Paul Turner

We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.

A spectral sequence converging to Khovanov homology is constructed which is applied to calculate the rational Khovanov homology of (3,q)-torus links.

We define a homology theory for a certain class of posets equipped with a representation. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the "cube" complex defined by Khovanov.

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.Comment: This is the version published by Algebraic & Geometric Topology on 24 August 200

These lecture notes, which were designed for the Summer School "Heegaard-Floer Homology and Khovanov Homology" in Marseilles, 29th May - 2nd June, 2006, provide an elementary introduction to Khovanov homology. The intended audience is graduate students with some minimal background in low-dimensional and algebraic topology. The lectures cover the basic definitions, important properties, a number of variants and some applications. At the end of each lecture the reader is referred to the relevant literature for further reading.

A spectral sequence converging to Khovanov homology is constructed which is applied to calculate the rational Khovanov homology of (3,q)-torus links.

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E_2-pages can be described in terms of groups arising from the action of a certain endomorphism on mod 2 Khovanov homology. Some simple consequences are discussed.

Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.

For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev's 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories.

For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev's 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-21.abs.htm