scholarly journals ON A BASIS FOR THE FRAMED LINK VECTOR SPACE SPANNED BY CHORD DIAGRAMS

2009 ◽  
Vol 18 (12) ◽  
pp. 1663-1680
Author(s):  
BRYAN BISCHOF ◽  
ROMAN KOGAN ◽  
DAVID N. YETTER
Keyword(s):  
Vector Space ◽  
Fundamental Theorem ◽  
Linear Span ◽  
Vassiliev Invariants ◽  
Chord Diagrams ◽  
Framed Link

In view of the result of Kontsevich, now often called "the fundamental theorem of Vassiliev theory", identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the "4T-relation" (and in the unframed case, originally considered by Vassiliev, the "1T-" or "isolated chord relation"), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation. We construct the basis for the vector space spanned by chord diagrams with n chords and m link components, modulo 4T relations for n ≤ 5.

The Fundamental Theorem of Vassiliev Invariants

2021 ◽  
pp. 101-134
Author(s):  
Dror Bar-Natan ◽  
Alexander Stoimenow
Keyword(s):  
Fundamental Theorem ◽  
Vassiliev Invariants

Linearized chord diagrams and an upper bound for vassiliev invariants

2000 ◽  
Vol 09 (07) ◽  
pp. 847-853 ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Upper Bound ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Chord Diagrams

Recently, Stoimenow [J. Knot Th. Ram. 7 (1998), 93–114] gave an upper bound on the dimension dn of the space of order n Vassiliev knot invariants, by considering chord diagrams of a certain type. We present a simpler argument which gives a better bound on the number of these chord diagrams, and hence on dn.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

scholarly journals Ribbon graphs and bialgebra of Lagrangian subspaces

2016 ◽  
Vol 25 (12) ◽  
pp. 1642006 ◽  
Author(s):  
Victor Kleptsyn ◽  
Evgeny Smirnov
Keyword(s):  
Vector Space ◽  
Symplectic Form ◽  
Chord Diagram ◽  
Ribbon Graph ◽  
Dimensional Vector Space ◽  
Ribbon Graphs ◽  
Chord Diagrams ◽  
Intersection Matrix ◽  
Lagrangian Subspace ◽  
Lagrangian Subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

ENUMERATION OF CHORD DIAGRAMS AND AN UPPER BOUND FOR VASSILIEV INVARIANTS

1998 ◽  
Vol 07 (01) ◽  
pp. 93-114 ◽  
Author(s):  
A. Stoimenow
Keyword(s):  
Upper Bound ◽  
Vassiliev Invariants ◽  
Chord Diagrams ◽  
Enumeration Problem

We treat an enumeration problem of chord diagrams, which is shown to yield an upper bound for the dimension of the space of Vassiliev invariants for knots. We give an asymptotical estimate for this bound. As an aside, we present a trivial proof for the bound D!.

scholarly journals THE INTERSECTION GRAPH CONJECTURE FOR LOOP DIAGRAMS

2000 ◽  
Vol 09 (02) ◽  
pp. 187-211 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Equivalence Class ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Single Loop ◽  
Vassiliev Invariants ◽  
Chord Diagrams

The study of Vassiliev invariants for knots can be reduced to the study of the algebra of chord diagrams modulo certain relations (as done by Bar-Natan). Chmutov, Duzhin and Lando defined the idea of the intersection graph of a chord diagram, and conjectured that these graphs determine the equivalence class of the chord diagrams. They proved this conjecture in the case when the intersection graph is a tree. This paper extends their proof to the case when the graph contains a single loop, and determines the size of the subalgebra generated by the associated "loop diagrams." While the conjecture is known to be false in general, the extent to which it fails is still unclear, and this result helps to answer that question.

Vassiliev invariants and the Hopf algebra of chord diagrams

1996 ◽  
Vol 119 (1) ◽  
pp. 55-65 ◽  
Author(s):  
Simon Willerton
Keyword(s):  
Hopf Algebra ◽  
Canonical Projection ◽  
Double Points ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Type Formula ◽  
Chord Diagrams ◽  
Multivariate Calculus

AbstractThis paper is closely related to Bar-Natan's work, and fills in some of the gaps in [1]. Following his analogy of the extension of knot invariants to knots with double points to the notion of multivariate calculus on polynomials, we introduce a new notation which facilitates the formulation of a Leibniz type formula for the product of two Vassiliev invariants. This leads us to see how Bar-Natan's co-product of chord diagrams corresponds to multiplication of Vassiliev invariants. We also include a proof that the multiplication in is a consequence of Bar-Natan's 4T relation.The last part of this paper consists of a proof that the space of weight systems is a sub-Hopf algebra of the space *, by means of the canonical projection.

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

2006 ◽  
Vol 15 (07) ◽  
pp. 853-868
Author(s):  
ROMAN S. AVDEEV
Keyword(s):  
Knot Theory ◽  
Classical Case ◽  
Coefficient Function ◽  
Planar Curves ◽  
Vassiliev Invariants ◽  
Virtual Knots ◽  
Chord Diagrams ◽  
Virtual Links ◽  
Kauffman Polynomial

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].

scholarly journals UNIVERSAL VASSILIEV INVARIANTS OF LINKS IN COVERINGS OF 3-MANIFOLDS

2004 ◽  
Vol 13 (04) ◽  
pp. 515-555 ◽  
Author(s):  
JENS LIEBERUM
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Finite Covering ◽  
Real Projective Plane ◽  
The Real ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M. We construct universal Vassiliev invariants of links in M, where M=P2×[0,1] is a cylinder over the real projective plane P2, M=Σ×[0,1] is a cylinder over a surface Σ with boundary, and M=S1×S2. A finite covering p:N→M induces a map π1(p)* between labeled chord diagrams that corresponds to taking the preimage p-1(L)⊂N of a link L⊂M. The maps p-1 and π1(p)* intertwine the constructed universal Vassiliev invariants.

scholarly journals FINITE TYPE INVARIANTS OF LINKS WITH A FIXED LINKING MATRIX

2002 ◽  
Vol 11 (07) ◽  
pp. 1017-1041 ◽  
Author(s):  
ELI APPLEBOIM
Keyword(s):  
Finite Type ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
Knots And Links

In this paper we introduce two theories of finite type invariants for framed links with a fixed linking matrix. We show that these theories are different from, but related to, the theory of Vassiliev invariants of knots and links. We will take special note of the case of zero linking matrix. i.e., zero-framed algebraically split links. We also study the corresponding spaces of "chord diagrams".

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