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Семинар "Комбинаторика инвариантов Васильева"

27 мая 2021
Grigory Chelnokov
Enumeration of coverings of compact 3-dimensional Euclidean manifolds
Для компактных 3-мерных евклидовых многообразий G2, G4, G5, G6, B1, B2, получена классификация их  конечнолистных накрытий, а также перечислены классы эквивалентности каждого типа накрытий как функции от числа слоев. Кроме того, для полученных комбинаторных последовательностей выписаны их производящие функции в терминах рядов Дирихле.

20 мая 2021

Polina Baron
Independents sets of matroids and log-concavity of $q$-state Potts model
A matroid M on a finite nonempty set E is a nonempty collection B of subsets of E, called the basis of the matroid, that satisfies the exchange property: for all b1,bin B for each ein b\ b2 there exists e2 in b2 \ b1 such that (b\ {e1}) Ս {e2} in B.
Matroids are a useful tool to study chromatic polynomials of graphs. We are going to prove the Mason’s conjecture:
Ik(M)2 ≥ (k+1)(n-k+1)/(k (n-k)) Ik-1(M) Ik+1(M), where Ik(M) stands for the number of k-element independent sets of an n-element matroid M. As a corollary of this conjecture, we will prove ultra log-concavity of q-state Potts models and, if time permits, discuss log-concavity of chromatic polynomials in more general cases.



13 мая 2021
Riya Dogra
Melvin—Morton—Rozansky Conjecture
The talk is based on the paper “On the Melvin-Morton-Rozansky Conjecture” by Dror Bar-Natan and Stavros Garoufalidis. Roughly, the conjecture states that the Alexander-Conway Polynomial can be read off the highest order part of the coloured Jones Polynomial. They invented a remarkable reduction of the conjecture to a certain identity on the corresponding weight systems via canonical invariants. Some preliminaries on canonical Vassiliev invariants will be given. The space of closed Jacobi diagrams modulo certain relations will be described with respect to invariants arising from some Lie algebras. All of the above will be used to prove the conjecture.


29 апреля 2021
Grigorii Yurgin
The Laplacian Spectrum Of Graphs II
Given a graph, one can consider its Laplacian matrix and its spectrum. This matrix is related to the adjacency matrix, but the Laplacian matrix seems to be much more deep and important. We are going to start with some basic properties of the Laplacian spectrum, and after that we shall discuss numerous relations between the Laplacian spectrum and graph invariants. Among these results are: the Matrix-Tree-Theorem about the number of spanning trees of graph; and some estimates on vertex connectivity and edge connectivity of graphs via second smallest eigenvalue of the Laplacian. We are going to prove several most important results, and some results will be discussed in the overview format. Also we shall mention some applications of Laplacians of graphs.
Video

22 апреля 2021
Grigorii Yurgin
The Laplacian Spectrum Of Graphs
Given a graph, one can consider its Laplacian matrix and its spectrum. This matrix is related to the adjacency matrix, but the Laplacian matrix seems to be much more deep and important. We are going to start with some basic properties of the Laplacian spectrum, and after that we shall discuss numerous relations between the Laplacian spectrum and graph invariants. Among these results are: the Matrix-Tree-Theorem about the number of spanning trees of graph; and some estimates on vertex connectivity and edge connectivity of graphs via second smallest eigenvalue of the Laplacian. We are going to prove several most important results, and some results will be discussed in the overview format. Also we shall mention some applications of Laplacians of graphs.
Video

15 апреля 2021
Dmitrii Rybin 
Lower and upper bounds for the number of Vasiliev knot invariants

Following works by S. Chmutov, I will prove inequalities on dimensions of subspaces of Vasiliev invariants of order at most n. I will also give an overview of other known asymptotic bounds and combinatorial objects that lead to them.
Video

8 апреля 2021

Yury Belousov
On the question of genericity of hyperbolic knots (based on a joint work with A. Malyutin) 
Thurston’s famous classification theorem, of 1978, states that a non-toric non-satellite knot is hyperbolic, that is, its complement admits a complete hyperbolic metric of finite volume. Until recently there was the conjecture (known as Adams conjecture) saying that the percentage of hyperbolic knots amongst all the prime knots of n or fewer crossings approaches 100 as n approaches infinity. In 2017 Malyutin showed that this statement contradicts several other plausible conjectures. Finally, in 2019 Adams conjecture was found to be false. In this talk we are going to discuss the key ingredients of the disproof of Adams conjecture.
Video on youtube



