Using the grid diagram formulation of knot Floer homology, Ozsváth, Szabó and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsváth, Stipsicz and Szabó defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.

Abstract. Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus version of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology.

We prove that the knot Floer homology of a fibered knot is nontrivial in its nextto-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with L-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots in the 3-sphere. We also generalize the latter result, proving a similar theorem for nullhomologous knots in any 3-manifold. We note that our method of proof inspired Baldwin and Sivek's recent proof that Khovanov homology detects the trefoils. As part of this work, we also introduce a numerical refinement of the Ozsváth-Szabó contact invariant. This refinement was the inspiration for Hubbard and Saltz's annular refinement of Plamenevskaya's transverse link invariant in Khovanov homology.

To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientationpreserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space. C 2013 American Institute of Physics.

We study 4-dimensional homology cobordisms without 3-handles via Heegaard and instanton Floer homologies, character varieties, and Thurston geometries. We provide obstructions to such cobordisms arising from each of these theories, and illustrate some topological applications.

No abstract

Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one further integer µ, the "triple linking number" of the title, which is well-defined modulo the greatest common divisor of p, q and r.To each such link L we associate a geometrically natural characteristic map g L from the 3-torus to the 2-sphere in such a way that link homotopies of L become homotopies of g L . Maps of the 3-torus to the 2-sphere were classified up to homotopy by Lev Pontryagin in 1941. A complete set of invariants is given by the degrees p, q and r of their restrictions to the 2-dimensional coordinate subtori, and by the residue class of one further integer ν, an "ambiguous Hopf invariant" which is well-defined modulo twice the greatest common divisor of p, q and r.

We provide the first example of a Legendrian knot with nonvanishing contact homology whose Thurston-Bennequin invariant is not maximal.