42 pages, 22 figures We added an example of an order one $J^-$-type invariant of fronts on a surface $F$ that does not come from an order one invariant of Legendrian knots in $PT^*F$, and corrected the typos
Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface $F$. All invariants satisfying the axioms which naturally generalize the axioms used by Arnold are explicitly described. We also give an explicit formula for the finest order one $J^+$-type invariant of fronts on an orientable surface $F\neq S^2$. We obtain necessary and sufficient conditions for an invariant of nongeneric fronts with one nongeneric singular point to be the Vassiliev-type derivative of an invariant of generic fronts. As a byproduct, we calculate all homotopy groups of the space of Legendrian immersions of $S^1$ into the spherical cotangent bundle of a surface.
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…
We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark-Zeeman systems and study their behaviour.
We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves.
We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.
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.
We give new functions of Legendrian knots derived from Legendrian fronts. These are integer-valued linear functions that are alike the Arnold basic invariant of plane curves. Various generalizations of the Arnold basic invariant have been known. In this paper, we give another extension of Arnold's idea.
We construct the infinite sequence of invariants for curves in surfaces by using word theory that V. Turaev introduced. For plane closed curves, we add some extra terms, e.g. the rotation number. From these modified invariants, we get the Arnold's basic invariants and some other invariants. We also express how these invariants classify plane closed curves. In addition, we consider other classes of plane curves: long curves and fronts.
In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invariants of degree 2, by Fujiwara. We give another proof of this result and show that those invariants do not provide the universal one of degree 2.
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-order invariants for plane immersed curves. The invariants in this theory are developed using the word theory proposed by Turaev.
Consider a real algebraic variety, $\R X$, of dimension $d$. If its complexification, $\C X$, is a rational homology manifold (at least in a neighborhood of $\R X$), then the intersection form in $\C X$ defines a bilinear form in $d$-homologies of $\R X$. Analizing it, one can obtain an information about $\R X$, as it was done by V.I.Arnold in the case of non-singular double planes and then generalized by O.Ya.Viro and V.M.Kharlamov to the nodal surfaces. I present an integration (based on the Euler characteristic) formula, which expresses this form in terms of a certain local inveriant of the…
We consider the general nonvanishing, divergence-free vector fields defined on a domain in three space and tangent to its boundary. Based on the theory of finite type invariants, we define a family of invariants for such fields, in the style of Arnold's asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.
We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves. We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under Ω3 moves depends only on the forward/backward character of the move, similar to how the change of …
We define an algebraic/combinatorial object on the front projection $Σ$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on $Σ$ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of $L_Σ$, where $L_Σ$ is the Ng resolution of $Σ$. In the case of Legendrian knot c…
In their paper "Integrating curvature: From Umlaufsatz to J+ invariant" Lanzat and Polyak introduced a polynomial invariant of generic curves in the plane as a quantization of Hopf's Umlaufsatz, and showed that Arnold's J+ invariant could be derived from their polynomial, leading to an integral formula for J+. Here we extend their invariant to the case of homologically trivial generic curves in closed oriented surfaces with Riemannian metric. The resulting invariant turns out to be a quantization of a new formula for the rotation number, which can be viewed as a form of the Gauss-Bonnet Theore…
Arnold introduced invariants $J^+$, $J^-$ and $St$ for generic planar curves. It is known that both $J^+ /2 + St$ and $J^- /2 + St$ are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves for uknotting. $J^- /2 + St$ works well for unmatched RII moves. However, it works only by halves for RI moves. Let $w$ denote the writhe for a knot diagram. We show that $J^- /2 + St \pm w/2$ works well also for RI moves, and demonstrate that it gives a precise estimation for a certain knot d…
In this paper we study the Haefliger invariant for long embeddings $\mathbb{R}^{4k-1}\hookrightarrow\mathbb{R}^{6k}$ in terms of the self-intersections of their projections to $\mathbb{R}^{6k-1}$, under the condition that the projection is a generic long immersion $\mathbb{R}^{4k-1}\looparrowright\mathbb{R}^{6k-1}$. We define the notion of "crossing changes" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in $\mathbb{R}^{4k-1}$. This formula is a higher-dimensional analogue to t…