18 марта 2021

Raisa Safronova
HOLOMORPHIC QUADRATIC DIFFERENTIALS ON GRAPHS AND THE CHROMATIC POLYNOMIAL (after RICHARD KENYON, WAI YEUNG LAM ), Continuation
Video on youtube
 

11 марта 2021
Raisa Safronova
HOLOMORPHIC QUADRATIC DIFFERENTIALS ON GRAPHS AND THE CHROMATIC POLYNOMIAL (after RICHARD KENYON, WAI YEUNG LAM )
In the classical theory holomorphic quadratic differentials (HQD) on Riemann surfaces are connected to many important things: harmonic functions, dynamical systems, minimal surfaces and so on. There exists a discrete analogue on graphs which preserves such rich relations. In this talk we will discuss some properties of HQD, which will allow us to evaluate chromatic polynomial of a graph at negative integer value.
Video on youtube


4 марта 2021

Mikhail Fedorov
Towards classification of embeddings of n-dimensional manifolds with boundary into (2n-1)-dimensional space
We study embeddings of n-manifolds with a nonempty boundary into R2n-1. This problem is interesting, in particular, because there is a similar classification of embeddings of k-connected manifolds with a nonempty boundary, for k>0, which cannot be extended to the case k=0. We introduce an analogue of the Seifert form for embeddings of punctured n-manifolds into R2n-1. We describe some properties of this invariant and restrictions on the set of its possible values. Our main conjecture asserts that this invariant yields a complete classification of embeddings of punctured n-manifolds into R2n-1.
Video on youtube


25 февраля 2021
Zhuoke Yang
Values of the sl3 weight system on chord diagrams whose intersection graph is complete bipartite K 2,n
The sl3 Lie algebra weight system takes values in the center of the universal enveloping algebra of the Lie algebra sl3 which is
isomorphic to the ring of polynomials in TWO variables (the Casimir elements of degrees 2 and 3). For the sl3 weight system, we do not
have a result similar to the Chmutov—Varchenko recurrence relations for the sl2 weight system which could help us to compute its value.
I will show the way to calculate the sl3 weight system on small chord diagrams and prove the theorem which is the formula for the
values of the sl3 weight system on chord diagrams whose intersection graph is complete bipartite K2,n.
Video on youtube


18 февраля 2021
Polina Fillipova
Values of the sl2 weight system on complete bipartite graphs (Continuation)
Video on youtube

11 февраля 2021
Polina Fillipova
Values of the sl2 weight system on complete bipartite  graphs
A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev's theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra sl2, but this definition is too cumbersome from the computational point of view, so that the values of this weight system are known only for some limited classes of chord diagrams. We give a formula for the values of the slweight system for a class of chord diagrams whose intersection graphs are complete bipartite graphs with no more than three vertices in one of the parts.

Our main computational tool is the Chmutov-Varchenko recurrence relation. Furthermore, complete bipartite graphs with no more than three vertices in one of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for the projection onto the subspace of primitive elements along the subspace of decomposable elements in these subalgebras. We compute the values of the sl2weight system for the projections of chord diagrams with such intersection graphs. Our results confirm certain conjectures due to S.K.Lando on the values of the weight system slat the projections of chord diagrams on the space of primitive elements.
Video on youtube


4 февраля 2021
Evgenii Zaikin
Jones-Krushkal polynomial
A virtual knot is a knot in a thickened surface. We recall the notion of Kauffman bracket for both classical and virtual knots. Then we use it to define the Jones polynomial for classical knots and the Jones-Krushkal polynomial for virtual knots. We discuss certain properties of this polynomial.
Video on youtube


28 января 2021

Igor Chaban
Edge colouring models for the Tutte polynomial and related graph invariants
(after A.Goodall, continuation)
For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley's symmetric monochrome polynomial. We exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q >= k each evaluate the number of proper edge k-colourings of G when G is Pfaffian. 
Video on youtube

