On a Basis for the Framed Link Vector Space Spanned by Chord Diagrams

TL;DR

This paper constructs a basis for the space of chord diagrams modulo 4T relations for diagrams with up to five chords, advancing the understanding of Vassiliev invariants in knot theory.

Contribution

It provides an explicit basis for the space of chord diagrams with up to five chords, addressing a key problem in the theory of Vassiliev invariants.

Findings

Basis constructed for diagrams with n ≤ 5 chords

Clarifies the structure of chord diagram spaces

Supports the fundamental theorem of Vassiliev theory

Abstract

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, 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 distinguishable link components, modulo 4T relations for n less than or equal to 5.