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Papers by Andrei Malyutin

arXiv: Geometric Topology, 2019

We show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of th... more We show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of the prime non-split links of nnn or fewer crossings does not converge to 000 as nnn approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of nnn or fewer crossings does not converge to 1 as nnn approaches infinity. We consider unoriented link types.

arXiv: Geometric Topology, 2019

We show that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer cros... more We show that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer crossings does not converge to 111 as nnn approaches infinity. Moreover, we show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of the prime knots of nnn or fewer crossings does not converge to 000 as nnn approaches infinity.

In this paper we study random walks on a f.g. group G which has a free action on a Z n-tree. We s... more In this paper we study random walks on a f.g. group G which has a free action on a Z n-tree. We show that if ΓG is a universal Z n-tree associated with G and ∂nΓG is the set of all ends of Z n-type of ΓG then for every non-degenerate probability measure µ on G and any µ-stationary measure ν, the space (∂nΓG, ν) is a µ-boundary. Moreover, if µ has finite first moment with respect to a finite word metric on G, then the measure space (∂nΓG, ν) is the Poisson(-Furstenberg) boundary of (G, µ).

We construct and study a new class of compact hyperbolic 3-manifolds with totally geodesic bounda... more We construct and study a new class of compact hyperbolic 3-manifolds with totally geodesic boundary. This class extends the line of research developed in [1, 2, 3, 4] and exhibits a number of remarkable properties. The members of this class are described by triples of Eulerian cycles in 4-regular graphs. Two Eulerian cycles are said to be compatible if no pair of adjacent edges are consecutive in both cycles. If a 4-regular graph G contains a triple θ of pairwise compatible Eulerian cycles, we say that G is 3-Eulerian and θ is a framing of G. Each finite 3-vertex-connected simple 4-regular graph is 3-Eulerian [5]. Let G be a 3-Eulerian graph, and let θ be a framing of G. A polyhedral realization of the pair (G, θ) is a 2-dimensional polyhedron P (G, θ) obtained from G by attaching a 2-cell along each cycle in θ.

International Mathematics Research Notices

A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of ... more A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer crossings approaches 111 as nnn approaches infinity. In this article, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.

Функциональный анализ и его приложения

Trudy Matematicheskogo Instituta imeni V.A. Steklova

В.М. Бухштабер, О.В. Карпов и С.И. Тертычный инициировали изучение эффекта целочисленного квантов... more В.М. Бухштабер, О.В. Карпов и С.И. Тертычный инициировали изучение эффекта целочисленного квантования числа вращения для класса динамических систем на торе, включающего динамические системы, моделирующие динамику джозефсоновского перехода. По аналогии с этим эффектом в настоящей работе инициировано изучение схожего эффекта квантования числа вращения для класса групп, действующих на окружности, включая группы кос Артина.

European Journal of Mathematics

In this paper we study random walks on a f.g. group GGG which has a free action on a Zn\Z^nZn-tree.... more In this paper we study random walks on a f.g. group GGG which has a free action on a Zn\Z^nZn-tree. We show that if GammaG\Gamma_GGammaG is a universal Zn\Z^nZn-tree associated with GGG and partialnGammaG\partial_n \Gamma_GpartialnGammaG is the set of all ends of Zn\Z^nZn-type of GammaG\Gamma_GGammaG then for every non-degenerate probability measure mu\mumu on GGG and any mu\mumu-stationary measure nu\nunu, the space (partialnGammaG,nu)(\partial_n \Gamma_G, \nu)(partialnGammaG,nu) is a mu\mumu-boundary. Moreover, if mu\mumu has finite first moment with respect to a finite word metric on GGG, then the measure space (partialnGammaG,nu)(\partial_n \Gamma_G, \nu)(partialnGammaG,nu) is the Poisson(--Furstenberg) boundary of (G,mu)(G, \mu)(G,mu).

Функциональный анализ и его приложения, 2015

Фазовый переход в задаче о границе-выход для случайных блужданий на группах * c 2015. A. М. Верши... more Фазовый переход в задаче о границе-выход для случайных блужданий на группах * c 2015. A. М. Вершик, А. В. Малютин Памяти Е. Б. Дынкина В работе описывается полная граница-выход случайных блужданий на однородных деревьях, в том числе на свободных группах. В этой модели существует фазовый переход, состоящий в потере эргодичности семейства марковских мер при изменении параметра случайного блуждания. Рассматриваемая проблема является частным случаем задачи об инвариантных (центральных) мерах на графах ветвления, охватывающей целый ряд проблем комбинаторики, теории представлений, теории вероятностей и в полном объеме поставленной в серии недавних работ первого автора [1]-[3]. С другой стороны, близкие задачи в контексте теории марковских процессов обсуждались еще в 60-х годах Е. Б. Дынкиным. * Работа поддержана грантом РНФ 14-11-00581. 1) По терминологии этих работ-пространств выходов (входов).

Известия Российской академии наук. Серия математическая, 2008

Topology and its Applications

We study decomposition into simple arcs (i. e., arcs without selfintersections) for diagrams of k... more We study decomposition into simple arcs (i. e., arcs without selfintersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph G is a knotted loop, then there exists a plane diagram D of G such that (i) each edge of G is represented by a simple arc of D and (ii) each vertex of G is represented by a point on the boundary of the convex hull of D. This generalizes the conjecture of S. Jablan and L. Radović stating that each knot has a meander diagram, i. e., a diagram composed of two simple arcs whose common endpoints lie on the boundary of the convex hull of the diagram. Also, we prove another conjecture of Jablan and Radović stating that each 2-bridge knot has a semimeander minimal diagram, i. e., a minimal diagram composed of two simple arcs.

arXiv: Geometric Topology, 2019

We show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of th... more We show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of the prime non-split links of nnn or fewer crossings does not converge to 000 as nnn approaches infinity. This implies in particular that the proportion of hyperbolic links among all of the prime non-split links of nnn or fewer crossings does not converge to 1 as nnn approaches infinity. We consider unoriented link types.

arXiv: Geometric Topology, 2019

We show that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer cros... more We show that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer crossings does not converge to 111 as nnn approaches infinity. Moreover, we show that if KKK is a nontrivial knot then the proportion of satellites of KKK among all of the prime knots of nnn or fewer crossings does not converge to 000 as nnn approaches infinity.

In this paper we study random walks on a f.g. group G which has a free action on a Z n-tree. We s... more In this paper we study random walks on a f.g. group G which has a free action on a Z n-tree. We show that if ΓG is a universal Z n-tree associated with G and ∂nΓG is the set of all ends of Z n-type of ΓG then for every non-degenerate probability measure µ on G and any µ-stationary measure ν, the space (∂nΓG, ν) is a µ-boundary. Moreover, if µ has finite first moment with respect to a finite word metric on G, then the measure space (∂nΓG, ν) is the Poisson(-Furstenberg) boundary of (G, µ).

We construct and study a new class of compact hyperbolic 3-manifolds with totally geodesic bounda... more We construct and study a new class of compact hyperbolic 3-manifolds with totally geodesic boundary. This class extends the line of research developed in [1, 2, 3, 4] and exhibits a number of remarkable properties. The members of this class are described by triples of Eulerian cycles in 4-regular graphs. Two Eulerian cycles are said to be compatible if no pair of adjacent edges are consecutive in both cycles. If a 4-regular graph G contains a triple θ of pairwise compatible Eulerian cycles, we say that G is 3-Eulerian and θ is a framing of G. Each finite 3-vertex-connected simple 4-regular graph is 3-Eulerian [5]. Let G be a 3-Eulerian graph, and let θ be a framing of G. A polyhedral realization of the pair (G, θ) is a 2-dimensional polyhedron P (G, θ) obtained from G by attaching a 2-cell along each cycle in θ.

