis is an extension of [BNS13].

A copy of the thesis can be found here: Thesis-HuanVo

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