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Vladimir Chernov

Publications and source records attributed to Vladimir Chernov.

At least 19 recordsLinked to original sources
arXiv2026

Detecting Causality with the Links--Gould Polynomial

The conjectures of Low and Natario--Tod, and Penrose's question on Arnold's Problem list ask if causality in spacetimes can be formulated in terms of linking of spheres of light rays in the manifold of all light rays. For $(2+1)$-dimensional spacetimes, this link happens in the manifold coverable by a solid torus $S^1\times \mathbb R^2$. This was solved positively by Chernov and Nemirovski, which raises the question of which link invariants can be used to study causality. Chernov, Martin and Petkova proved that Heegaard--Floer and Khovanov homology completely capture causality. Allen--Swenberg conjectured that the Jones polynomial, which is obtained as an alternating Euler characteristic from Khovanov homology, is also sufficient. But they constructed complicated examples of links $\mathrm{AS}(n)_{n=1}^{\infty}$ that suggest that the Alexander--Conway polynomial -- which is the Euler characteristic of Heegaard--Floer homology -- is not enough. The Links--Gould polynomial is a quantum invariant that specializes to the classical Alexander--Conway polynomial in two different ways and somewhat surprisingly inherits some of its characteristic classical features. We show that it distinguishes all the Allen-Swenberg links from the link of causally unrelated events and hence detects causality in all known examples where the Alexander--Conway polynomial is not sufficient. This suggests that it may completely capture causality. The work on the categorification of the Links--Gould Polynomial is an ongoing and hard problem, and it is not a subject of this paper. As a corollary, we also compute the Seifert genus of all Allen--Swenberg links.

math.GT↗
arXiv2025

Pseudo-Legendrian and Legendrian Simplicity of Links in 3-Manifolds

We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and have Legendrian-isotopic components and (2) a pair of Legendrian links that are not Legendrian isotopic, but are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and which are link-homotopic as Legendrian links. Moreover, we construct examples showing that both of these non-simplicity phenomena can occur in the same smooth isotopy class. To construct these examples, we develop the theory of links transverse to a nowhere-zero vector field in a 3-manifold, and construct analogous examples in the category of links transverse to a vector field.

math.GT↗
arXiv2025

Manturov Projection for Virtual Legendrian Knots in $ST^*F$

Kauffman virtual knots are knots in thickened surfaces $F\times R$ considered up to isotopy, stabilizations and destabilizations, and diffeomorphisms of $F\times R$ induced by orientation preserving diffeomorphisms of $F$. Similarly, virtual Legendrian knots, introduced by Cahn and Levi~\cite{CahnLevi}, are Legendrian knots in $ST^*F$ with the natural contact structure. Virtual Legendrian knots are considered up to isotopy, stabilization and destabilization of the surface away from the front projection of the Legendrian knot, as well as up to contact isomorphisms of $ST^*F$ induced by orientation preserving diffeomorphisms of $F$. We show that there is a projection operation $proj$ from the set of virtual isotopy classes of Legendrian knots to the set of isotopy classes of Legendrian knots in $ST^*S^2$. This projection is obtained by substituting some of the classical crossings of the front diagram for a virtual crossing. It restricts to the identity map on the set of virtual isotopy classes of classical Legendrian knots. In particular, the projection $proj$ extends invariants of Legendrian knots to invariants of virtual Legendrian knots. Using the projection $proj$, we show that the virtual crossing number of every classical Legendrian knot equals its crossing number. We also prove that the virtual canonical genus of a Legendrian knot is equal to the canonical genus. The construction of $proj$ is inspired by the work of Manturov.

math.SG↗
arXiv2024

Conditions when the problems of linear programming are algorithmically unsolvable

We study the properties of the constructive linear programing problems. The parameters of linear functions in such problems are constructive real numbers. To solve such a problem is to find the optimal plan with the constructive real number components. We show that it is impossible to have an algorithm that solves an arbitrary constructive real programming problem.

math.OC↗
arXiv2024

Affine linking number estimates for the number of times an observer sees a star

Affine linking numbers are the generalization of linking numbers to the case of nonzero homologous linked submanifolds. They were introduced by Rudyak and the first author who used them to study causality in globally hyperbolic spacetimes. In this paper we use affine linking numbers to estimate the number of times an observer sees light from a star, that is how many copies of the star do they see on the sky due to gravitational lensing.