 

21 января 2021

Igor Chaban
Edge colouring models for the Tutte polynomial and related graph invariants
(after A.Goodall)
For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley's symmetric monochrome polynomial. We exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q >= k each evaluate the number of proper edge k-colourings of G when G is Pfaffian. 
Video on youtube


17 декабря 2020
Grigorii Yurgin

Bijective proofs of proper coloring theorems
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. In this talk, we are going to establish three different ways to express these invariants: as sums over all spanning subgraphs, as sums over some special spanning subgraphs (namely, ones with no broken circuits), and in terms of acyclic orientations with compatible colorings. We will do it constructing combinatorial bijections. One needs only two essentially different bijections to obtain all these expressions for both invariants. 

Video on youtube


10 декабря 2020
Polina Baron
Applications of link invariants and chromatic polynomial to statistical mechanics and topological quantum theory
In the focus of the talk are connections between link invariants and chromatic polynomial and geometric representations of several models of statistical mechanics. We define the chromatic algebra, which occurs in the low temperature expansion of the Q-state Potts model. The Markov trace of the chromatic algebra is the chromatic polynomial of the associated graph. We relate, from the point of view of both mathematics and physics, the chromatic algebra to the  SO(3)  Birman-Murakami-Wenzl algebra, which is an algebra-level analogue of the correspondence between the  SO(3)  Kauffman polynomial and the chromatic polynomial. This relation implies that correlators in the  SO(3)  topological quantum field theory can be expressed in terms of the chromatic polynomial. If time permits, we are going to conclude by listing some open questions in the field.
Video on youtube

 

3 декабря 2020
Vadim Retinskii
On sequences of polynomials arising from graph invariants
 
The talk is based on the paper ‘On Sequences of Polynomials Arising from Graph Invariants ’, by T. Koteka, J.A. Makowskyb, E.V. Ravvec. Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. We will discuss characteristic and matching polynomials and find out the connection between graph invariants and orthogonal polynomials.
Video on youtube

26 ноября 2020
Danil Gubarevich
Snarl invariant for long knots
The talk is based on the paper A polynomial time knot polynomial by Dror Bar-Natan and Roland van der Veen (2018). A long knot is just a usual knot, but with a point on it chosen. The snarl invariant is a polynomial invariant of long knots that has some advantages when compared to other polynomial knot invariants. Namely, it is closely related to Alexander polynomial and in contrast to other quantum invariants, such as Jones polynomial, can be computed in polynomial time. I will try to explain how it can be computed for a long knot corresponding to trefoil and to show how it can be built with the help of Weyl algebra to state the connection with Alexander polynomial. If time permits, I will speculate on the proof of the fact that it is indeed invariant with respect to Turaev moves for oriented tangles (an analogue of Reidemeister  moves).
Video on youtube


19 ноября 2020
Riya Dogra
Stanley’s chromatic polynomial for delta-matroids
The talk summarizes the results of the paper, ‘An extension of Stanley’s chromatic symmetric function to binary delta matroids’, by M. Dudina. V. Zhukov. The goal of the talk is to extend Stanley’s chromatic polynomial, a well known graph invariant, to delta matroids. The talk will begin with introducing the necessary background on graphs and delta matroids. The structure of Hopf algebras on the spaces under consideration will be given. Later, Stanley’s chromatic function for graphs will be defined, and with suitable ‘characters’ extended for delta matroids.
Video on youtube


12 ноября 2020
Mark Alekseev
Invariants of knots II
My talk will be devoted to a description of certain classical knot and link invariants, including the linking number, fundamental group, diagram coloring.
Video on youtube

5 ноября 2020
Mark Alekseev
Invariants of knots
My talk will be devoted to a description of certain classical knot and link invariants, including the linking number, fundamental group, diagram coloring.
Video on youtube

29 октября 2020
Sergei Lando
Graph invariants and Hopf algebra structure
Many graph invariants allow an extension to Hopf algebra homomorphisms from the Hopf algebra of graphs to other Hopf algebras, like polynomial ones. Such a homomorphism is uniquely determined by its values on primitive elements in the Hopf algebra of graphs. These values are very simple, since they also are primitive elements, whence linear polynomials. Hence, understanding primitive elements plays a crucial role in understanding polynomial graph invariants. In particular, we will discuss the projection of the Hopf algebra of graphs to its primitive subspace.
Video on youtube

