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Room P3.10, Mathematics Building
Louis Kauffman, University of Illinois, Chicago
Virtual Knot Theory and Oriented Extensions of the Jones Polynomial
The original Jones polynomial for classical knots depends only
weakly on the orientation of the knot or link to which it is
applied. Careful examination of the orientations assigned to states
in an oriented bracket polynomial model for the Jones polynomial
reveals that there is much more topological information available
from orientation when the knot or link is embedded in a thickened
surface or regarded as a virtual link. This talk will discuss a
large scale generalization that we call the extended bracket
polynomial (taking polynomial and graphical values) and the arrow
polynomial (joint work with Heather Dye) taking polynomial values
with infinitely many new variables. The arrow polynomial is related
to the Miyazawa polynomials for virtual knots. The talk will
discuss applications of these new invariants to finding virtual
crossing number and genus of virtual knots, and we shall discuss
extensions of Khovanov homology related to these invariants.