gr-qc↗
arXiv2023

Conjectures about virtual Legendrian knots and links

We formulate conjectures generalizing some known results to the category of virtual Legendrian knots. This includes statements relating virtual Legendrian knots to ordinary Legendrian knots, non-existence of positive virtual Legendrian self isotopy for the class of the fiber of $ST^*M$ and the conjectural relation of virtual Legendrian isotopy to causality in generalized spacetimes. We prove the conjectures in the case of $2$-dimensional $M$ and $(2+1)$-dimensional spacetimes. We also formulate and prove the version of the Arnold's $4$ cusp conjecture for virtual isotopies.

math.GT↗
arXiv2022

Conjectures on the Khovanov Homology of Torus Knots, Twist Knots, and Legendrian Simple Knots

A theorem of Kronheimer and Mrowka states that Khovanov homology is able to detect the unknot. That is, if a knot has the Khovanov homology of the unknot, then it is equivalent to it. Similar results hold for the trefoils and the figure-eight knot. We conjecture that Khovanov homology is able to distinguish all torus and twist knots. Numerical evidence has been gathered by examining all prime knots with 20 or fewer crossings, a total of 2,199,471,680 knots (not including mirrors). We found that all knots with the same Khovanov polynomial (the Poincar\'{e} polynomial of Khovanov homology) as a torus or twist knot are indeed torus or twist knots themselves. Since torus knots are known to be Legendrian simple, and since all twist knots $K_{m}$ with $m\geq{-3}$ are Legendrian simple, this provides evidence for the claim that Khovanov homology and Legendrian simplicity may be connected. We conjecture that indeed Khovanov homology is able to distinguish Legendrian simple knots and use the (conjectured) Legendrian simple knots from the Legendrian knot atlas to test this claim. A similar observation was made, and no knots with 20 or fewer crossing share their Khovanov polynomial with the knots in the Legendrian knot atlas (except for the knots that are a part of this atlas).

math.GT↗
arXiv2021

Graded Poisson algebras on bordism groups of garlands

Let $M$ be an oriented manifold and let $\frak N$ be a set consisting of oriented closed manifolds of the same odd dimension. We consider the topological space $G_{\frak N, M}$ of commutative diagrams. Each commutative diagram consists of a few manifolds from $\frak N$ that are mapped to $M$ and a few one point spaces $pt$ that are each mapped to a pair of manifolds from $\frak N$. We consider the oriented bordism group $Ω_*(G_{\frak N, M})=\oplus_{i=0}^{\infty} Ω_i(G_{\frak N, M}).$ We introduce the operations $\star$ and $[\cdot, \cdot]$ on $Ω_*(G_{\frak N,M})\otimes \mathbb Q,$ that make $Ω_*(G_{\frak N,M})\otimes \mathbb Q$ into a graded Poisson algebra (Gerstenhaber-like algebra). For ${\frak N}=\{S^1\}$ and a surface M=$F^2,$ the subalgebra $Ω_0(G_{\{S^1\}, F^2})\otimes \mathbb Q$ of our algebra is related to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams.

math.GT↗
arXiv2019

Loose Legendrian and Pseudo-Legendrian Knots in 3-Manifolds

We prove a complete classification theorem for loose Legendrian knots in an oriented 3-manifold, generalizing results of Dymara and Ding-Geiges. Our approach is to classify knots in a $3$-manifold $M$ that are transverse to a nowhere-zero vector field $V$ up to the corresponding isotopy relation. Such knots are called $V$-transverse. A framed isotopy class is simple if any two $V$-transverse knots in that class which are homotopic through $V$-transverse immersions are $V$-transverse isotopic. We show that all knot types in $M$ are simple if any one of the following three conditions hold: $1.$ $M$ is closed, irreducible and atoroidal; or $2.$ the Euler class of the $2$-bundle $V^{\perp}$ orthogonal to $V$ is a torsion class, or $3.$ if $V$ is a coorienting vector field of a tight contact structure. Finally, we construct examples of pairs of homotopic knot types such that one is simple and one is not. As a consequence of the $h$-principle for Legendrian immersions, we also construct knot types which are not Legendrian simple.

math.GT↗
arXiv2019

Interval topology in contact geometry

A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.

