Knot Theory

Vassily Olegovich Manturov is professor of Geometry and Topology at Bauman Moscow State Technical University.

Knots, links, and invariant polynomials. Introduction. Reidemeister moves. Knot arithmetics. Links in 2-surfaces in R3.Fundamental group; the knot group. The knot quandle and the Conway algebra. Kauffman's approach to Jones polynomial. Properties of Jones polynomials. Khovanov's complex. Theory of braids. Braids, links and representations of braid groups. Braids and links. Braid construction algorithms. Algorithms of braid recognition. Markov's theorem; the Yang-Baxter equation. Vassiliev's invariants. Definition and Basic notions of Vassiliev invariant theory. The chord diagram algebra. The Kontsevich integral and formulae for the Vassiliev invariants. Atoms and d-diagrams. Atoms, height atoms and knots. The bracket semigroup of knots. Virtual knots. Basic definitions and motivation. Invariant polynomials of virtual links. Generalised Jones-Kauffman polynomial. Long virtual knots and their invariants. Virtual braids. Other theories. 3-manifolds and knots in 3-manifolds. Legendrian knots and their invariants. Independence of Reidemeister moves.