arXiv research

David Gabai proves a 4D light bulb theorem using smooth techniques.

Constructs a concordance between standard surfaces in $S^2 imes S^2$.

New theorem shows when spheres with duals can be smoothly deformed.

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a neg…

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…

David Gabai showed that disk decomposable knot and link complements carry taut foliations of depth one. In an arbitrary sutured 3-manifold M, such foliations F, if they exist at all, are determined up to isotopy by an associated ray [F] issuing from the origin in H^1(M;R) and meeting points of the integer lattice H^1(M…

New evidence supports the Euler class one conjecture for tight contact structures.

An algorithm calculates Gabai width for thousands of knots.

Defines new metric space sections with Ahlfors-David regularity.

We present a new, completely three-dimensional proof of the fact, due to Gabai-Eliashberg-Thurston, that every closed, oriented, irreducible 3-manifold with nonzero second homology carries a universally tight contact structure.

The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.

Any Haken 3--manifold (possibly with boundary consisting of tori) can be transformed into a $\mathrm{surface}\times S^1$ by a series of splitting and regluing along incompressible surfaces. This fact was proved by Gabai as an application of his sutured manifold theory. The first half of this paper provides a few techni…

We show that if a knot has a minimal spanning surface that admits certain Gabai disks, then this knot has Property P. As one of the applications we extend and simplify a recent result of Menasco and Zhang that closed 3-braid knots have Property P. Other applications are given.

New invariant detects more elements in 4D diffeomorphism group.

In this article we extend a euclidean result of David and Semmes to the Heisenberg group by giving a sufficient condition for a $k$-Ahlfors-regular subset to have big pieces of bilipschitz images of subsets of $\R^k$. This Carleson type condition measures how well the set can be approximated by the Heisenberg $k$-plane…

Classifies fibered ribbon pretzels, except for a few cases.

Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positi…

Study extends surface embedding theorems to non-orientable cases.

New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.

In this technical note, we adapt an idea of Gabai to construct non-uniquely ergodic, non-geometric, arational trees.

4D theorem for disks, generalizing previous work.

Discussing hybrid models in Bayesian networks.

Critical surfaces can be regarded as topological index 2 minimal surfaces which was introduced by David Bachman. In this paper we give a sufficient condition and a necessary condition for self-amalgamated Heegaard surfaces to be critical.

The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.

We review a construction of hyperkahler metrics proposed in joint work of Davide Gaiotto, Greg Moore and the author. A key ingredient in this construction is a collection of integer "DT invariants" obeying the wall-crossing formula of Kontsevich-Soibelman.

We give a sufficient condition for a branched surface in a 3 dimensional manifold to fully carry a lamination, giving a piece of answer to a classical question of D. Gabai.

We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants …

Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalised knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and…

We show that for any nontrivial knot in $S^3$, there is an open interval containing zero such that a Dehn surgery on any slope in this interval yields a 3-manifold with taut foliations. This generalizes a theorem of Gabai on zero frame surgery.

Survey explores interactions between four conformal dynamics branches.

A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.

We present a short elementary proof of an existence theorem of certain CAT(-1)-surfaces in open hyperbolic 3-manifolds. The main construction lemma in Calegari and Gabai's proof of Marden's Tameness Conjecture can be replaced by an applicable version of our theorem.

These are problems on Heegaard splittings, that were raised at the Workshop, listed according to their contributors: David Bachman, Mario Eudave-Munoz, John Hempel, Tao Li, Yair Minsky, Yoav Moriah and Richard Weidmann. On pages 285-298 of this monograph (arxiv:0904.0017) Hyam Rubinstein gives a personal collection of …

Marden's Tameness Conjecture predicts that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. It was recently established by Agol and Calegari-Gabai. We will survey the history of work on this conjecture and discuss its many applications.

ICML workshop on making machine learning models more understandable.

One of the more memorable moments of last summer's credit crunch came when the CFO of Goldman Sachs, David Viniar, announced in August that Goldman's flagship GEO hedge fund had lost 27% of its value since the start of the year. As Mr. Viniar explained, "We were seeing things that were 25-standard deviation moves, seve…

New algorithm learns multiclass concepts with finite Littlestone dimension.

New method finds unique branched surfaces in 3-manifolds.

Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduce…

We enumerate the small-volume manifolds that can be obtained by Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai, Meyerhoff, and the author. In so doing we complete the proof that the Weeks manifold is the minimum-volume compact hyperbolic 3-manifold, as well as enumerating the 10 smallest one-cusped hyper…

Ozsváth and Szabó conjectured that knot Floer homology detects fibred links. We will verify this conjecture for closed 3-braids, by classifying fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is fibred, or it differs from a fibred link by a half twist. The proof uses Gabai's method o…

We correct and complete a conjecture of D. Gabai, R. Meyerhoff and N. Thurston on the classification and properties of thin tubed closed hyperbolic 3-manifolds. We additionally show that if N is a closed hyperbolic 3-manifold, then either N=Vol3 or N contains a closed geodesic that is the core of an embedded tube of ra…

Any two compact, complete, one-dimensional geodesic spaces with identical marked length spectrum have isometric $π_1$-hull. The present version contains errors, notably in Lemmas 2.2 and 2.3 (path cancellations can be more complicated), which then propagate through the paper. The main result is correct as stated, and a…

We study Dehn surgeries on null-homotopic knots that yield fibred $3$--manifolds when an additional (but natural) homological restriction is imposed. The major tool used is Gabai's theory of sutured manifold decomposition. Such surgeries are negative examples to a question of Michel Boileau. Another result we will prov…

It has been pointed out to the author by David Glickenstein that the proof of the (closely related) Lemmas 1.2 and 3.2 in the title paper is incorrect. The statements of both Lemmas are correct, and the purpose of this note is to give a correct argument. The argument is of some interest in its own right.

Improved mistake bounds for transductive online learning.

Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ …

Blackwell's theorems influence modern AI through information compression and decision making.