math.SG↗
arXiv2018

Minimizing intersection points of curves under virtual homotopy

A flat virtual link is a finite collection of oriented closed curves $\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersection points of curves in the virtual homotopy class of $(L_1, L_2)$ equals to the number of terms of a generalization of the Anderson--Mattes--Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.

math.GT↗
arXiv2018

Causality and Legendrian linking for higher dimensional spacetimes

Let $(X^{m+1}, g)$ be an $(m+1)$-dimensional globally hyperbolic spacetime with Cauchy surface $M^m$, and let $\widetilde M^m$ be the universal cover of the Cauchy surface. Let $\mathcal N_{X}$ be the contact manifold of all future directed unparameterized light rays in $X$ that we identify with the spherical cotangent bundle $ST^*M.$ Jointly with Stefan Nemirovski we showed when $\widetilde M^m$ is {\bf not\/} a compact manifold, then two points $x, y\in X$ are causally related if and only if the Legendrian spheres $\mathfrak S_x, \mathfrak S_y$ of all light rays through $x$ and $y$ are linked in $\mathcal N_{X}.$ In this short note we use the contact Bott-Samelson theorem of Frauenfelder, Labrousse and Schlenk to show that the same statement is true for all $X$ for which the integral cohomology ring of a closed $\widetilde M$ is {\bf not} the one of the CROSS (compact rank one symmetric space). If $M$ admits a Riemann metric $\overline g$, a point $x$ and a number $\ell>0$ such that all unit speed geodesics starting from $x$ return back to $x$ in time $\ell$, then $(M, \overline g)$ is called a $Y^x_{\ell}$ manifold. Jointly with Stefan Nemirovski we observed that causality in $(M\times \mathbb R, \overline g\oplus -t^2)$ is {\bf not} equivalent to Legendrian linking. Every $Y^x_{\ell}$-Riemann manifold has compact universal cover and its integral cohomology ring is the one of a CROSS. So we conjecture that Legendrian linking is equivalent to causality if and only if one can {\bf not} put a $Y^x_{\ell}$ Riemann metric on a Cauchy surface $M.$

math.DG↗
arXiv2018

Conjectures on the Relations of Linking and Causality in Causally Simple Spacetimes

We formulate the generalization of the Legendrian Low conjecture of Natario and Tod (proved by Nemirovski and myself before) to the case of causally simple spacetimes. We prove a weakened version of the corresponding statement. In all known examples, a causally simple spacetime $(X, g)$ can be conformally embedded as an open set into some globally hyperbolic $(\widetilde X, \widetilde g)$ and the space of light rays in $(X, g)$ is an open submanifold of the space of light rays in $(\widetilde X, \widetilde g)$. If this is always the case, this provides an approach to solving the conjectures relating causality and linking in causally simples spacetimes.

math.DG↗
arXiv2017

Redshift and contact forms

It is shown that the redshift between two Cauchy surfaces in a globally hyperbolic spacetime equals the ratio of the associated contact forms on the space of light rays of that spacetime.

math.SG↗
arXiv2014

The number of framings of a knot in a 3-manifold

In view of the self-linking invariant, the number $|K|$ of framed knots in $S^3$ with given underlying knot $K$ is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that $|K|$ is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of $S^1\times S^2$; the knot $K$ need not be zero-homologous and the manifold is not required to be compact. We show that when $M$ is orientable, the number $|K|$ is infinite unless $K$ intersects a non-separating sphere at exactly one point, in which case $|K|=2$; the existence of a non-separating sphere implies that $M$ contains a connected sum factor of $S^1\times S^2$. For knots in nonorientable manifolds we show that if $|K|$ is finite, then $K$ is disorienting, or there is an isotopy from the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded $S^2$ or $\mathbb R P^2$ at exactly one point, or it intersects some embedded $S^2$ at exactly two points in such a way that a closed curve consisting of an arc in $K$ between the intersection points and an arc in $S^2$ is disorienting.

math.GT↗
arXiv2014

Universal orderability of Legendrian isotopy classes

It is shown that non-negative Legendrian isotopy defines a partial order on the universal cover of the Legendrian isotopy class of the fibre of the spherical cotangent bundle of any manifold. This result is applied to Lorentz geometry in the spirit of the authors' earlier work on the Legendrian Low conjecture.

math.SG↗