22 октября 2020
Raphael Fesler
Embedded graphs and twisted Hurwitz theory

Let’s take a permutation of n elements with k independent cycles. This permutation can be represented by a product of m transpositions. Once we have this representation we can build a surface, of genus g=1+(m-n-k)/2, with k boundary components. Furthermore, there is an embedded graph with n vertices and m edges, such that all the vertices are on the boundary and their positons are given by the cyclic structure, and the edges correspond the transpositions. The surface is obtained by gluing ribbons along the edges. This is the oriented Hurwitz theory.
We can extend this construction to the case of twisted ribbons. It will give us a possibly non orientable surface. We can at the same time construct the orientation cover of this surface, with also an embedded graph. We thus can build an analogue of the oriented Hurwitz theory, which is called twisted Hurwitz theory. In this talk we will explain the geometric construction above, then recall what is the oriented Hurwitz theory (Hurwitz numbers, cut-and-join, Schur functions) and explain the analogue for the twisted Hurwitz theory. The talk is based on a joint work with Y.Burman.
Video on youtube

15 октября 2020
Boris Bychkov

Algebraic structures associated to chord diagrams and graphs II
The chromatic polynomial is multiplicative: its value on a disjoint union of connected graphs is the product of its values on the components. There are many graph invariants possessing this property, which hints that disjoint union can be considered as a multiplication operation on graphs. Moreover, it makes sense to allow adding linear combinations of graphs, and extending multiplication by linearity to linear combinations we obtain an algebra of graphs. Probably even more interesting is the operation of comultiplication on this algebra, which makes it into a Hopf algebra. These algebraic structures and their relationship with 4-term relations will be discussed in the talk. 
Video on youtube

8 октября 2020
Boris
Bychkov

Algebraic structures associated to chord diagrams and graphs I
The chromatic polynomial is multiplicative: its value on a disjoint union of connected graphs is the product of its values on the components. There are many graph invariants possessing this property, which hints that disjoint union can be considered as a multiplication operation on graphs. Moreover, it makes sense to allow adding linear combinations of graphs, and extending multiplication by linearity to linear combinations we obtain an algebra of graphs. Probably even more interesting is the operation of comultiplication on this algebra, which makes it into a Hopf algebra. These algebraic structures and their relationship with 4-term relations will be discussed in the talk.
Video on youtube.


1 октября 2020
Polina Fillipova

Examples of graph invariants satisfying the 4-term relations
We will discuss some examples of graph invariants that satisfy the so-called 4-term relations introduced in the previous talk. Namely, I will define the chromatic polynomial, the Stanley chromatic polynomial, the number of vertex quadrangles, the number of edge polygons modulo 2, the number of perfect matchings and the corank of the adjacency matrix. I will also discuss some of these invariants in more detail.
The talk is based on the book: S.K.Lando and A.K.Zvonkin. Graphs on Surfaces and their Applications.
Video on youtube.


24 сентября 2020
Sergei Lando
The 4-term relation: where it comes from

We will explain the main argument of Vassiliev’s theory of finite type knot invariants, which leads to the notions of chord diagram, weight systems, and  4-term relations. This will be an introductory talk. No preliminary knowledge is expected.
Video on youtube.


17 сентября 2020
Sergei Lando

Main characters of Vassiliev’s theory
Main combinatorial and topological objects playing crucial roles in the theory of finite order knot invariants will be described. Knots, chord diagrams, graphs, surfaces, and embedded graphs among them. This will be an introductory talk. No preliminary knowledge is expected.
Video: https://www.youtube.com/watch?v=YRRnaOBzYUs

Summer break. The seminar will continue in September 2020.

