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

Tuesday, June 23, 2015
11:10 a.m.

PhD Candidate: Oleg Chterental
Supervisor:  Dror Bar-Natan
Thesis title: Virtual braids and virtual curve diagrams



There is a well known injective homomorphism $\phi:{\mathcal {B}}_n \rightarrow {\rm Aut}(F_n)$ from the classical braid group ${\mathcal {B}}_n$ into the automorphism group of the free group $F_n$, first described by Artin \cite{A}. This homomorphism induces an action of ${\mathcal {B}}_n$ on $F_n$ that can be recovered by considering the braid group as the mapping class group of $H_n$ (an upper half plane with $n$ punctures) acting naturally on the fundamental group of $H_n$.

Kauffman introduced virtual links \cite{Ka} as an extension of the classical notion of a link in ${\mathbb {R}}^3$. As in the classical case, there is a corresponding group ${\mathcal {VB}}_n$ of virtual braids. In this thesis, we will generalize the above action to ${\mathcal {VB}}_n$. We will define a set, ${\mathcal {VCD}}_n$, of “virtual curve diagrams” and define an action of ${\mathcal {VB}}_n$ on ${\mathcal {VCD}}_n$. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in ${\mathcal {VB}}_n$.

In the papers \cite{B,M}, an extension $\psi:{\mathcal {VB}}_n\rightarrow {\rm Aut}(F_{n+1})$ of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that $\psi$ is not injective by exhibiting a non-trivial virtual braid in the kernel when $n=4$.

The thesis can be found in the following link: http://www.math.toronto.edu/ochteren/thesis.pdf



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