Research objectives and content
The discovery of the Jones polynomial led to a renaissance in knot theory but was problematic in that it was not understood in terms of classical ideas. Vassiliev invariants give one way of relating the 'quantum' knot theory to pre-Jones knot theory and topology.
My intention is to continue my programme of understanding connections between Vassiliev theory and classical knot theory. The method will be by studying cabling operations on Vassiliev invariants. Cabling is a natural, classical operation on knots, and it is known that Vassiliev invariants behave reasonably well under cabling. Detailed study of this behaviour should reveal further insight into the algebraic structure of the Vassiliev invariants. It should also, for instance, allow the universal Vassiliev invariants of torus knots to be calculated.
The Alexander-Conway polynomial is a classically very well understood knot invariant; it appears to be related to cabling operations on Vassiliev invariants. I will closely examine this relationship to try to unveil further connections between classical and quantum notions.
Training content (objective, benefit and expected impact)
I will be working in a World-class mathematics department with active researchers in my own and also in closely allied fields: this will provide a stimulating atmosphere and a broad mathematical background for my mathematical development.
Links with industry / industrial relevance (22)