Categorification of the Kauffman bracket skein module of I-bundles over surfaces

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients. After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefficients of L in the standard basis of the skein module of M. Therefore, our homology groups provide a `categorification' of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro's exact sequence for our homology groups. Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L.Comment: Version 2 was obtained by merging math.QA/0403527 (now removed) with Version 1. This version is published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-52.abs.htm

Persistent clinical efficacy and safety of anti-tumour necrosis factor \textgreeka therapy with infliximab in patients with ankylosing spondylitis over 5 years: evidence for different types of response

Background: There is insufficient evidence for the long-term efficacy and safety of anti-tumour necrosis factor therapy in patients with ankylosing spondylitis (AS). This is the first report on the treatment with infliximab over 5 years.Methods: As part of a multicentre randomised trial, 69 patients with active AS at baseline (BL) have been continuously treated with infliximab (5 mg/kg i.v. every 6 weeks)---except for a short discontinuation after 3 years (FU1). The primary outcome of this extension was remission according to the ASsessment in Ankylosing Spondylitis (ASAS) criteria at the end of year 5 of the study (FU2).Results: Of the 43 patients who completed year 3, 42 agreed to continue, 38 of which (90.5%) finished year 5 (55% of 69 initially). Partial clinical remission was achieved in 13 of 38 patients (34.2%) at FU1 and FU2. At FU2, the mean Bath Ankylosing Spondylitis Disease Activity Index (BASDAI) was 2.5±1.9 (BL:6.4, FU1:2.5). BASDAI values <4 were seen in 79% of patients at both, FU1 and FU2. ASAS 20% and 40% responses were seen in 32 (84%) and 24 (63%) patients at FU2, respectively. Most patients classified as non-responders at FU2 were part-time responders, as all but one patient achieved an ASAS 20% response at least once within the last 2 years. Three types of responders were identified. No major side effects occurred during years 4 and 5 of infliximab therapy.Conclusions: Infliximab is safe and efficacious in AS patients over 5 years. The majority of the patients remained on treatment and had rather persistent levels of low disease activity. Different response types could be identified

Conformally Einstein Products and Nearly K\"ahler Manifolds

In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella classification admitting a parallel vector field and show that (under some regularity assumption) they are obtained as mapping tori of isometries of compact Sasaki-Einstein 5-dimensional manifolds. In particular, we obtain examples of inhomogeneous locally (non-globally) conformal nearly K\"ahler compact manifolds