Vassiliev Invariants Research Articles

Genera, band sum of knots and Vassiliev invariants

Recently Stoimenow showed that for every knot K and any n∈N and u0⩾u(K) there is a prime knot Kn,uo which is n-equivalent to the knot K and has unknotting number u(Kn,uo) equal to u0. The similar result has been obtained for the 4-ball genus gs of a knot. Stoimenow also proved that any admissible value of the Tristram–Levine signature σξ can be realized by a knot with the given Vassiliev invariants of bounded order. In this paper, we show that for every knot K with genus g(K) and any n∈N and m⩾g(K) there exists a prime knot L which is n-equivalent to K and has genus g(L) equal to m.

Finite-type invariants of 3-manifolds and the dimension subgroup problem

For a certain class of compact oriented 3-manifolds, Goussarov and Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of ‘cut-and-paste’ operations defined by the lower central series of the Torelli group of a surface. In this paper, we observe that this is a variation of a classical problem in group theory, namely the ‘dimension subgroup problem’. This viewpoint allows us to prove, by purely algebraic methods, an analog of the Goussarov–Habiro conjecture for finite-type invariants with values in a fixed field. We deduce that their original conjecture is true at least in a weaker form.

Brunnian links, claspers and Goussarov–Vassiliev finite type invariants

AbstractGoussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

Jacobi identities in low-dimensional topology

The Jacobi identity is the key relation in the definition of a Lie algebra. In the last decade, it has also appeared at the heart of the theory of finite type invariants of knots, links and 3-manifolds (and is there called the IHX relation). In addition, this relation was recently found to arise naturally in a theory of embedding obstructions for 2-spheres in 4-manifolds in terms of Whitney towers. This paper contains the first proof of the four-dimensional version of the Jacobi identity. We also expose the underlying topological unity between the three- and four-dimensional IHX relations, deriving from a beautiful picture of the Borromean rings embedded on the boundary of an unknotted genus 3 handlebody in 3-space. This picture is most naturally related to knot and 3-manifold invariants via the theory of grope cobordisms.

On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials

AbstractIt is known that the Brandt–Lickorish–Millett–Ho polynomialQcontains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable fromQis an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degreed≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

ARRANGEMENTS OF NESTED CURVES

Motivated by Arnold's theory of invariants of plane curves, we introduce the semi-group of equivalence classes of arrangements of nested curves. There exists a natural invariant of plane curves without inverse self-tangencies with values in this semi-group. We show that the associated Grothendieck group is ℤ × ℤ. These two factors correspond to previously known invariants of plane curves without inverse self-tangencies, namely Whitney's index and Arnold's J- invariant. We show that arrangements of nested curves are not classified by their finite type invariants.

THE COEFFICIENTS OF THE HOMFLYPT AND THE KAUFFMAN POLYNOMIALS ARE POINTWISE LIMITS OF VASSILIEV INVARIANTS

The Vassiliev conjecture states that the Vassiliev invariants are dense in the space of all numerical link invariants in the sense that any link invariant is a pointwise limit of Vassiliev invariants. In this article, we prove that the Vassiliev conjecture holds in the case of the coefficients of the HOMFLY and the Kauffman polynomials.

On parabolic submonoids of a class of singular Artin monoids

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

On surgery along Brunnian links in 3–manifolds

We consider surgery moves along (n+1)-component Brunnian links in compact connected oriented 3-manifolds, where the framing of the each component is 1/k for k in Z. We show that no finite type invariant of degree < 2n-2 can detect such a surgery move. The case of two link-homotopic Brunnian links is also considered. We relate finite type invariants of integral homology spheres obtained by such operations to Goussarov-Vassiliev invariants of Brunnian links.

TREE DIAGRAMS FOR STRING LINKS

In previous work, the author defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

CLASPER-MOVES AMONG RIBBON 2-KNOTS CHARACTERIZING THEIR FINITE TYPE INVARIANTS

Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's Ck-moves. Ck-moves are defined by using his claspers. In this paper we define "oriented" claspers and RCk-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing.

VASSILIEV INVARIANTS AND SIMILARITY INDICES OF KNOTS, LINKS AND TANGLES

Whether Vassiliev invariants can distinguish all knots or not is a well-known open problem which is equivalent to the question whether the similarity index of any two different knots is finite or not. We give relations between the degrees of Vassiliev invariants and the similarity indices of knots, links and tangles. From these, we get necessary conditions for a knot invariant to be a Vassiliev invariant and get methods to detect the similarity index of two knots or tangles.

