USA Topology Seminar
Twisted Duality and its Applications
Joanna Ellis-Monaghan, Saint Michael's College
Thursday, October 7, 2010
We consider two operations on the edge of an embedded (i.e., ribbon) graph: giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of
ribbon group of G
. We show that this ribbon group action gives a complete characterization of duality in that if
is any cellularly embedded graph with medial graph
, then the orbit of
under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to
. We then show how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials such as the generalized transition polynomial, an extension of the Penrose polynomial to embedded graphs, and the topological Tutte polynomials of Las Vergnas and also Bollobás and Riordan, as well as various knot and link invariants. We make a brief excursion into a motivating application arising from emergent nanotechnology design strategies.
This is joint work with Iain Moffatt.