Finite type invariants for knotoids

We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complete theory of type-1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.     

Invariant tori in perturbed three vortex motion

The motion of three point vortices in an ideal fluid in a plane comprises a Hamiltonian dynamical system – one that is completely integrable, so it exhibits numerous periodic orbits, and quasiperiodic orbits on invariant tori. Certain perturbations of three vortex dynamics, such as three vortex motion in a half-plane, are also Hamiltonian, but not completely integrable. Yet these perturbed systems may still have periodic trajectories and invariant tori close to those for the unperturbed dynamics. Extending recent work by the authors [4], invariant 2-tori approximating those for the unperturbed system are located and analyzed using a combination of classical analysis, asymptotics, and Hamiltonian methods. It is shown that the results and approximation methods used are applicable to several perturbations of three vortex dynamics such as three vortices in a half-plane, the restricted four vortex problem in the plane, and three coaxial vortex rings in 3-space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hamiltonian systems and Vasil’ev invariants

It is proved that a dynamical system related to the first-order Vasil’ev invariants exactly coincides with the system of Cartesian vortices on the plane. Also, the general form of Hamiltonian systems corresponding to the second-order Vasil’ev invariant is written and some properties of such systems are indicated Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

Finite-order invariants of ornaments

Anornament is a collection of oriented closed curves in a plane, no three of which intersect at the same point. We consider homotopy invariants of ornaments. Thefinite-order invariants of ornaments are a natural analog of the Vassiliev invariants of links. The calculation of them is based on the homological study of the corresponding space of singular objects. We perform the “local” part of these calculations and a part of the “global” one, which allows us to estimate the dimensions of the spaces of invariants of any order. We also construct explicity two large series of such invariants and establish some new algebraic structures in the space of invariants. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 35, Algebraicheskaya Geometriya-6, 1996.     

Discrete Moving Frames and Discrete Integrable Systems

Group-based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group-based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulas. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices, respectively, under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.     

Isomorphism of the Groups of Vassiliev Invariants of Legendrian and Pseudo-Legendrian Knots

The study of the Vassiliev invariants of Legendrian knots was started by D. Fuchs and S. Tabachnikov who showed that the groups of C-valued Vassiliev invariants of Legendrian and of framed knots in the standard contact R³ are canonically isomorphic. Recently we constructed the first examples of contact 3-manifolds where Vassiliev invariants of Legendrian and of framed knots are different. Moreover in these examples 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.     

Vassiliev invariants and the Poincaré conjecture

This article examines the relationship between 3-manifold topology and knot invariants of finite type. We prove that in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. If, on the other hand, Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true (i.e. every homotopy 3-sphere is homeomorphic to the standard 3-sphere).     

Vassiliev invariants for braids on surfaces

We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.

     

Some applications of Tristram-Levine signatures and relation to Vassiliev invariants

Tristram and Levine introduced a continuous family of signature invariants for knots. We show that any possible value of such an invariant is realized by a knot with given Vassiliev invariants of bounded degree. We also show that one can make a knot prime preserving Alexander polynomial and Vassiliev invariants of bounded degree. Finally, the Tristram-Levine signatures are applied to obtain a condition on (signed) unknotting number.     

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

We show that for a large class of contact three-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.     

Calculating the first nontrivial 1-cocycle in the space of long knots

For spaces of knots in ℝ³, the Vassiliev theory defines the so-called cocycles of finite order. The zero-dimensional cocycles are finite-order invariants. The first nontrivial cocycle of positive dimension in the space of long knots is one-dimensional and is of order 3. We apply the combinatorial formula given by Vassiliev in his paper     

Invariants of knots,surfaces in R<Superscript>3</Superscript>, and foliations

We give a survey of some known results related to combinatorial and geometric properties of finite-order invariants of knots in a three-dimensional space. We study the relationship between Vassiliev invariants and some classical numerical invariants of knots and point out the role of surfaces in the investigation of these invariants. We also consider combinatorial and geometric properties of essential tori in standard position in closed braid complements by using the braid foliation technique developed by Birman, Menasco, and other authors. We study the reductions of link diagrams in the context of finding the braid index of links. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1239–1252, September, 2007.     

On the Melvin–Morton–Rozansky conjecture

We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander–Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the “colored” Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin–Morton–Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Oblatum 28-VII-1994 & 5-XI-1995 & 22-XI-1995     

For a fixed Turaev shadow Jones-Vassiliev invariants depend polynomially on the gleams

We use Turaev's technique of shadows and gleams to parametrize the set of all knots in S ³ with the same Hopf projection. We show that the Vassiliev invariants arising from the Jones polynomial J _t (K) are polynomials in the gleams, i.e., for , the n-th order Vassiliev invariant u _n , defined by , is a polynomial of degree 2n in the gleams. Received: April 30, 1996     

Periodic orbits in the restricted three-body problem and Arnold’s <Emphasis Type="Italic">J</Emphasis><Superscript>+</Superscript>-invariant

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 behavior.     

Invariants de Vassiliev et conjecture de PoincaréVassiliev invariants and the Poincaré conjecture

We prove the following result: if Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true, in other words every homotopy sphere is homeomorphic to the standard sphere. On the other hand, in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. To cite this article: M. Eisermann, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1005–1010.     

First Order Semi-Local Invariants of Stable Maps of 3-Manifolds into the Plane

In the late 1980s, Vassiliev introduced new graded numericalinvariants of knots, which are now called Vassiliev invariantsor finite-type invariants. Since he made this definition, manypeople have been trying to construct Vassiliev type invariantsfor various mapping spaces. In the early 1990s, Arnold and Goryunovintroduced the notion of first order (local) invariants of stablemaps. In this paper, we define and study first order semi-localinvariants of stable maps and those of stable fold maps of aclosed orientable 3-dimensional manifold into the plane. Weshow that there are essentially eight first order semi-localinvariants. For a stable map, one of them is a constant invariant,six of them count the number of singular fibers of a given typewhich appear discretely (there are exactly six types of suchsingular fibers), and the last one is the Euler characteristicof the Stein factorization of this stable map. Besides theseinvariants, for stable fold maps, the Bennequin invariant ofthe singular value set corresponding to definite fold pointsis also a first order semi-local invariant. Our study of unstablefold maps with codimension 1 provides invariants for the connectedcomponents of the set of all fold maps. 2000 Mathematics SubjectClassification 57R45 (primary), 32S20, 58K15 (secondary).     

Chromatic weight systems and the corresponding knot invariants

We prove that chromatic weight systems, introduced by Chmutov, Duzhin and Lando, can be expressed in terms of weight systems associated with direct sums of the Lie algebras and . As a consequence the Vassiliev invariants of knots corresponding to the chromatic weight systems distinguish exactly the same knots as a one-variable specialization of the Homfly and Kauffman polynomial. Received: 22 December 1997 / Revised: 18 March 1999 / Published online: 8 May 2000     

Arrow-diagram formulas for fourth-order invariants of knots

In this paper, we study explicit arrow-diagram formulas for fourth-order Vassiliev invariants of knots announced by several authors. We show that, in fact, these formulas do not determine any knot invariants. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 3–17, 2005.     

Les invariants rationnels de type fini ne distinguent pas les nœuds dans SS2×SS1

We introduce geometric sequences of knots and establish the following criterion: if v is a rational invariant of degree ≤m in the sense of Vassiliev, then v is a polynomial of degree ≤m on every geometric sequence of knots. The torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: we construct knots in SS²×SS¹ which cannot be distinguished by rational invariants of finite type. They can, however, be distinguished by invariants of finite type with values in a finite abelian group.