*Date:* Tuesday, February 18, 3:00 Eastern Standard Time *[Note the new
time!]*
*Speaker:* Louis H. Kauffman
*Title:* Trivalent Graphs, Perfect Matchings and Virtual Knot Theory
*Zoom Link:* Meeting ID: 463-430-866, https://wpi.zoom.us/j/463430866
*Abstract: * (This talk is joint work with Scott Baldridge and William
Rushworth.) We review knot invariants such as the bracket state sum model
for the Jones polynomial and its relation to the Tutte polynomial. We
recall trivalent graphs immersed in the plane and the well-known Penrose
evaluation formula for them. We generalize the Penrose formula to a Penrose
polynomial that is defined on trivalent graphs with perfect matchings (G,
M) where M is a perfect matching on the trivalent graph G. Remarkably,
these perfect matching polynomials are directly related to the Jones
polynomial of virtual knots and links via a mapping K from trivalent
perfect matchings to virtual link diagrams. The talk discusses how the
mapping K induces structural correspondences between graph theory and knot
theory. These include the above relationship between the Penrose matching
polynomial and the Jones polynomial and new relationships with Khovanov
homology and other knot invariants. Reformulations of coloring problems
arise in this context and are intertwined with virtual knot theory.
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*Jo*
Jo Ellis-Monaghan
Chair, Department of Mathematics and Statistics (on sabbatical 2019-2020)
Editor-in-Chief, PRIMUS
Saint Michael's College
One Winooski Park
Colchester, VT 05439
office: Jeanmarie 279
website: http://academics.smcvt.edu/jellis-monaghan/
phone: 802 654 2660
fax: 802 654 2960
e-mail: je <jellis-monaghan@smcvt.edu>llismonaghan@gmail.com
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