Семинар "Комбинаторика инвариантов Васильева"
16 декабря 2021
Dmitry Chebasov
The Descendant colored Jones polynomial
The colored Jones polynomial is a generalization of the well-known Jones polynomial. We shall review a realization of colored Jones polynomial that arose in the recent work of D. Zagier and S. Garoufalidis. More precisely, a colored Jones polynomial J_n(q) is a family of polynomials parametrized by n. We generalize it further to the descendant colored Jones polynomial. It is a two-parameter family of polynomials from which we will obtain J_n(q) and other related invariants.
Video
9 декабря 2021
Vladislav Pokidkin
A new approach to quantum knot invariants
There is an algebraic approach to constructing universal quantum knot invariants via generating function technique. We consider the construction for Heisenberg algebra and the link of the obtained invariants with the equivalence classes of tangle diagrams by Reidemeister moves. The same method applies to some ribbon Hopf algebra to build the universal invariant, connected with the equivalence classes of rotational tangle diagrams by rotational Reidemeister moves. The main theorem claims that the universal invariant determines the universal quantum SL(2) invariant and, hence, all colored Jones polynomials.
Video
2 декабря 2021
Grisha Taroyan
Khovanov homology and categorification of the Jones polynomial
One can functorially associate a bigraded complex C(D) of abelian groups to a planar diagram D of a link L. Complex C(D) can be shown to be independent up quasi-isomorphism from a specific planar projection of the link L. Thus homology groups H(L) of the complex C(L) yield a new invariant of the link L known as Khovanov homology. A celebrated result of Khovanov is the following
Theorem. The graded Poincare polynomial Kh(L)(t,q)=Σ t^i q^j dim H^{i,j}(L) is a link invariant which specifies to the unnormalized Jones polynomial of the link L at t=-1.
The bulk of my talk will be dedicated to proving invariance of C(L) under Reidemeister moves of planar diagrams. Then I will prove the theorem stated above. Time permitting I will also talk about functoriality of H(L) with respect to link cobordism and Khovanov homology as 4-dimensional TQFT.
The talk follows several papers. The first part can be deduced from [1]. The original paper of Khovanov [2] is also an invaluable source in understanding the theory although a slightly more difficult and unpolished one. Finally, Jacobsson's paper provides a necessary background and proof of Khovanov homology's invariance under link cobordism. Definition of 4-dimensional TQFT can be found in standard sources like the following entry on nLab [4]. References: [1] Bar-Natan, Dror. “On Khovanov’s categorification of the Jones polynomial.” Algebraic & Geometric Topology 2.1 (2002): 337-370. [2] Khovanov, Mikhail. “A categorification of the Jones polynomial.” Duke Mathematical Journal 101.3 (2000): 359-426. [3] Jacobsson, Magnus. “An invariant of link cobordisms from Khovanov homology.” Algebraic \& Geometric Topology 4.2 (2004): 1211-1251. [4] nLab authors. “4d-TQFT.” (November 2021).
Video25 ноября 2021
Raisa Sofronova
Twisted Neumann–Zagier Matrices and twisted 1-loop invariant
(After the paper "Twisted Neumann–Zagier Matrices" by S. Garoufalidis and S. Yoon)
Neumann–Zagier matrices encode information about ideal triangulations of 3-manifolds and their gluing equations. They have some remarkable properties which allow one to construct quantum invariants of these manifolds.
Garoufalidis and Yoon wanted to study these invariants under cyclic cover and the natural way is to define twisted NZ matrices - NZ matrices of infinite cyclic cover. In this talk all required definitions will be given, main results stated and an example of 1-loop invariant of a hyperbolic knot will be computed.
Video
18 ноября 2021
Yury Belousov
On enumeration of meanders
A meander is an open curve on the plane that intersects a given line transversally at a finite number of points. There is a deep connection between meanders and Temperley-Lieb algebras, statistical physics models, and moduli spaces of complex curves. Despite the high interest in this area, the central questions remain open. The number of meanders with a given number of intersections, as well as the asymptotic behaviour of these numbers are unknown. We are going to discuss some basic facts about meanders, in particular the problems of meander enumeration. We also will describe the recently discovered geometric decomposition of the meanders into irreducible components. This decomposition leads us to a new approach to the meander enumeration problem.
Video
11 ноября 2021
Svetlana Gavrilova
A-polynomials for infinite families of knots
In order to find an effective way to compute the A-polynomial, which is a powerful knot invariant, we consider a family ofknots which can be obtained by Dehn filling of a 2-component link. After certain change of variables our problem can bereduced to the problem of solving equations of degree two. It turns out that these equations, up to sign, are the equationsbetween cluster variables in the cluster algebra. Also, the combinatorial construction of perfect matchings of weighted laddergraphs allows one to simplify the computations. The talk is based on the paper Twisting, ladder graphs and A-polynomials by Emily K. Thompson.
Video
28 октября 2021
Alexander Dunaykin
A-polynomial from a manifold with a torus boundary
We present the notion of an A-polynomial that is a polynomial in two variables associated to a compact 3-manifold with boundary consisting of a single torus. We follow the article "Plane Curves Associated to Character Varieties of 3-Manifolds" by D.Cooper, M. Culler, H. Gillet, D. D. Long and P. B. Shalen
Video
14 октября 2021
Boris Bychkov
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, hence 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
7 октября 2021
Boris Bychkov
Algebraic structures associated to chord diagrams and graphs
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
30 сентября 2021
Polina Zakorko
An algorithm for calculating the values of the sl(2)-weight system on chord diagrams whose intersection graphs are complete graphs
A special case of a weight system on chord diagrams is the sl(2)-weight system. In addition to 4-term relations, it satisfies so-called 6-term relations. Algorithms based on the 6-term relations in practice compute only values on chord diagrams of small order and for some simple sequences of diagrams, due to the rapid growth of the number of intermediate chord diagrams. S.K.Lando formulated a conjecture about the form of the values of the sl(2)-weight system on chord diagrams with complete intersection graphs. We suggest a simple inductive algorithm for calculating these values, which is based on calculation of values on very few intermediate chord diagrams. The results agree with the predictions of Lando’s conjecture.
Video
23 сентября 2021
Sergei Lando
Graph invariants and weight systems
If a graph invariant satisfies 4-term relations for graphs, then it defines a weight system: a function on chord diagrams satisfying 4-term relations. The converse is not true generically. The talk will present examples of graph invariants satisfying 4-term relations. We will also discuss the weight system associated to the Lie algebra sl(2) and its presumable relationship with graph invariants. This talk is a preparatory one for the next week talk by Polina Zakorko.
Video
16 сентября 2021
Sergei Lando
Main heroes in the combinatorics of knot invariants
The talk will be devoted to a description of objects studied by the seminar (knots, graphs, embedded graphs, knot diagrams,delta-matroids, and their invariants) and various relations between them.
Video
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.
Video
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,b2 in B for each e1 in b1 \ b2 there exists e2 in b2 \ b1 such that (b1 \ {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.
Video
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.
Video
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
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
Raisa Safronova
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
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 sl2 weight 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 sl2 at 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
Video on youtube
21 января 2021
Video on youtube
17 декабря 2020
Grigorii Yurgin
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
Vadim Retinskii
On sequences of polynomials arising from graph invariants
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
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
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
Video on youtube.
17 сентября 2020
Sergei Lando
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.
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
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
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
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 sl2 on 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=niy8kPq5Whw23 января 2020
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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