Paul Turner

The homotopy theory of Khovanov homology

We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology. 57M27; 55P42 Motivation and introduction In order to apply the methods of homotopy theory to Khovanov homology there are several natural approaches. One is to build a space or spectrum whose classical invariants give Khovanov homology, then show its homotopy type is a link invariant, and finally study this space using homotopy theory. Ideally this approach would begin with some interesting geometry and lead naturally to Khovanov homology. One also might hope to construct something more refined than Khovanov homology in this way (see Lipshitz and Sarkar [12] for a combinatorial approach to this). Another approach is to interpret the existing constructions of Khovanov homology in homotopy theoretic terms. By placing the constructions into a homotopy setting one makes Khovanov homology amenable to the methods and techniques of homotopy theory. In this paper our interest is with the second of these approaches. Our aim is to show that Khovanov homology can be interpreted in a homotopy theoretic way using homotopy limits and to subsequently develop a number of results about the specific type of homotopy limit arising. The latter will provide homotopy tools appropriate for studying Khovanov homology.

Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.

THE HOMOLOGY OF DIGRAPHS AS A GENERALIZATION OF HOCHSCHILD HOMOLOGY

Przytycki has established a connection between the Hochschild homology of an algebra A and the chromatic graph homology of a polygon graph with coefficients in A. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary A–A bimodule, for A possibly non-commutative, which on polygons agrees with Hochschild homology through a range of dimensions.