Page 311 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 311
GRAPHS, POLYNOMIALS, SURFACES, AND KNOTS (MS-49)
Tutte characters for combinatorial coalgebras
Alex Fink, a.fink@qmul.ac.uk
Queen Mary University of London, United Kingdom
The Tutte polynomial is a favourite invariant of matroids and graphs. So when one is working
in a generalisation of these settings, for example arithmetic matroids or ribbon graphs, it is a
tempting question to find a counterpart of the Tutte polynomial; answers have been given in
many cases. Work of Krajewski, Moffatt, and Tanasa found a framework unifying the Tutte-
like polynomials arising from graphs in surfaces using Hopf algebras. Our contribution, besides
adding some more examples and observing that the machinery is useful for producing convolu-
tion formulae in the style of Kook-Reiner-Stanton-Etienne-Las Vergnas, is a generalisation of
the formalism using comonoids in linear species.
Joint with Clément Dupont and Luca Moci.
Tutte’s dichromate for signed graphs
Andrew Goodall, goodall.aj@gmail.com
Charles University, Czech Republic
Coauthors: Bart Litjens, Guus Regts, Lluís Vena
A signed graph is a graph with signed edges (positive or negative). Two signed graphs are
considered equivalent if their edge signs differ on a cutset of the graph. Proper colourings and
nowhere-zero flows of signed graphs are defined analogously to those of graphs. For graphs,
these are both enumerated by evaluations of the Tutte polynomial. For signed graphs, Zaslavsky
enumerated proper colourings, and recently DeVos–Rollová–Šámal showed that the number of
nowhere-zero flows satisfies a deletion-contraction recurrence, and, independently, Qian–Ren
and Goodall–Litjens–Regts–Vena gave a subset expansion formula. We construct a trivariate
polynomial invariant of signed graphs that contains both the number of proper colourings and
the number of nowhere-zero flows as evaluations: for this three variables are needed, giving a
“trivariate Tutte polynomial" for signed graphs. Specializations include Zaslavsky’s bivariate
rank-generating polynomial of the (frame matroid of the) signed graph and the Tutte polynomial
of the (cycle matroid of the) underlying graph.
A colored version of Brylawski’s tensor product formula and its
applications
Gábor Hetyei, ghetyei@uncc.edu
University of North Carolina at Charlotte, United States
The tensor product of a graph and a pointed graph is obtained by replacing each edge of the first
graph with a copy of the second. In this talk we outline a simple proof of Brylawski’s formula
for the Tutte polynomial of the tensor product which can be generalized to the colored Tutte
polynomials introduced by Bollobás and Riordan. Consequences include formulas for Jones
polynomials of (virtual) knots and for invariants of composite networks in which some major
links are identical subnetworks in themselves.
309
Tutte characters for combinatorial coalgebras
Alex Fink, a.fink@qmul.ac.uk
Queen Mary University of London, United Kingdom
The Tutte polynomial is a favourite invariant of matroids and graphs. So when one is working
in a generalisation of these settings, for example arithmetic matroids or ribbon graphs, it is a
tempting question to find a counterpart of the Tutte polynomial; answers have been given in
many cases. Work of Krajewski, Moffatt, and Tanasa found a framework unifying the Tutte-
like polynomials arising from graphs in surfaces using Hopf algebras. Our contribution, besides
adding some more examples and observing that the machinery is useful for producing convolu-
tion formulae in the style of Kook-Reiner-Stanton-Etienne-Las Vergnas, is a generalisation of
the formalism using comonoids in linear species.
Joint with Clément Dupont and Luca Moci.
Tutte’s dichromate for signed graphs
Andrew Goodall, goodall.aj@gmail.com
Charles University, Czech Republic
Coauthors: Bart Litjens, Guus Regts, Lluís Vena
A signed graph is a graph with signed edges (positive or negative). Two signed graphs are
considered equivalent if their edge signs differ on a cutset of the graph. Proper colourings and
nowhere-zero flows of signed graphs are defined analogously to those of graphs. For graphs,
these are both enumerated by evaluations of the Tutte polynomial. For signed graphs, Zaslavsky
enumerated proper colourings, and recently DeVos–Rollová–Šámal showed that the number of
nowhere-zero flows satisfies a deletion-contraction recurrence, and, independently, Qian–Ren
and Goodall–Litjens–Regts–Vena gave a subset expansion formula. We construct a trivariate
polynomial invariant of signed graphs that contains both the number of proper colourings and
the number of nowhere-zero flows as evaluations: for this three variables are needed, giving a
“trivariate Tutte polynomial" for signed graphs. Specializations include Zaslavsky’s bivariate
rank-generating polynomial of the (frame matroid of the) signed graph and the Tutte polynomial
of the (cycle matroid of the) underlying graph.
A colored version of Brylawski’s tensor product formula and its
applications
Gábor Hetyei, ghetyei@uncc.edu
University of North Carolina at Charlotte, United States
The tensor product of a graph and a pointed graph is obtained by replacing each edge of the first
graph with a copy of the second. In this talk we outline a simple proof of Brylawski’s formula
for the Tutte polynomial of the tensor product which can be generalized to the colored Tutte
polynomials introduced by Bollobás and Riordan. Consequences include formulas for Jones
polynomials of (virtual) knots and for invariants of composite networks in which some major
links are identical subnetworks in themselves.
309
