Page 312 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 312
GRAPHS, POLYNOMIALS, SURFACES, AND KNOTS (MS-49)
All results presented are joint work with Yuanan Diao, some of them are also joint work
with Kenneth Hinson.
Hopf Algebras in Studying Graph and Embedded Graph Polynomials
Sergei Lando, lando@hse.ru
HSE University, Skolkovo Institute of Science and Technology, Russian Federation
Study of Hopf algebra structures on spaces spanned by graphs was initiated by S. Joni and G.-
C. Rota in 1979 and was later unified with umbral calculus. Since then, a lot of combinatorial
objects similar to graphs were shown to generate natural Hopf algebras. Embedded graphs are
not among them, but this is true for closely related to them binary delta-matroids as defined by
A. Bouchét in 1987. These general Hopf algebras have interesting Hopf subalgebras the study
of which is sometimes easier and leads to effective explicit computations.
Many polynomial invariants of graphs, embedded graphs, and binary delta-matroids demon-
strate a nice behavior with respect not only to the multiplicative structure, but to comultipli-
cation as well. Examples include chromatic polynomial, characteristic polynomial, matching
polynomial, Stanley’s symmetrized chromatic polynomial, and many others.
Invariants of abstract graphs are closely related to those of chord diagrams (which are em-
bedded graphs with a single vertex). In the framework of Vassiliev’ theory of finite order knot
invariants, chord diagrams serve as a tool to describe the latter. Similarly, certain invariants of
binary delta-matroids and embedded graphs produce finite invariants of links. The Hopf algebra
point of view leads to unexpected approaches to extending graph invariants to embedded graphs
and binary delta-matroids.
The talk will be based on recent results of my students, colleagues, and myself.
A new enumerator polynomial with a smart derivative
Serge Lawrencenko, lawrencenko@hotmail.com
Russian State University of Tourism and Service, Russian Federation
Let Sn be a given set of n-vertex simplicial complexes; e.g., a set of n-vertex paths, cycles,
trees, or 2-cell embeddings of graphs, etc. We solve the problem of determining the cardinality
|Sn| in a double sense: (1) the labeled sense; all n vertices are mapped bijectively onto the set of
labels {1, 2, . . . , n} where different maps (labelings) may produce different complexes, (2) the
unlabeled sense, that is, up to isomorphism (labels removed). A new enumerator polynomial,
Pn(x), will be introduced. It has interesting properties: The value Pn(1) is equal to |Sn| in the
labeled sense while the value of the derivative Pn(1) is equal to n! times |Sn| in the unlabeled
sense. For example, for paths with n vertices Pn(x) = (n!/2)x2. More examples and properties
of the enumerator polynomial will be presented.
310
All results presented are joint work with Yuanan Diao, some of them are also joint work
with Kenneth Hinson.
Hopf Algebras in Studying Graph and Embedded Graph Polynomials
Sergei Lando, lando@hse.ru
HSE University, Skolkovo Institute of Science and Technology, Russian Federation
Study of Hopf algebra structures on spaces spanned by graphs was initiated by S. Joni and G.-
C. Rota in 1979 and was later unified with umbral calculus. Since then, a lot of combinatorial
objects similar to graphs were shown to generate natural Hopf algebras. Embedded graphs are
not among them, but this is true for closely related to them binary delta-matroids as defined by
A. Bouchét in 1987. These general Hopf algebras have interesting Hopf subalgebras the study
of which is sometimes easier and leads to effective explicit computations.
Many polynomial invariants of graphs, embedded graphs, and binary delta-matroids demon-
strate a nice behavior with respect not only to the multiplicative structure, but to comultipli-
cation as well. Examples include chromatic polynomial, characteristic polynomial, matching
polynomial, Stanley’s symmetrized chromatic polynomial, and many others.
Invariants of abstract graphs are closely related to those of chord diagrams (which are em-
bedded graphs with a single vertex). In the framework of Vassiliev’ theory of finite order knot
invariants, chord diagrams serve as a tool to describe the latter. Similarly, certain invariants of
binary delta-matroids and embedded graphs produce finite invariants of links. The Hopf algebra
point of view leads to unexpected approaches to extending graph invariants to embedded graphs
and binary delta-matroids.
The talk will be based on recent results of my students, colleagues, and myself.
A new enumerator polynomial with a smart derivative
Serge Lawrencenko, lawrencenko@hotmail.com
Russian State University of Tourism and Service, Russian Federation
Let Sn be a given set of n-vertex simplicial complexes; e.g., a set of n-vertex paths, cycles,
trees, or 2-cell embeddings of graphs, etc. We solve the problem of determining the cardinality
|Sn| in a double sense: (1) the labeled sense; all n vertices are mapped bijectively onto the set of
labels {1, 2, . . . , n} where different maps (labelings) may produce different complexes, (2) the
unlabeled sense, that is, up to isomorphism (labels removed). A new enumerator polynomial,
Pn(x), will be introduced. It has interesting properties: The value Pn(1) is equal to |Sn| in the
labeled sense while the value of the derivative Pn(1) is equal to n! times |Sn| in the unlabeled
sense. For example, for paths with n vertices Pn(x) = (n!/2)x2. More examples and properties
of the enumerator polynomial will be presented.
310