28 мая 2020
Natalia Babina
A categorification for the chromatic polynomial
For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. The talk is based on the paper by Laure Helme-Guizon and Yongwu Rong (2005).
Video: https://www.youtube.com/watch?v=We2MRoQtWoI

21 мая 2020

Polina Baron
Encoding Knots by Clasp Diagrams

Knot invariants are commonly counted using Gauss diagrams, which are, loosely speaking, generated by crossings of a knot. We will introduce Clasp diagrams, which are an analogue of Gauss diagrams that instead of crossings exploit full twists on two strings. Two Clasp diagrams represent the same knot if and only if they can be transformed into each other by a sequence of moves from a well-defined set. We will show that the Clasp diagrams are a convenient tool to encode Seifert surfaces, calculate Alexander polynomials and study Vassiliev invariants of knots. The talk is based on the paper by Jacob Mostovoy and Michael Polyak (2019).
Video: https://www.youtube.com/watch?v=lpkH_UIs6JY

14 мая 2020
Elizaveta Shuvaeva
The Colored Jones Weight System
The colored Jones weight system on chord diagrams can be expressed in terms of the chord diagram's intersection matrix and in terms of its labeled intersection graph. We will speak about the formulas for these expressions and give the outlines of their proofs. The talk is based on the 2002 paper by S. Garoufalidis and M. Loebl.
Video: https://www.youtube.com/watch?v=I7MNVbYyF0E


7 мая 2020
Sergei Chmutov (Ohio State University, USA)
B-symmetric chromatic function of signed graphs.

First I review Stanley’s symmetric chromatic function of graphs and its appearance in Vassiliev knot invariants. In the second part of the talk I explain the generalization of this theory to signed graphs. These functions naturally possess the symmetry of B-type. Thus, the case of signed graphs can be considered as the B-type analog of the Stanley A-type symmetrical functions of usual graphs. This is a joint work with my students James Enouen,  Eric Fawcett, Rushil Raghavan, and Ishaan Shah.
Video: https://www.youtube.com/watch?v=khA7rP84sYY

30 апреля 2020
Bogdan Parshukov
Valuation of foams
We consider geometric-combinatorial objects called foams equipped with additional structures such as marked points and colorings. We introduce Robert-Wagner valuation invariant and show that it takes values in the ring of symmetric polynomials in three variables over the field F2 consisting of two elements. The talk will follow the paper https://arxiv.org/abs/1808.09662
Video: https://www.youtube.com/watch?v=rNRTGPGVTo4

23 апреля 2020
Sergei Lando
POLYAK-VIRO TYPE FORMULAS FOR VASSILIEV INVARIANTS

The distance between weight systems (= functions on chord diagrams satisfying 4-term relations) to finite type knot invariants is rather big. It requires computation of the so-called Kontsevich integral of a knot, which nobody knows how to compute. However, there is a way, due to M.Polyak and O.Viro, to construct finite type invariants of knots from their diagrams. This way expresses the value of a finite type invariant in terms of the Gauss diagram associated to the knot diagram. Although it becomes impractical for knot invariants of big order, for small orders it works well, and efficient formulas are known.
Video: https://www.youtube.com/watch?v=8XNv9WvD_6s

16 апреля 2020
Petr Dunin-Barkowski
Constructing the Jones polynomial of knots from quantum R-matrices
We consider the most basic example of the theory of quantum knot invariants: the case of the Jones polynomial, which corresponds to the standard two-dimensional representation of the sl(2) algebra. We discuss how to obtain the Jones polynomial from a solution (called the quantum R-matrix) of the quantum Yang-Baxter equation in this simplest case.
A good overview of this subject can be found in the "Introduction to Vassiliev Knot Invariants" textbook by Chmutov, Duzhin and Mostovoy (section 2.6).
Video: https://www.youtube.com/watch?v=1CK_Y4WiTNQ

9 апреля 2020
Sergei Lando
CUNNINGHAM’S DECOMPOSITION OF GRAPHS

In 1982, W. H. Cunnigham suggested a canonical decomposition of graphs into so-called prime graphs. For a connected graph with more than 5 vertices, this decomposition is unique, and it is closely related to the property of graphs to be intersection graphs of chord diagrams. In particular, a graph is an intersection graph if and only if each prime graph entering its canonical decomposition is one. In fact, if a prime graph is an intersection graph, then there is only one (up to reflection) chord diagram having it as the intersection graph, which allow the usage of Cunningham’s decomposition to describe all chord diagrams having a given graph as the intersection graph. Cunningham’s decomposition can be used to prove the following theorem (S.Chmutov, S.Lando, 2007) The value of the weight system associated to a Vassiliev knot invariant depends only on the intersection graph of a chord diagram rather than on the diagram itself if and only if this knot invariant cannot tell any knot from its mutant knot.(Mutation is an operation on knots, which consists in cutting off a ball from S3 intersecting the knot at 4 points and regluing it back after rotation.)

