arXiv subjects

Vladimir Chernov

Publications and source records attributed to Vladimir Chernov.

33 records · Page 2Linked to original sources

arXiv2012

Some Corollaries of Manturov's projection Theorem

In our works with Stoimenow, Vdovina and with Byberi, we introduced the virtual canonical genus $g_{vc}(K)$ and the virtual bridge number $vb(K)$ invariants of virtual knots. One can see from the definitions that for an classical knot $K$ the values of these invariants are less or equal than the classical canonical genus $g_c(K)$ and the bridge number $b(K)$ respectively. We use Manturov's projection from the category of virtual knot diagrams to the category of classical knot diagrams, to show that for every classical knot type $K$ we have $g_{vc}(K)=g_c(K)$ and $vb(K)=b(K)$.

math.GT↗

arXiv2012

Cosmic censorship of smooth structures

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard $\R^4$. Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold $N$ and $\R$ and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to $N\times \R$. Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on $(3+1)$-dimensional spacetimes.

math.GT↗

arXiv2012

Intersections of Loops and the Andersen-Mattes-Reshetikhin Algebra

Given two free homotopy classes $α_1, α_2$ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points $m(α_1, α_2)$ of loops in these two classes. We show that for $α_1\neqα_2$ the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of $α_1$ and $α_2$ is equal to $m(α_1, α_2)$. Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of $α_1$ and $α_2$. The main result of this paper in the case where $α_1, α_2$ do not contain different powers of the same loop first appeared in the unpublished preprint of the second author. In order to prove the main result for all pairs of $α_1\neq α_2$ we had to use the techniques developed by the first author in her study of operations generalizing Turaev's cobracket of loops on a surface.

math.GT↗

arXiv2010

Topological properties of manifolds admitting a $Y^x$-Riemannian metric

A complete Riemannian manifold $(M, g)$ is a $Y^x_l$-manifold if every unit speed geodesic $γ(t)$ originating at $γ(0)=x\in M$ satisfies $γ(l)=x$ for $0\neq l\in \R$. Bérard-Bergery proved that if $(M^m,g), m>1$ is a $Y^x_l$-manifold, then $M$ is a closed manifold with finite fundamental group, and the cohomology ring $H^*(M, \Q)$ is generated by one element. We say that $(M,g)$ is a $Y^x$-manifold if for every $ε>0$ there exists $l>ε$ such that for every unit speed geodesic $γ(t)$ originating at $x$, the point $γ(l)$ is $ε$-close to $x$. We use Low's notion of refocussing Lorentzian space-times to show that if $(M^m, g), m>1$ is a $Y^x$-manifold, then $M$ is a closed manifold with finite fundamental group. As a corollary we get that a Riemannian covering of a $Y^x$-manifold is a $Y^x$-manifold. Another corollary is that if $(M^m,g), m=2,3$ is a $Y^x$-manifold, then $(M, h)$ is a $Y^x_l$-manifold for some metric $h.$

math.DG↗

arXiv2009

Legendrian links, causality, and the Low conjecture

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Natário and Tod says that two events $x,y\inß$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial in the contact manifold of all light geodesics in $X$. The Low conjecture says that for $m=2$ the events $x,y$ are causally related if and only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic $(X^{m+1}, g)$ such that a cover of its Cauchy surface is diffeomorphic to an open domain in $\mathbb R^m.$

math.SG↗

arXiv2009

Non-negative Legendrian isotopy in $ST^*M$

It is shown that if the universal cover of a manifold $M$ is an open manifold, then two different fibres of the spherical cotangent bundle $ST^*M$ cannot be connected by a non-negative Legendrian isotopy. This result is applied to the study of causality in globally hyperbolic spacetimes. It is also used to strengthen a result of Eliashberg, Kim, and Polterovich on the existence of a partial order on $\widetilde{\mathrm{Cont}}_0 (ST^*M)$.

