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. During this process we will prove some of the more significant results from the topics and briefly justify their motivation. Once we have done this, we will then consider how Jacobi diagrams relate to singular knots and introduce Habiro’s clasper to provide a mapping between these two spaces. 1. The Vassiliev Knot Invariant and Classical Knot Polynomials: Definition of Vassiliev Invariants: Let K be the vector space over freely spanned by oriented knots in S. Then a singular knot is an immersion of S into S whose singularities are transversal double points. Furthermore, we can think of a singular knot as an element in K by removing the singularity with the following relation: