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

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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