On a Basis for the Framed Link Vector Space Spanned by Chord Diagrams
نویسندگان
چکیده
In view of the result of Kontsevich, [5] 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 in [7], [5], and [1], 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.
منابع مشابه
Vassiliev Invariants, Chord Diagrams, and Jacobi Diagrams
The goal of this paper is to understand the topological meaning of Jacobi diagrams in relation to knot theory and finite type knot invariants, otherwise known as Vassiliev invariants. To do so, we will first set up the notions of: the Vassiliev invariant and its respective space, the space of chord diagrams, a weight system on chord diagrams, and Jacobi diagrams and their respective space. Duri...
متن کاملFinite Type Invariants of Knots via Their Seifert Matrices∗
We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev’s filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial. The theory of finite type invariants (Vassiliev invariants) for knots was first introduced by V. Vassiliev [13] and reformula...
متن کاملChord diagram invariants of tangles and graphs
The notion of a chord diagram emerged from Vassiliev's work Vas90], Vas92] (see also Gusarov Gus91], Gus94] and Bar-Natan BN91], BN95]). Slightly later, Kontsevich Kon93] deened an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo the four-term relation. This knot invariant establishes an isomorphism of a projective limit ...
متن کاملOn Free Knots
The aim of the present paper is to study free knots, a dramatic simplification of the notion of virtual knots also connected to finite type invariants, curves on surfaces and other objects of low-dimensional topology. It turns out that the objects of such sort (a factorization of arbitrary graphs modulo three formal Reidemeister moves) can be studied by using some very much simpler objects: in ...
متن کاملVector Space semi-Cayley Graphs
The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V},S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1,v_2$ join by an edge whenever...
متن کامل