math.SG↗

arXiv2007

Virtual Bridge Number One Knots

We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\leq u(K), vb(K)\leq b(K).$ There are no ordinary knots $K$ with $b(K)=1.$ We show there are infinitely many homotopy classes of virtual knots each of which contains infinitely many isotopy classes of $K$ with $vb(K)=1.$ In fact for each $i\in \N$ there exists $K$ virtually homotopic (but not virtually isotopic) to the unknot with $vb(K)=1$ and $vu(K)=i.$

math.GT↗

arXiv2007

Linking and causality in globally hyperbolic spacetimes

The linking number $lk$ is defined if link components are zero homologous. Our affine linking invariant $alk$ generalizes $lk$ to the case of linked submanifolds with arbitrary homology classes. We apply $alk$ to the study of causality in Lorentz manifolds. Let $M^m$ be a spacelike Cauchy surface in a globally hyperbolic spacetime $(X^{m+1}, g)$. The spherical cotangent bundle $ST^*M$ is identified with the space $N$ of all null geodesics in $(X,g).$ Hence the set of null geodesics passing through a point $x\in X$ gives an embedded $(m-1)$-sphere $S_x$ in $N=ST^*M$ called the sky of $x.$ Low observed that if the link $(S_x, S_y)$ is nontrivial, then $x,y\in X$ are causally related. This motivated the problem (communicated by Penrose) on the Arnold's 1998 problem list to apply link theory to the study of causality. The spheres $S_x$ are isotopic to fibers of $(ST^*M)^{2m-1}\to M^m.$ They are nonzero homologous and $lk(S_x,S_y)$ is undefined when $M$ is closed, while $alk(S_x, S_y)$ is well defined. Moreover, $alk(S_x, S_y)\in Z$ if $M$ is not an odd-dimensional rational homology sphere. We give a formula for the increment of $\alk$ under passages through Arnold dangerous tangencies. If $(X,g)$ is such that $alk$ takes values in $\Z$ and $g$ is conformal to $g'$ having all the timelike sectional curvatures nonnegative, then $x, y\in X$ are causally related if and only if $alk(S_x,S_y)\neq 0$. We show that $x,y$ in nonrefocussing $(X, g)$ are causally unrelated iff $(S_x, S_y)$ can be deformed to a pair of $S^{m-1}$-fibers of $ST^*M\to M$ by an isotopy through skies. Low showed that if $(ß, g)$ is refocussing, then $M$ is compact. We show that the universal cover of $M$ is also compact.

math.GT↗

arXiv2007

Mapped Null Hypersurfaces and Legendrian Maps

For an $(m+1)$-dimensional space-time $(X^{m+1}, g),$ define a mapped null hypersurface to be a smooth map $ν:N^{m}\to X^{m+1}$ (that is not necessarily an immersion) such that there exists a smooth field of null lines along $ν$ that are both tangent and $g$-orthogonal to $ν.$ We study relations between mapped null hypersurfaces and Legendrian maps to the spherical cotangent bundle $ST^*M$ of an immersed spacelike hypersurface $μ:M^m\to X^{m+1}.$ We show that a Legendrian map $\wt λ: L^{m-1}\to (ST^*M)^{2m-1}$ defines a mapped null hypersurface in $X.$ On the other hand, the intersection of a mapped null hypersurface $ν:N^m\to X^{m+1}$ with an immersed spacelike hypersurface $μ':M'^m\to X^{m+1}$ defines a Legendrian map to the spherical cotangent bundle $ST^*M'.$ This map is a Legendrian immersion if $ν$ came from a Legendrian immersion to $ST^*M$ for some immersed spacelike hypersurface $μ:M^m\to X^{m+1}.$

math.DG↗

arXiv2006

On generalized winding numbers

Let $M^m$ be an oriented manifold, let $N^{m-1}$ be an oriented closed manifold, and let $p$ be a point in $M^m$. For a smooth map $f:N^{m-1} \to M^m, p \not\in Im f,$ we introduce an invariant $awin_p(f)$ that can be regarded as a generalization of the classical winding number of a planar curve around a point. We show that $awin_p$ estimates from below the number of times a wave front on $M$ passed through a given point $p\in M$ between two moments of time. Invariant $awin_p$ allows us to formulate the analogue of the complex analysis Cauchy integral formula for meromorphic functions on complex surfaces of genus bigger than one.

math.GT↗

arXiv2004

Affine Linking Numbers and Causality Relations for Wave Fronts

Two wave fronts $W_1$ and $W_2$ that originated at some points of the manifold $M^n$ are said to be causally related if one of them passed through the origin of the other before the other appeared. We define the causality relation invariant $CR (W_1, W_2)$ to be the algebraic number of times the earlier born front passed through the origin of the other front before the other front appeared. Clearly, if $CR(W_1, W_2)\neq 0$, then $W_1$ and $W_2$ are causally related. If $CR(W_1, W_2)= 0$, then we generally can not make any conclusion about fronts being causally related. However we show that for front propagation given by a complete Riemannian metric of non-positive sectional curvature, $CR(W_1, W_2)\neq 0$ if and only if the two fronts are causally related. The models where the law of propagation is given by a metric of constant sectional curvature are the famous Friedmann Cosmology models. The classical linking number $lk$ is a $Z$-valued invariant of two zero homologous submanifolds. We construct the affine linking number generalization $AL$ of the $lk$ invariant to the case of linked $(n-1) $-spheres in the total space of the unit sphere tangent bundle $(STM)^{2n-1}\to M^n$. For all $M$, except of odd-dimensional rational homology spheres, $AL$ allows one to calculate the value of $CR(W_1, W_2)$ from the picture of the two wave fronts at a certain moment. This calculation is done without the knowledge of the front propagation law and of their points and times of birth. Moreover, in fact we even do not need to know the topology of $M$ outside of a part $\bar M$ of $M$ such that $W_1$ and $W_2$ are null-homotopic in $\bar M$.

math.GT↗

arXiv2003

Relative Framing of Transverse knots

It is well-known that a knot in a contact manifold $(M,C)$ transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class $e_C\in H^2(M)$ of $C$ is nonzero, then $C$ is nontrvivializable and the natural framing of transverse knots does not exist. We construct a new framing-type invariant of transverse knots called relative framing. It is defined for all tight $C$, all closed irreducible atoroidal $M$, and many other cases when $e_C\neq 0$ and the classical framing invariant is not defined. We show that the relative framing distinguishes many transverse knots that are isotopic as unframed knots. It also allows one to define the Bennequin invariant of a zero-homologous transverse knot in contact manifolds $(M,C)$ with $e_C\neq 0$. Our recent result is that the groups of Vassiliev-Goussarov invariants of transverse and of framed knots are canonically isomorphic, when $C$ is trivialized and transverse knots have the natural framing. We show that the same result is true whenever the relative framing is well-defined.

math.SG↗

arXiv2003

Framed knots in 3-manifolds and affine self-linking numbers

The number $|K|$ of non-isotopic framed knots that correspond to a given unframed knot $K\subset S^3$ is infinite. This follows from the existence of the self-linking number $\slk$ of a zerohomologous framed knot. We use the approach of Vassiliev-Goussarov invariants to construct ``affine self-linking numbers'' that are extensions of $\slk$ to the case of nonzerohomologous framed knots. As a corollary we get that $|K|=\infty$ for all knots in an oriented (not necessarily compact) 3-manifold $M$ that is not realizable as a connected sum $(S^1\times S^2) # M'$. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge the proof of this fundamental fact was not given in literature. Our proof is based on different ideas. For $M=(S^1\times S^2) # M'$ we construct $K$ in $M$ such that $|K|=2\neq \infty$.

math.GT↗

arXiv2003

Algebraic structures on generalized strings

A garland based on a manifold $P$ is a finite set of manifolds homeomorphic to $P$ with some of them glued together at marked points. Fix a manifold $M$ and consider a space $\NN$ of all smooth mappings of garlands based on $P$ into $M$. We construct operations $\bullet$ and $[-,-]$ on the bordism groups $\bor_*(\NN)$ that give $\bor_*(\NN)$ the natural graded commutative assosiative and graded Lie algebra structures. We also construct two auto-homomorphisms $\pr$ and $\li$ of $\bor_*(\NN)$ such that $\pr(\li α_1\bullet \li α_2)= [α_1, α_2]$ for all $α_1, α_2 \in \bor_*(\NN)$. If $P$ is a boundary, then $\pr \circ \li=0$ and thus $Δ^2=0$ for $Δ=\li \circ \pr$. We show that under certain conditions the operations $Δ$ and $\bullet$ give rise to Batalin-Vilkoviski and Gerstenhaber algebra structures on $\bor_*(\NN)$. In a particular case when $P=S^1$, the algebra $\bor_*(\NN)$ is related to the string-homology algebra constructed by Chas and Sullivan.

math.GT↗

arXiv2003

The universal order one invariant of framed knots in most S^1-bundles over orientable surfaces

It is well-known that self-linking is the only Z valued Vassiliev invariant of framed knots in S^3. However for most 3-manifolds, in particular for the total spaces of S^1-bundles over an orientable surface F not S^2, the space of Z-valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant I of framed knots in orientable total spaces of S^1-bundles over an orientable not necessarily compact surface F not S^2. We show that if F is not S^2 or S^1 X S^1, then I is the universal order one invariant, i.e. it distinguishes every two framed knots that can be distinguished by order one invariants with values in an Abelian group.

math.GT↗