The Kontsevich Integral

The paper contains a detailed exposition of the construction and properties of the Kontsevich integral invariant, crucial in the study of Vassiliev knot invariants.     

Algorithms in low-dimensional topology: holonomic parametrization of knots

In this work, we reintroduce the so-called AFL (Arkaden-Faden-Lage) representation of knots introduced by Kurt Reidemeister and show how it can be used to develop efficient algorithms in low-dimensional topology. In particular, we develop an algorithm to calculate the functions for the holonomic parametrization of knots introduced by Vassiliev in 1997, who proved that each knot type has a holonomic parametrization (but no method to find such a parametrization was known). Further, we show that the result of Vassiliev can be easily derived from the AFL representation of knots. This is one of the first practical results of the application of the AFL representation of knots that can open new perspectives in the field of low dimensional topology such as computation of the Kontsevich integral and some operators of quantum groups.     

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.     

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).     

Topology of Maximally Writhed Real Algebraic Knots

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in ??³ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree d with the maximal possible value of this invariant. We show that for a given d all such knots are topologically isotopic and explicitly identify their knot type.     

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.     

Grope cobordism of classical knots

Motivated by the lower central series of a group, we define the notion of a grope cobordism between two knots in a 3-manifold. Just like an iterated group commutator, each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants.The derived commutator series of a group also has a three-dimensional analogy, namely knots modulo symmetric grope cobordism. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot concordance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at h=2, Casson-Gordon invariants at h=3, and for h=4 one finds the new von Neumann signatures of a knot.     

Reidemeister–Turaev torsion of 3-dimensional¶Euler structures with simple boundary tangency¶and pseudo-Legendrian knots

We generalize Turaev's definition of torsion invariants of pairs (M,&\xi;), where M is a 3-dimensional manifold and &\xi; is an Euler structure on M (a non-singular vector field up to homotopy relative to ∂M and modifications supported in a ball contained in Int(M)). Namely, we allow M to have arbitrary boundary and &\xi; to have simple (convex and/or concave) tangency circles to the boundary. We prove that Turaev's H ₁(M)-equivariance formula holds also in our generalized context. Using branched standard spines to encode vector fields we show how to explicitly invert Turaev's reconstruction map from combinatorial to smooth Euler structures, thus making the computation of torsions a more effective one. Euler structures of the sort we consider naturally arise in the study of pseudo-Legendrian knots (i.e.~knots transversal to a given vector field), and hence of Legendrian knots in contact 3-manifolds. We show that torsion, as an absolute invariant, contains a lifting to pseudo-Legendrian knots of the classical Alexander invariant. We also precisely analyze the information carried by torsion as a relative invariant of pseudo-Legendrian knots which are framed-isotopic. Received: 3 October 2000 / Revised version: 20 April 2001     

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.     

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.     

New perspectives on self-linking

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.     

On denominators of the Kontsevich integral and the universal perturbative invariant of 3-manifolds

 The integrality of the Kontsevich integral and perturbative invariants is discussed. It is shown that the denominator of the degree n part of the Kontsevich integral of any knot or link is a divisor of (2!3!…n!)⁴(n+1)!. We prove this by establishing the existence of a Drinfeld's associator in the space of chord diagrams with special denominators. We also show that the denominator of the degree n part of the universal perturbative invariant of homology 3-spheres is not divisible by any prime greater than 2n+1. Oblatum 20-VI-1997 & 28-IV-1998 / Published online: 12 November 1998     

Finite type invariants of nanowords and nanophrases

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree 1 and degree 2 respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.     

Wheels, wheeling, and the Kontsevich integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT]. This paper is available electronically     

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).     

Vassiliev invariants of type one for links in the solid torus

In Bataineh (2003) [2] we studied the type one invariants for knots in the solid torus. In this research we study the type one invariants for n-component links in the solid torus by generalizing Aicardi's invariant for knots in the solid torus to n-component links in the solid torus. We show that the generalized Aicardi's invariant is the universal type one invariant, and we show that the generalized Aicardi's invariant restricted to n-component links in the solid torus with zero winding number for each component is equal to an invariant we define using the universal cover of the solid torus. We also define and study a geometric invariant for n-component links in the solid torus. We give a lower bound on this invariant using the type one invariants, which are easy to calculate, which helps in computing this geometric invariant, which is usually hard to calculate.     

The Alexander polynomial and finite type 3-manifold invariants

Using elementary counting methods, we calculate a universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M, satisfying , in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga]. Received: 27 April 1998 / in final form: 8 August 1999     

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     

Goussarov-Habiro theory for string links and the Milnor-Johnson correspondence

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's μ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the Milnor-Johnson correspondence: in particular, an analogue of the Birman-Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.     

Finite type invariants of cyclic branched covers

Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parameterized by the positive integers), namely the cyclic branched coverings of the knot. In this paper, we give a formula for the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its signature function.