Algebraic Structures Associated with Embedded Graphs

Abstract

This chapter is devoted mainly to the description of algebraic and combinatorial constructions related to Vassiliev’s theory of knot invariants [283], [284], [285]. The main combinatorial object of the theory is a chord diagram, or, which is the same, a one-vertex map. We start with a description of this notion and of the famous 4-term relation for for the chord diagrams. Our presentation follows the main lines of Bar-Natan [18], but we also pay a lot of attention to the recent development in the analysis of intersection graphs of chord diagrams. We do not present the proof of Kontsevich’s theorem for Vassiliev invariants since its machinery is too far from the subject of our book. Good explanations are available in [50], [195].