arXiv subjects

Vladimir Tchernov

Publications and source records attributed to Vladimir Tchernov.

7 recordsLinked to original sources

arXiv2000

Isomorphism of the groups of Vassiliev Invariants of Legendrian and of Pseudo Legendrian Knots in Contact 3-manifolds

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of complex-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact $R^3$ are canonically isomorphic. Recently we constructed the first examples where Vassiliev invariants of Legendrian and of framed knots are different, and Vassiliev invariants of Legendrian knots distinguish Legendrian knots that are isotopic as framed knots and homotopic as Legendrian immersions. This raised the question what information about Legendrian knots can be captured using Vassiliev invariants. Here we answer this question by showing that for any contact 3-manifold with a cooriented contact structure the groups of Vassiliev invariants of Legendrian knots and of knots that are nowhere tangent to a vector field that coorients the contact structure are canonically isomorphic.

math.GT↗

arXiv2000

Vassiliev invariants of Legendrian, of transverse and framed knots in contact 3-manifolds

We show that for a large class of contact 3-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's $J^+$-type invariants of wave fronts on a surface $F$ is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle $ST^*F$ of $F$. On the other hand we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as Legendrian immersions.

math.SG↗

arXiv1999

Finite Order Invariants of Legendrian, Transverse, and Framed Knots in Contact 3-manifolds

We show that for a big class of contact manifolds the groups of order $\leq n$ invariants (with values in an arbitrary Abelian group) of Legendrian, of transverse and of framed knots are canonically isomorphic. On the other hand for an arbitrary cooriented contact structure on $S^1\times S^2$ with the nonzero Euler class of the contact bundle we construct examples of Legendrian homotopic Legendrian knots $K_1$ and $K_2$ such that they realize isotopic framed knots but can be distinguished by finite order invariants of Legendrian knots in $S^1\times S^2$. We construct similar examples for a big class of contact manifolds $M$ such that $M$ is a total space of a locally trivial $S^1$-fibration over a nonorientable surface. We show that in some of these examples the complements of $K_1$ and of $K_2$ are overtwisted.

math.SG↗

arXiv1999

The Most Refined Invariant of Degree One of Knots and Links in $R^1$-Fibrations Over a Surface

As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants of a knot in an $R^1$-fibration over a surface $F$. They take values in the free $Z$-module generated by all the free homotopy classes of loops in $F$. Here, we generalize them to the most refined Vassiliev invariant of degree one. The ranges of values of all these invariants are explicitly described. We also construct a similar invariant of a two-component link in an $\R^1$-fibration. It generalizes the linking number.

math.GT↗

arXiv1999

Shadows of Wave Fronts and Arnold-Bennequin Type Invariants of Fronts on Surfaces and Orbifolds

A first order Vassiliev invariant of an oriented knot in an $S^1$-fibration and a Seifert fibration over a surface is constructed. It takes values in a quotient of the group ring of the first homology group of the total space of the fibration. It gives rise to an invariant of wave fronts on surfaces and orbifolds related to the Bennequin-type invariants of the Legendrian curves studied by F. Aicardi, V. Arnold, M. Polyak, and S. Tabachnikov. Formulas expressing these relations are presented. We also calculate Turaev's shadow for the Legendrian lifting of a wave front. This allows to use in the case of wave fronts all invariants known for shadows.

math.GT↗

arXiv1999

Homotopy Groups of the Space of Curves on a Surface

We explicitly calculate the fundamental group of the space $\mathcal F$ of all immersed closed curves on a surface $F$. It is shown that $π_n(\mathcal F)=0$, n>1 for $F\neq S^2, RP^2$. It is also proved that $π_2(\mathcal F)=\Z$, and $π_n(\mathcal F)=π_n(S^2)\oplusπ_{n+1}(S^2)$, n>2, for $F$ equal to $S^2$ or $RP^2$.

math.GT↗

arXiv1999

Arnold-type Invariants of Curves on Surfaces

Recently V. Arnold introduced Strangeness and $J^{\pm}$ invariants of generic immersions of an oriented circle to $\R^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface $F$. We explicitly describe all the invariants satisfying axioms, which naturally generalize the axioms used by V. Arnold.

math.GT↗