Date and Time: Tuesday, April 13, 2010, 11:10 a.m., in BA 6183, 40 St. George Street

PhD Candidate: Jana Archibald

PhD Advisor: Dror Bar-Natan

Thesis Title: The Multivariable Alexander Polynomial on Tangles

(http://www.math.toronto.edu/jfa/jana_thesis.pdf)

Thesis Abstract:

The multivariable Alexander polynomial (MVA) is a classical invariant

of knots and links. We give an extension to regular virtual knots which

has simple versions of many of the relations known to hold for the

classical invariant.

By following the previous proofs that the MVA is of finite type we

give a new definition for its weight system which can be computed as

the determinant of a matrix created from local information. This is an

improvement on previous definitions as it is directly computable (not

defined recursively) and is computable in polynomial time. We also

show that our extension to virtual knots is a finite type invariant of

virtual knots.

We further explore how the multivariable Alexander polynomial takes

local information and packages it together to form a global knot

invariant, which leads us to an extension to tangles. To define this

invariant we use so-called circuit algebras, an extension of planar

algebras which are the `right’ setting to discuss virtual knots. Our

tangle invariant is a circuit algebra morphism, and so behaves well

under tangle operations and gives yet another definition for the

Alexander polynomial. The MVA and the single variable Alexander

polynomial are known to satisfy a number of relations, each of which

has a proof relying on different approaches and techniques. Using our

invariant we can give simple computational proofs of many of these

relations, as well as an alternate proof that the MVA and our virtual

extension are of finite type.

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