## Fourientation activities and the Tutte polynomial

## Sam Hopkins (MIT)

#### Abstract :

This talk is based on joint work with Spencer Backman and Lorenzo
Traldi. The Tutte polynomial, a kind of two-variable generalization of the
chromatic polynomial, is one of the most important polynomials associated to a
graph. One straightforward definition, the corank-nullity expansion, expresses the
Tutte polynomial as a sum over all (spanning) subgraphs. Tutte's original
definition, however, was as a sum over spanning trees weighted by internal and
external activity. In 1990 Gordon and Traldi found a "generalized activities"
expansion of the Tutte polynomial that recovers both of these definitions.
Meanwhile, it has been known that the Tutte polynomial is intimately related to
graph orientations since at least the seminal work of Stanley expressing the
number of acyclic orientations as a Tutte polynomial evaluation. Las Vergnas
generalized Stanley's result by expressing the Tutte polynomial as a sum over all
orientations weighted by "orientation activities." We provide a common
generalization of the Gordon-Traldi and Las Vergnas formulas by expressing the
Tutte polynomial as a sum over fourientations weighted by their activities. A
fourientation is a hybrid between a subgraph and a orientation: it is a choice for
each edge whether to orient that edge in either direction, leave it unoriented, or
biorient it. Via this fourientation activities formula we recapture early work
with Spencer Backman which says that various "min edge" classes of fourientations
are enumerated by the Tutte polynomial. These min edge classes arise in connection
with many geometric and algebraic objects associated to a graph including
bigraphical hyperplane arrangements, cycle-cocycle reversal systems, graphic
Lawrence ideals and zonotopal algebras. Time permitting, I will discuss some of
these connections.