From Vassiliev to Khovanov: Finite-type Invariants
and Categorification for Knotted Objects

VasKho is a french research project on finite-type invariants and categorification of knotted objects supported by Agence Nationale de la Recherche.

The project involves :
  • Benjamin Audoux (CMI),
  • Paolo Bellingeri (LMNO),
  • Jean-Baptiste Meilhan (IF) - coordinator,
  • Emmanuel Wagner (IMB)

 Research Area

The broad area of research of the VasKho project is knot theory, and more generally the study of knotted objects, such as links, braids, virtual/welded knots and - even more generally - surfaces and 3-manifolds.

The project articulates around two spectacular developments which revolutionized knot theory over the past two decades : finite-type invariants (such as Vassiliev invariants) and the concept of categorification (starting with Khovanov homology) provide indeed completely new approaches to the study of knot invariants. The notion of finite-type invariant provides a unified framework for the study of many invariants, that includes polynomial invariants, and enables a wide range of new technics to study them. Categorification generalises polynomial invariants by interpreting them as the graded Euler characteristic of some richer homological invariants.

Both topics led to some of the most important developments of recent research in knot theory.

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. 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.

VasKho involves four french researchers, and aims at creating a french pole of expertise in this boiling field of research.