Ivan Dynnikov

Ivan Dynnikov
Lomonosov Moscow State University | MSU · Division of Mathematics

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107
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Publications

Publications (107)
Article
At the beginning of the 80s, H. Masur and W. Veech started the study of generic properties of interval exchange transformations (IETs) proving that almost every such transformation is uniquely ergodic. About the same time, S. Novikov’s school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliat...
Article
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the composition by an exotic matrix multiplication. An efficient solution of the word problem for mapping class grou...
Preprint
Full-text available
We consider here quasiperiodic potentials on the plane, which can serve as a "transitional link" between ordered (periodic) and chaotic (random) potentials. As can be shown, in almost any family of quasiperiodic potentials depending on a certain set of parameters, it is possible to distinguish a set (in the parameter space) where, according to a ce...
Article
We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of “simpler” moves not increasing the complexity of the diagra...
Article
In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere S3 by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to prove a Reidemeister type theorem for rectangular diagrams of surfaces.
Preprint
At the beggining of the 80's, H.Masur and W.Veech started the study of generic properties of interval exchange transformations proving that almost every such transformation is uniquely ergodic. About the same time, S.Novikov's school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on s...
Preprint
In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to prove a Reidemeister type theorem for rectangular diagrams of surfaces.
Preprint
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the composition by an exotic matrix multiplication.
Preprint
We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of `simpler' moves not increasing the complexity of the diagra...
Preprint
In recent joint works of the present author with M.Prasolov and V.Shastin a new technique for distinguishing Legendrian knots has been developed. In this paper the technique is extended further to provide a tool for distinguishing transverse knots. It is shown that the equivalence problem for transverse knots with trivial orientation-preserving sym...
Article
We classify Legendrian knots of topological type [Formula: see text] having maximal Thurston–Bennequin number confirming the corresponding conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997].
Preprint
We classify Legendrian knots of topological type $7_6$ having maximal Thurston--Bennequin number confirming the corresponding conjectures of Chongchitmate--Ng.
Preprint
In a recent work of I.\,Dynnikov and M.\,Prasolov a new method of comparing Legendrian knots is proposed. In general, to apply the method requires a lot of technical work. In particular, one needs to search all rectangular diagrams of surfaces realizing certain dividing configurations. In this paper, it is shown that, in the case when the orientati...
Article
It is shown that, for the discretization of complex analysis introduced earlier by S. P. Novikov and the present author, there exists a rich family of bounded discrete holomorphic functions on the hyperbolic (Lobachevsky) plane endowed with a triangulation by regular triangles whose vertices have even valence. Namely, it is shown that every discret...
Article
In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce moves that allow transition between diagrams representing isotopic surfaces. We also introduce more general com...
Article
We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on $\mathbb S^3$ and to rectangular diagrams of links. By using rectangular diagrams of surfaces we are going, in particular, to develop a method...
Article
Full-text available
It is known since 40 years old paper by M. Keane that minimality is a generic (i.e. holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this...
Article
Full-text available
In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution and have the smallest possible rank. This allows for constructing a 3-periodic surface in the three-space wit...
Article
This paper develops an approach to discretization of complex analysis proposed by S.P. Novikov and the author in 2003. Under this approach discrete analytic functions are real-valued. It is shown that a large class of such functions on a lattice admits a canonical multiplication by the imaginary unit. Arbitrary lattices are considered for a triangu...
Article
It is proved that the pseudocharacter defined on the braid group by the signature of braid closures is linearly independent of all pseudocharacters obtained from the twist number via the Malyutin operators, provided that the number of strands is greater than 4. This pseudocharacter is shown to have a nontrivial kernel part. It is observed that the...
Article
Full-text available
It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is self-similar, then a typical leaf has exactly one topological end. We also construct the first example of a f...
Article
Full-text available
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the monotonic simplification theorem for the unknot is given. It is shown that a minimal rectangular diagram maxi...
Article
Full-text available
The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In this paper we present in detail...
Article
Interval identification systems is a notion that, on the one hand, generalizes interval exchange transformations and, on the other hand, describes special cases of such transformations. In the present paper we overview some elementary facts, address a few questions about interval identification systems, and describe explicitly systems that allow on...
Article
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Article