LOCAL MOVES AND GORDIAN COMPLEXES

By the works of Gusarov [2] and Habiro [3], it is known that a local move called the Cnmove is strongly related to Vassiliev invariants of order less than n. The coefficient of the znterm in the Conway polynomial is known to be a Vassiliev invariant of order n. In this note, we will consider to what degree the relationship is strong with respect to Conway polynomial. Let K be a knot, and KCnthe set of knots obtained from a knot K by a single Cnmove. Let [Formula: see text] be the set of the Conway polynomials [Formula: see text] for a set of knots [Formula: see text]. Our main result is the following: There exists a pair of knots K1, K2such that ∇K1= ∇K2and [Formula: see text]. In other words, the CnGordian complex is not homogeneous with respect to Conway polynomial.

Vassiliev invariants of quasipositive knots

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Quasipositive knots are transverse intersections of complex plane curves with the standard sphere $S^3 \subset {\mathbb C}^2$. It is known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot $K$ and any natural number $n$ there exists a quasipositive knot $Q$ whose Vassiliev invariants of order less than or equal to $n$ coincide with those of $K$.

ON EXTREME COEFFICIENTS OF THE JONES–KAUFFMAN POLYNOMIAL FOR VIRTUAL LINKS

An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can be reduced to the case when the extreme coefficient function is expressible in terms of chord diagrams (previous authors consider only d-diagrams which correspond to the classical case [9]). We find the maximal absolute values for generic chord diagrams, thus, for generic virtual knots. Also we consider the "next" coefficient of the Jones–Kauffman polynomial in terms of framed chord diagrams and find its maximal value for a given number of chords. These two functions on chord diagrams are of their own interest because there are related to the Vassiliev invariants of classical knots and J-invariants of planar curves, as mentioned in [10].

Cn-MOVE AND ITS DUPLICATED MOVE OF LINKS

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, Cn-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by Cn-moves. We define a local move, βn-move, which is obtained from a Cn-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a βn-move and that a βn-move is realized by twice Cn-moves. In this note we study the relation between Cn-move and βn-move, and in particular, give answers to the following questions: (1) Is a βn-move realized by a finite sequence of Cn+1-moves? (2) Is Cn-move realized by a finite sequence of βn-moves?

The Lagrangian filtration of the mapping class group and finite-type invariants of homology spheres

In a recent paper we defined a new filtration of the mapping class group – the Lagrangian filtration. We here determine the successive quotients of this filtration, up to finite index. As an application we show that, for any additive invariant of finite-type (e.g. the Casson invariant) and any level of the Lagrangian filtration, there is a homology 3-sphere which has a Heegaard decomposition whose gluing diffeomorphism lies at that level, on which this invariant is non-zero. In a final section we examine the relationship between the Johnson and Lagrangian filtrations.

Generalization of Braids and Homologies

In this paper, we give a survey of recent results devoted to the homology of generalizations of braids: the homological properties of virtual braids and the generalized homology of Artin groups studied by C. Broto and the author. Virtual braid groups VBn correspond to virtual knots in the same way that classical braids correspond to usual knots. Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The Burau representation to GLnℤ[t, t−1] is extended from classical braids to virtual ones. Its homological properties are also studied. The following splitting of infinite loop spaces for the plus-construction of the classifying space of the virtual braid group on an infinite number of strings exists: $$\mathbb{Z} \times BV{\kern 1pt} B_\infty ^ + \simeq \Omega ^\infty S^\infty \times S^1 \times Y,$$ where Y is an infinite loop space. Connections with K*ℤ are discussed. In the last section, information on Morava K-theory and the Brown-Peterson homology of Artin groups and braid groups in handlebodies is collected.

Diagram Invariants of Knots and the Kontsevich Integral

This paper is devoted to the solution of the integration problem for higher derivatives of finite-type invariants (so-called weight systems) up to real invariants defined on the space of knots.

On injective homomorphisms for pure braid groups, and associated Lie algebras

The purpose of this article is to record the center of the Lie algebra obtained from the descending central series of Artin's pure braid group, a Lie algebra analyzed in work of Kohno [T. Kohno, Linear representations of braid groups and classical Yang–Baxter equations, in: Contemp. Math., vol. 78, 1988, pp. 339–363; T. Kohno, Vassiliev invariants and the de Rham complex on the space of knots, in: Symplectic Geometry and Quantization, in: Contemp. Math., vol. 179, Amer. Math. Soc., Providence, RI, 1994, pp. 123–138; T. Kohno, Série de Poincaré–Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985) 57–75], and Falk and Randell [M. Falk, R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985) 77–88]. The structure of this center gives a Lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criterion used to test whether certain morphisms out of free groups are faithful [F.R. Cohen, J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, in: Algebraic Topology: Categorical Decomposition Techniques, in: Progr. Math., vol. 215, Birkhäuser, Basel, 2003; Braid groups, free groups, and the loop space of the 2-sphere, math.AT/0409307]. However, it is as unclear whether this criterion for faithfulness can be applied to any open cases concerning representations of P n such as the Gassner representation.