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Two Generalizations of the Multivariable Alexander Polynomial
by
Iva Halacheva
University of Toronto
An extension of the Multivariable Alexander Polynomial (MVA) for links to virtual tangles, taking values in a tensor product of exterior algebras, was defined by J. Archibald in her thesis (arXiv:0710.4885v1). The computations involve and exponential-time algorithm but are relatively straightforward and provide an easy verification of almost all the relations satisfied by the MVA and its weight system. Another generalization of the MVA is given by D. Bar-Natan and comes from a reduction of an invariant of knotted copies of S2 and S1 in four-dimensional space. It has the advantage that it is matrix-valued and computable in polynomial time, but is only defined on tangles with no closed components, i.e. pure tangles. We will show that after some repackaging, the two invariants coincide on the level of pure tangles, and we will discuss a partial extension of Bar-Natans invariant, arising from this connection, which allows closed components.
Date received: October 22, 2015
Copyright © 2015 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cblw-17.