Abstract
The Tutte equations are ported (or set-pointed) when the equations F(N) = g\_e
F(N/e) + r\_e F(N\e) are omitted for elements e in a distinguished set called
ports. Solutions F can distinguish different orientations of the same matroid.
A ported extensor with ground set is a decomposible element in the exterior
algebra over a vector space with a given basis, called the ground set,
containing a distinguished subset called ports. These can represent ported
matroids and have analogous dualization, deletion and contraction operations. A
ported extensor function is defined using dualization, port element renaming,
exterior multiplication, and contraction of non-ports. We prove that this
function satisfies a sign-corrected variant of the Tutte equations over
exterior algebra. For non-ported unimodular, i.e., regular matroids, our
function reduces to the basis generating function and for graphs the Laplacian
(or Kirchhoff) determinant. In general, the function value, as an extensor,
signifies the space of solutions to Kirchhoff's and Ohm's electricity equations
after projection to the variables associated to the ports. Combinatorial
interpretation of various determinants (the Plucker coordinates) generalize the
matrix tree theorem and forest enumeration expressions for electrical
resistance. The corank-nullity polynomial, basis expansions with activities,
and a geometric lattice expansion generalize to ported Tutte functions of
oriented matroids. The ported Tutte functions are parametrized, which raises
the problem of how to generalize known characterizations of parameterized
non-ported Tutte functions.
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