arXiv research

New proof for sphere recognition algorithm.

New algorithm for recognizing 3-spheres using Groebner basis methods.

In this paper we describe a procedure to simplify any given triangulation of the 3-sphere using Pachner moves. We obtain an explicit exponential-type bound on the number of Pachner moves needed for this process. This leads to a new recognition algorithm for the 3-sphere.

Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognit…

A triangulation of a $3$-manifold can be shown to be homeomorphic to the $3$-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a …

Integer homology 3-spheres have irreducible SU(2) representations.

It is a major unsolved problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time algorithm. In practice, trivial knots (which can be untangled) are typically easy to identify using fast simplification techniques, wherea…

The crushing operation of Jaco and Rubinstein is a powerful technique in algorithmic 3-manifold topology: it enabled the first practical implementations of 3-sphere recognition and prime decomposition of orientable manifolds, and it plays a prominent role in state-of-the-art algorithms for unknot recognition and testin…

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension …

Study on 3-sphere Goeritz group's twisted first homology group.

This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite di…

Stable 2-lobed Delaunay tori found in 3-sphere.

Characterizes a specific type of convex curves on a 3-sphere.

Real algebraic structures help classify overtwisted contact 3-spheres.

Disk complexes show 3-sphere surfaces are topologically minimal.

This paper shows how to unknot 2-manifolds in 3-sphere using twistings.

We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-…

The paper examines conditions for contact surgeries on rational homology 3-spheres.

5 minimal tori found in 3-spheres with positive Ricci curvature.

For any knot, a 3-sphere triangulation exists with a knotted edge.

New rational homology 3-spheres found that can't bound definite 4-manifolds.

Minimal surfaces in spheres found for any genus.

The paper studies invariants of surfaces in the 3-sphere using handlebody-links.

The paper constructs scalar invariants for surfaces in a 3-sphere using Minkowski spacetime.

Paper finds infinitely many 3-spheres without nice orderings.

Proving NP-hardness of embedding 3-manifolds with boundary tori into R^3.

Flat tori in 3-sphere have π extrinsic diameter under specific conditions.

Classifies Legendrian Hopf links in 3-sphere.

Diagrammatic method characterizes non-split surfaces in 3-sphere.

The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.

We prove Mayberry-Murasugi's formula for links in homology 3-spheres, which was proved before only for links in the 3-sphere. Our proof uses Franz-Reidemeister torsions.

Classifies SL(2;C) representations of a Brieskorn homology 3-sphere.

First examples of 3-spheres smoothly embedding in S^2 x S^2 but not S^4.

Study shows knots in 3-manifolds imply 3-sphere if certain conditions met.

Perelman's proof confirmed, new method uses 4D topology.

Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.

Constructs fat, shellable 3-spheres with specific $f$-vectors.

Study invariants of $\mathbb{Z}/p$-homology 3-spheres from abelianization of mapping class groups.

Researchers compute Reidemeister torsion for a specific 3-sphere.

Paper connects hypersurfaces in 4D to surfaces in 3-sphere.

New rack and multiple group rack cohomology for surfaces in 3-sphere.

New invariant fully describes finite type invariants of knots in homology 3-spheres.

Authors prove a conjecture about the Goeritz group of 3-sphere Heegaard splittings.

New steady Euler flows found on 3-sphere and Sasakian manifolds.

Study of cosmetic contact surgeries on knots, excluding some cases.

We show that the perturbative ${\frak g}$ invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra ${\frak g}$, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …

We show that on a Sasakian 3-sphere the Sasaki-Ricci flow initiating from a Sasakian metric of positive transverse scalar curvature converges to a gradient Sasaki- Ricci soliton. We also show the existence and uniqueness of gradient Sasaki-Ricci soliton on each Sasakian 3-sphere.

For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.