Khovanov Homology for Tangles in Connected Sums

TL;DR

This paper extends Khovanov homology to links in certain 3-manifolds, constructing invariants for tangles via type D and type A structures that are compatible with gluing operations.

Contribution

It introduces a novel extension of Khovanov homology to tangles in connected sum 3-manifolds using type D and type A structures, generalizing Roberts' work.

Findings

Constructed type D and type A structures as invariants of tangles.

Gluing these structures recovers the classical Khovanov homology.

Extended the applicability of Khovanov homology to new 3-manifold settings.

Abstract

Khovanov homology is an invariant for links in the three sphere that categorizes the Jones polynomial. We extend Khovanov's construction to links in 3-manifolds that are connected sums of orientable interval bundles over surfaces. Cutting the 3-manifold along a separating sphere, we construct type D and type A structures that are invariants of tangles in the two halves following the work of Roberts. Gluing the type D and type A structures along the common boundary recovers the Khovanov homology of the link.