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

Wednesday, June 20 2018
11:10 a.m.

PhD Candidate:  Huan Vo
Supervisor:   Dror Bar-Natan
Thesis title:  Alexander Invariants of Tangles via Expansions



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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