The main goal of the talk is approbation of techniques of distant seminar talks.
Video: https://www.youtube.com/watch?v=bzz1MdrdypI


12 марта 2020
Seminar Test

5 марта 2020
Dogra Riya
Acyclic orientation polynomials  (after B.H. Hwang, W.S. Jung, K.J. Lee, J. Oh, and S.H. Yu)
Continuation


27 февраля 2020
Dogra Riya
 
Acyclic orientation polynomials (after B.H. Hwang, W.S. Jung, K.J. Lee, J. Oh, and S.H. Yu)

We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at −1 up to sign. Motivated by this result, we develop “acyclic orientation” analogues for theorems concerning the chromatic polynomial by Birkhoff, Whitney, and Greene-Zaslavsky.

20 февраля 2020
No seminar

13 февраля 2020
Sergei Lando
Vassiliev knot invarinats and other knot invariants
Video: https://www.youtube.com/watch?v=8c6kWR0pvdc

6 февраля 2020
Dmitrii Zhurbenko
Graphs, links, and duality on surfaces (after V.Krushkal, the end of the talk on December 12, 2019)

I will talk about the polynomial associated to a graph on surface introduced by V.Krushkal. Both Tutte and Bollobash-Riordan polynomials can be obtained as its specializations; I will also consider duality relations for this polynomial and its connection with generalized Kauffman bracket. Definitions from the beginning of the talk will be recalled.
Video: https://www.youtube.com/watch?v=DPIxNh8MSUQ

30 января 2020 
Polina Filippova
Values of the sl2 weight system on some series of complete bipartite and split graphs

To each chord diagram its intersection graph is assigned. In 2007, S.V.Chmutov and S.K.Lando proved that value of the weight system associated to the Lie algebra sl2 on a chord diagram depends only on the intersection graph of this chord diagram, so we may speak about values of this weight system on intersection graphs.

A conjecture of S.K.Lando states that the value of the weight system sl2 on the projection of a chord diagram C to the space of primitive elements is a polynomial of degree k such that 2k is at most the number of vertices in the circumference of the intersection graph of this chord diagram. In particular, in the case of split graph Sm,n or complete bipartite graph Km,n, k is at most min(m,n).

We will discuss some results that confirm this conjecture. Namely, we will consider Hopf subalgebras generated by complete bipartite graphs Km,n, n = 1,2,3,… and by split graphs Sm,n, n = 1,2,3,… in cases m = 1,2,3. All the necessary definitions will be given. I will give equations which express exponential generating functions for projections of Km,n and of Sm,n, n = 1, 2, 3,... in cases m = 1,2,3 to the space of primitive elements through exponential generating functions for these graphs. Then I will discuss values of the weight system sl2on graphs Km,n and Sm,n, m = 1,2,3. Combining these results, we will get exponential and ordinary generating functions for the values of the weight system slon the projections of these graphs to the space of primitive elements. Finally I will say some words about a more general case. 

Video: https://www.youtube.com/watch?v=niy8kPq5Whw

23 января 2020
Nadezhda Kodaneva
Interlace polynomial for graphs and delta-matroids

In the talk, we will discuss the interlace polynomial for graphs and its generalization to delta-matroids. The interlace polynomial was introduced for a certain type of graphs and then for arbitrary graphs. It satisfies the 4-term relation for graphs and therefore determines a knot invariant. This result can be generalized to links by defining the interlace polynomial for binary delta-matroids and considering the corresponding 4-term relation. Binary delta-martoids span a Hopf algebra, and the dimension of the interlace polynomial's values on its subspace of primitive elements is n in degree n.
Video: https://www.youtube.com/watch?v=z2RMayPJ4FI




Архив семинара: https://math.hse.ru/combinv/seminar




 

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