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.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/26151 |
| Date | 15 February 2011 |
| Creators | Archibald, Jana |
| Contributors | Bar-Natan, Dror |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis, Other |
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