arXiv research

The paper proves the existence of at least 4 embedded minimal tori in a three-sphere with positive Ricci curvature.

Proves complex homology three-spheres can be bounded by many handles.

Manolescu correction terms are numerical invariants of homology three-spheres arising from $\mathrm{Pin}(2)$-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…

We show that a free period three action on the three-sphere is standard, i.e. the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts.

We determine the lengths of all closed sub-Riemannian geodesics on the three-sphere. Our methods are elementary and allow us to avoid using explicit formulas for the sub-Riemannian geodesics.

Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of those links is zero. We use this criterion to give a new proof of a result of Yano…

Study shows moduli space of fibrations has specific homotopy types.

We construct embedded closed minimal surfaces in the round three-sphere, resembling two parallel copies of the Clifford torus, joined by m^2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus.

Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.

In this short note, we prove that on the three-sphere with any bumpy metric there exist at least four solutions of the Allen-Cahn equation with spherical interface and index at most two. The proof combines several recent results from the literature.

Study shows volume and genus unrelated for hyperbolic fibred knots.

Contact group retracts to unitary subgroup.

We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a constant mean curvature $H$-sphere in the space that is unique up to an ambient isometry.

The paper solves problems related to curvature on a 3-sphere.

We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.

We present another proof of the sharp inequality for Paneitz operator on the standard three sphere, in the spirit of subcritical approximation for the classical Yamabe problem. To solve the perturbed problem, we use a symmetrization process which only works for extremal functions. This gives a new example of symmetriza…

Rectangular diagrams help analyze foliations in 3-sphere.

How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon-Sm…

We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olev…

Proves conjecture about sphere widths under rotational symmetry.

A smooth foliation of a Riemannian manifold is metric when its leaves are locally equidistant and is homogenous when its leaves are locally orbits of a Lie group acting by isometries. Homogenous foliations are metric foliations, but metric foliations need not be homogenous foliations. We prove that a homogenous three-s…

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology…

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on…

We study supergravity solutions corresponding to fivebranes wrapped on a three-sphere inside a G_2 holonomy manifold. By changing a parameter the solutions interpolate between a G_2 manifold X_i \cong S^3 x R^4 with flux on a three-sphere and a distinct G_2 manifold X_j \cong S^3 x R^4 with branes on another three-sphe…

The study examines conditions for Haken 3-manifolds and their fundamental groups.

We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a manifold with more than two connected summands, we show that r is bounded in absolute v…

Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.

A 3-manifold $M$ is said to be $p$-periodic ($p\geq 2$ an integer) if and only if the finite cyclic group of order $p$ acts on $M$ with a circle as the set of fixed points. This paper provides a criterion for periodicity of rational homology three-spheres. Namely, we give a necessary condition for a rational homology t…

We explore certain restrictions on knots in the three-sphere which admit non-trivial Seifert fibered surgeries. These restrictions stem from the Heegaard Floer homology for Seifert fibered spaces, and hence they have consequences for both the Alexander polynomial of such knots, and also their knot Floer homology. In pa…

Study strong inversions on knots using Khovanov homology.

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with Jones- Witten-Reshetikhin-Turaev invariant arising from Chern-Simons filed theory for the same Lie algebra and the same root of unity on all integer homol- ogy three-spheres, at roo…

Knots generating infinite subgroup bound rational homology balls.

We discuss a concordance invariant constructed from Heegaard Floer homology "correction terms" and +/- 1 surgeries on knots in the three-sphere.

It is shown that if the exterior of a link L in the three sphere admits a genus 2 Heegaard splitting, then L has Generalized Property R.

We show that if a prime homology sphere has the same Floer homology as the standard three-sphere, it does not contain any incompressible tori.

We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$.…

We prove a semi-global result on the existence of conformal embeddings of the two-sphere into the round three-sphere S^3(1) with prescribed mean curvature.

CR Yamabe flow fails to converge on small deformations of the standard CR three-sphere.

We calculate the weak homotopy type of the group of contactomorphisms of the three-sphere which coincide with the identity on (a neighborhood of) an overtwisted disk.

We prove various results on contact structures obtained by contact surgery on a single Legendrian knot in the standard contact three--sphere. Our main tool are the contact Ozsvath--Szabo invariants.

We describe a contact analog of the symplectic cut construction. As an application we show that the group of contactomorphisms for a particular overtwisted contact structure on the three sphere contains countably many nonconjugate two tori.

Study non-orientable 4-genus for 11-crossing non-alternating knots.

Study shows infinite-rank summand in homology concordance group of knots.

Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. Thi…

We prove a surgery formula for the renormalized Euler characteristic of Ozsvath and Szabo. Equality between this Euler cahracteristic and the Seiberg-Witten invariant follows for rational homology three-spheres.

We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the equator, or along the equatorial circle and the poles. To carry out these constructions…

This note is an exposition of Waldhausen's proof of Waldhausen's Theorem: the three-sphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister-Singer Theorem.

As a generalization of a classical result on the Alexander polynomial for fibered knots, we show in this paper that the Reidemeister torsion associated to a certain representation detects fiberedness of knots in the three sphere.