From Vassiliev to Khovanov – Finite-type invariants and Categorification for knotted objects – VasKho

The project VasKho articulates around the two major theories which revolutionized knot theory over the past twenty years, namely the notions of finite-type invariants and categorification. On one hand, the theory of finite-type invariants, initiated by Goussarov and Vassiliev, provides a unified framework for the study of invariants of knots and knotted objects which includes, in particular polynomial invariants. On the other hand, the theory of categorification considerably enhanced polynomial invariants by interpretating them as the graded Euler characteristic of some richer invariants of homological nature. It includes, for instance, the Khovanov and Heegaard-Floer homologies.

There are two main parts in this project. The first one proposes to pursue the study of some of the central problems raised in each of these theories (Tasks 1 and 2). They concern knotted objects, such as usual/virtual/welded links and braids, as well as 3-manifolds. The second part aims at the study of the yet widely open problem of the nature of the connections between finite-type invariants and categorification (Task 3).

The project involves four freshly hired Maîtres de Conférence who are all specialized in finite-type invariants or categorification, and aims at creating a french network on these subjects and their interactions.