International Mathematics Research Notices

A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of ... more A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of nnn or fewer crossings approaches 111 as nnn approaches infinity. In this article, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.

Функциональный анализ и его приложения

Trudy Matematicheskogo Instituta imeni V.A. Steklova

В.М. Бухштабер, О.В. Карпов и С.И. Тертычный инициировали изучение эффекта целочисленного квантов... more В.М. Бухштабер, О.В. Карпов и С.И. Тертычный инициировали изучение эффекта целочисленного квантования числа вращения для класса динамических систем на торе, включающего динамические системы, моделирующие динамику джозефсоновского перехода. По аналогии с этим эффектом в настоящей работе инициировано изучение схожего эффекта квантования числа вращения для класса групп, действующих на окружности, включая группы кос Артина.

European Journal of Mathematics

In this paper we study random walks on a f.g. group GGG which has a free action on a Zn\Z^nZn-tree.... more In this paper we study random walks on a f.g. group GGG which has a free action on a Zn\Z^nZn-tree. We show that if GammaG\Gamma_GGammaG is a universal Zn\Z^nZn-tree associated with GGG and partialnGammaG\partial_n \Gamma_GpartialnGammaG is the set of all ends of Zn\Z^nZn-type of GammaG\Gamma_GGammaG then for every non-degenerate probability measure mu\mumu on GGG and any mu\mumu-stationary measure nu\nunu, the space (partialnGammaG,nu)(\partial_n \Gamma_G, \nu)(partialnGammaG,nu) is a mu\mumu-boundary. Moreover, if mu\mumu has finite first moment with respect to a finite word metric on GGG, then the measure space (partialnGammaG,nu)(\partial_n \Gamma_G, \nu)(partialnGammaG,nu) is the Poisson(--Furstenberg) boundary of (G,mu)(G, \mu)(G,mu).

Функциональный анализ и его приложения, 2015

Фазовый переход в задаче о границе-выход для случайных блужданий на группах * c 2015. A. М. Верши... more Фазовый переход в задаче о границе-выход для случайных блужданий на группах * c 2015. A. М. Вершик, А. В. Малютин Памяти Е. Б. Дынкина В работе описывается полная граница-выход случайных блужданий на однородных деревьях, в том числе на свободных группах. В этой модели существует фазовый переход, состоящий в потере эргодичности семейства марковских мер при изменении параметра случайного блуждания. Рассматриваемая проблема является частным случаем задачи об инвариантных (центральных) мерах на графах ветвления, охватывающей целый ряд проблем комбинаторики, теории представлений, теории вероятностей и в полном объеме поставленной в серии недавних работ первого автора [1]-[3]. С другой стороны, близкие задачи в контексте теории марковских процессов обсуждались еще в 60-х годах Е. Б. Дынкиным. * Работа поддержана грантом РНФ 14-11-00581. 1) По терминологии этих работ-пространств выходов (входов).

Известия Российской академии наук. Серия математическая, 2008

Topology and its Applications

We study decomposition into simple arcs (i. e., arcs without selfintersections) for diagrams of k... more We study decomposition into simple arcs (i. e., arcs without selfintersections) for diagrams of knots and spatial graphs. In this paper, it is proved in particular that if no edge of a finite spatial graph G is a knotted loop, then there exists a plane diagram D of G such that (i) each edge of G is represented by a simple arc of D and (ii) each vertex of G is represented by a point on the boundary of the convex hull of D. This generalizes the conjecture of S. Jablan and L. Radović stating that each knot has a meander diagram, i. e., a diagram composed of two simple arcs whose common endpoints lie on the boundary of the convex hull of the diagram. Also, we prove another conjecture of Jablan and Radović stating that each 2-bridge knot has a semimeander minimal diagram, i. e., a minimal diagram composed of two simple arcs.