Instanton Floer homology and the Alexander polynomial

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots.Comment: 25 pages, 6 figures. Revised version, correcting errors concerning mod 2 gradings in the skein sequenc

Gauge theory and Rasmussen's invariant

A previous paper of the authors' contained an error in the proof of a key claim, that Rasmussen's knot-invariant s(K) is equal to its gauge-theory counterpart. The original paper is included here together with a corrigendum, indicating which parts still stand and which do not. In particular, the gauge-theory counterpart of s(K) is not additive for connected sums.Comment: This version bundles the original submission with a 1-page corrigendum, indicating the error. The new version of the corrigendum points out that the invariant is not additive for connected sums. 23 pages, 3 figure

Filtrations on instanton homology

In earlier work of the authors, the Khovanov complex of a knot or link appeared as the first page in a spectral sequence abutting to the instanton homology. The quantum and (co)homological gradings on Khovanov homology do not survive as gradings, but we show that they survive as filtrations.Comment: 40 pages, 4 figures. Revised version, with corrected typos and extended introductio

Khovanov homology is an unknot-detector

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.Comment: 124 pages, 13 figure

Micron-Scale Resolution Radiography of Laser-Accelerated and Laser-Exploded Foils Using an Yttrium X-Ray Laser

The authors have imaged laser-accelerated foils and exploding foils on the few-micron scale using an yttrium x-ray laser (155 {angstrom}, 80 eV, {approximately}200 ps duration) and a multilayer mirror imaging system. At the maximum magnification of 30, resolution was of order one micron. The images were side-on radiographs of the foils. Accelerated foils showed significant filamentation on the rear-side (away from the driving laser) of the foil, although the laser beam was smoothed. In addition to the narrow rear-side filamentation, some shots revealed larger-scale plume-like structures on the front (driven) side of the Al foil. These plumes seem to be little-affected by beam smoothing and are likely a consequence of Rayleigh-Taylor instability. The experiments were carried out at the Nova two-beam facility