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In this paper the topological theory of quasi-periodic functions on the plane is presented. The development of this theory was started (in another terminology) by the Moscow topology group in the early 1980s, motivated by needs of solid state physics which led to the necessity of investigating a special (non-generic) case of Hamiltonian foliations...
Article
In this paper the problem of constructing algorithms for comparing knots and links is discussed. A survey of existing approaches and basic results in this area is given. In particular, diverse combinatorial methods for representing links are discussed, the Haken algorithm for recognizing a trivial knot (the unknot) and a scheme for constructing a g...
Article
Full-text available
The article describes a topological theory of quasiperiodic functions on the plane. The development of this theory was started (in different terminology) by the Moscow topology group in early 1980s. It was motivated by the needs of solid state physics, as a partial (nongeneric) case of Hamiltonian foliations of Fermi surfaces with multivalued Hamil...
Article
We construct finitely presented semigroups whose center is identified with the set of all isotopy classes of oriented links. Previously, similar semigroups were constructed for nonoriented links.
Article
Full-text available
We define a measure of "complexity" of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators $\Delta\_{ij}$, which are Garside-like half-twists involving strings $i$ through $j$, and by counting powered generators $\De...
Article
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In this paper we present the implementation of a partial knot recognition algorithm as a mathematical web service on the internet. Knots may interactively be loaded and edited, and then checked for being unknottet.
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A non-traditional approach to the discretization of differential-geometrical connections was suggested by the authors in 1997. At the same time we started studying first order difference ``black and white triangle operators (equations)'' on triangulated surfaces with a black and white coloring or triangles. In this work, we develop the theory of th...
Article
Full-text available
In the early 90's J. Birman and W. Menasco worked out a nice technique for studying links presented in the form of a closed braid. The technique is based on certain foliated surfaces and uses tricks similar to those that were introduced earlier by D. Bennequin. A few years later P. Cromwell adapted Birman-Menasco's method for studying so-called arc...
Article
Full-text available
A difference q-analogue of the dressing chain is discussed in this paper.
Article
An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. The group presented by two generators and three relations [[a,b],a 2ba −2]=[[a,b],b 2ab −2]=[[a,b],[a −1,b −1...
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Contents §1. Introduction: history of the problem, the one-dimensional Schrödinger operator §2. The non-stationary one-dimensional Schrödinger equation §3. One-dimensional difference operators §4. The Laplace transformation and the two-dimensional Schrödinger operator in a magnetic field §5. Difference equations and Laplace transformations. The hyp...
Article
An explicit construction of a finitely presented semigroup whose central elements are in a one-to-one correspondence with the isotopy classes of unoriented links in the three-space is given, together with a finite presentation for the group of invertible elements of the semigroup. The group is presented by two generators and three relations. The co...
Article
The trefoil knot is the simplest non-trivial knot. Its simplest planar diagram has 3 crossings. A list of other knots with small crossing number can be found in [D. Rolfsen, Knots and links, Publish or Perish, Houston (1990; Zbl 0339.55004), 2nd ed. (1990; Zbl 0854.57002); C. Cerf, Atlas of oriented knots and links, http://at.yorku.ca/t/a/i/c/31.di...
Article
In the present paper, we suggest a new combinatorial approach to knot theory based on embeddings of knots and links into a union of three half-planes with the same boundary. The idea to embed knots into a “book” is quite natural and was considered already in [1]. Among recent papers on embeddings of knots into a book with infinitely many pages, we...
Article
The structure of nonclosed trajectories of semiclassical electron motion in a crystal in a weak constant and uniform magnetic field of irrationality degree 3 is considered. It is proved that two cases can exist. In the first case the set of energy levels which contain nonclosed trajectories is a closed interval and any regular nonclosed trajectory...
Article
Contents § 1. Introduction1.1. Statement of results § 2. The algebraic density of closed trajectories and the Euler characteristic2.1. Types of trajectories. Degree of irrationality2.2. Stability of closed trajectories2.3. The algebraic density of closed trajectories2.4. The heights of critical points2.5. The heights of cylinders of closed trajecto...
Article
The paper considers the topological characteristics of dispersion functions ɛs(p) in energy bands in single crystals related to classical electron trajectories in uniform magnetic fields. Specifically, the topological properties of open trajectories in p-space on various energy levels within one energy band and related physical effects are describe...
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The definition of an invariant of finite order for links devised by V. A. Vassiliev [Adv. Sov. Math. 1, 23-69 (1990; Zbl 0727.57008)] has led to the construction of a beautiful theory that includes almost all the known polynomial invariants. D. Bar-Natan [On the Vassiliev knot invariants, Topology 34, No. 2, 423-472 (1995)] and J. S. Birman and X.-...
Article
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Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the homology requires, and the corre...

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