Related papers: The Hopf invariant of a Haefliger knot

The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as…

In an earlier paper, we used the absolute grading on Heegaard Floer homology to give restrictions on knots in $S^3$ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising…

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends…

We extend the notion of thin multiple Heegaard splittings of a link in a 3-manifold to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly compressible bridge…

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the…

We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space…

We introduce and study bridge decompositions for 3-manifolds embedded in the 5-sphere. These generalize both the classical notion of bridge position for knots in the 3-sphere and the bridge trisections of surfaces in the 4-sphere due to…

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we…

Hedden defined two knots in each lens space that, through analogies with their knot Floer homology and doubly pointed Heegaard diagrams of genus one, may be viewed as generalizations of the two trefoils in S^3. Rasmussen shows that when the…

In ``Knots in lattice homology", Ozsv\'ath, Stipsicz, and Szab\'o showed that knot lattice homology satisfies a surgery formula similar to the one relating knot Floer homology and Heegaard Floer homology, and in previous work, I showed that…

We construct a new family of knot concordance invariants $\theta^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg-Witten-Floer cohomology, constructed by the author and Hekmati, applied to the…

In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that…

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a…

Regular homotopy classes of immersions of a 3-sphere in 5-space constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an…

We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable $3$-manifold of the form (non-orientable surface) $\times$ $[0, 1]$.…

By extending a result of Kronheimer-Mrowka to the family setting, we prove a gluing formula for the family Seiberg-Witten invariant. This formula allows one to compute the invariant for a smooth family of 4-manifolds by cutting it open…

Let M be a closed 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C^N. Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than…

We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in…

Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is…