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.

scholarly journals TREE DIAGRAMS FOR STRING LINKS II: DETERMINING CHORD DIAGRAMS

2008 ◽  
Vol 17 (06) ◽  
pp. 649-664
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
String Links

In previous work [7], we 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 look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

scholarly journals Genus Ranges of Chord Diagrams

2015 ◽  
Vol 24 (04) ◽  
pp. 1550022 ◽  
Author(s):  
Jonathan Burns ◽  
Nataša Jonoska ◽  
Masahico Saito
Keyword(s):  
Outer Boundary ◽  
Orientable Surface ◽  
Fixed Number ◽  
Chord Diagram ◽  
Boundary Circle ◽  
Line Segments ◽  
Chord Diagrams ◽  
Computer Calculations

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can be, and those that cannot be, realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.

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 WEIGHT SYSTEMS FOR MILNOR INVARIANTS

2008 ◽  
Vol 17 (02) ◽  
pp. 213-230 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Spanning Tree ◽  
Recursive Relation ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Link Invariants ◽  
Finite Type Invariants ◽  
Milnor Invariants ◽  
Skein Relation

We use Polyak's skein relation to give a new proof that Milnor's string link invariants μ12⋯n are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram.

BRAIDED CHORD DIAGRAMS

1998 ◽  
Vol 07 (01) ◽  
pp. 1-22 ◽  
Author(s):  
Joan S. Birman ◽  
Rolland Trapp
Keyword(s):  
Finite Type ◽  
Equivalence Relation ◽  
Chord Diagram ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
Braid Index

The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which can be represented as closed 3-braids. Partial results are obtained about spanning sets for the algebra of chord diagrams of braid index 3.

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