arXiv subjects

Andrei Malyutin

Publications and source records attributed to Andrei Malyutin.

5 recordsLinked to original sources

arXiv2021

$3$-manifolds represented by $4$-regular graphs with three Eulerian cycles

We construct and study a new class $\mathscr{M}=\{\mathscr{M}_n\}_{n\ge 4}$ of compact hyperbolic $3$-manifolds with totally geodesic boundary. The members of $\mathscr{M}_n$ are defined via triples of pairwise compatible Eulerian cycles in $4$-regular $n$-vertex graphs. We show that each $M$ in $\mathscr{M}_n$ is of Matveev complexity $n$ and has a unique minimal ideal triangulation, which consists of $n$ tetrahedra. We exploit these properties to show that $n!\,4^n > |\mathscr{M}_n| > n!$ for each sufficiently large $n\in\mathbb{N}$.

math.GT↗

arXiv2019

Hyperbolic knots are not generic

We show that the proportion of hyperbolic knots among all of the prime knots of $n$ or fewer crossings does not converge to $1$ as $n$ approaches infinity. Moreover, we show that if $K$ is a nontrivial knot then the proportion of satellites of $K$ among all of the prime knots of $n$ or fewer crossings does not converge to $0$ as $n$ approaches infinity.

math.GT↗

arXiv2018

Meander diagrams of knots and spatial graphs: proofs of generalized Jablan--Radovi\'{c} conjectures

We study decomposition into simple arcs (i. e., arcs without self-intersections) 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. Radovi\'{c} 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 Radovi\'{c} stating that each 2-bridge knot has a semi-meander minimal diagram, i. e., a minimal diagram composed of two simple arcs.

math.GT↗

arXiv2017

Boundaries of $\mathbb{Z}^n$-free groups

In this paper we study random walks on a finitely generated group $G$ which has a free action on a $\mathbb{Z}^n$-tree. We show that if $G$ is non-abelian and acts minimally, freely and without inversions on a locally finite $\mathbb{Z}^n$-tree $Γ$ with the set of open ends ${\rm Ends}(Γ)$, then for every non-degenerate probability measure $μ$ on $G$ there exists a unique $μ$-stationary probability measure $ν_μ$ on ${\rm Ends}(Γ)$, and the space $({\rm Ends}(Γ), ν_μ)$ is a $μ$-boundary. Moreover, if $μ$ has finite first moment with respect to the word metric on $G$ (induced by a finite generating set), then the measure space $({\rm Ends}(Γ), ν_μ)$ is isomorphic to the Poisson--Furstenberg boundary of $(G, μ)$.

math.GR↗

arXiv2016

On the question of genericity of hyperbolic knots

A well-known conjecture in knot theory says that the percentage of hyperbolic knots amongst all of the prime knots of $n$ or fewer crossings approaches $100$ as $n$ approaches infinity. In this paper, 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.

math.GT↗