arXiv research

The paper develops surgery theories for foliations and solves a problem posed by Weinberger.

Proves Poincaré surgery theorem using homotopy theory.

In this paper, we prove that there are no truly cosmetic surgeries on genus one classical knots. If the two surgery slopes have the same sign, we give the only possibilities of reflectively cosmetic surgeries. The result is an application of Heegaard Floer theory and number theory.

Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.

This paper is an introduction to the use of the cobordism of chain complexes with Poincaré duality in surgery theory. It is a companion to the author's paper "An introduction to algebraic surgery" math.AT/0008071 (to appear in Volume 2 of Surveys in Surgery Theory, Ann. of Maths. Studies, Princeton, 2001) which is an i…

This paper uses Steenrod homology to simplify surgery on generalized manifolds.

The abstract discusses constructing 3d N=2 gauge theories using surgeries and M5-branes.

A surgery classification theory is introduced for manifolds of bounded geometry up to quasi-isometry. The Borel conjecture for this theory is proven for flat Euclidean space.

Graphs describe contact surgery on 3-manifolds.

Two knot families meet cosmetic surgery conjecture.

Browder-Novikov-Sullivan-Wall surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory (such as the Wall surgery obstruction groups), without losing sight of the geometric…

New proof for a knot type not admitting certain surgeries.

Study calculates instanton Floer homology for surgeries on L-space knots.

Paper restricts chirally cosmetic surgeries on knots.

An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.

Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.

Dehn surgery homeomorphic pairs contradict a conjecture.

Contact surgeries yield algebraically overtwisted manifolds.

Study proves nontrivial knots can't undergo cosmetic surgeries.

The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small seifert $\mathbb{Z}/2\mathbb{Z}$-homology sphere or a toroidal irreducible non-Seifert surgery then t…

Proves exact triangle linking knot instanton Floer homology to surgeries.

We use sutured manifold theory, essential laminations and essential branched surfaces to establish the upper bounds of distances between certain types of nonsimple Dehn surgery slopes. This is the revised version of an earlier preprint {\it Dehn surgery and simple manifolds.}

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…

New group theory insights on knot surgery results.

Study uses instanton Floer theory to obstruct knot unknotting operations.

New formula for dual knots using involutions.

We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally …

The study calculates and analyzes alternating surgeries for various knots.

The validity of Freedman's disk theorem is known to depend only on the fundamental group. It was conjectured that it fails for nonabelian free fundamental groups. If this were true then surgery theory would work in dimension four. Recently, Krushkal and Lee proved a surprising result that surgery theory works for a lar…

Special knots with many twists have no certain type of surgery.

Study shows most odd pretzel knots don't allow chirally cosmetic surgeries.

The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.

We apply the geometric-topology surgery theory on spacetime manifolds to study the constraints of quantum statistics data in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable creating anyon excitations of particles and strings, well-defined in gapped st…

Paper proves knots satisfy a conjecture using Jones polynomial.

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive …

Bi-contact surgery operations can be applied to Anosov flows.

Generates sutured manifolds from a surface using surgery.

It is well-known that an n-dimensional Poincaré complex $X^n$, $n \ge 5$, has the homotopy type of a compact topological $n$-manifold if the total surgery obstruction $s(X^n)$ vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary algebraic an…

Refined 3D index uses surgery and gradings to distinguish 3-manifolds.

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(s) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a …

Satellite formula connects knot concordance invariants to surgery.

Characterizes $(1,1)$ knots with large Dehn surgeries.

We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is…

Extends string-net theory to 3D TQFT via surface graphs and surgery.

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N)\ge min (σ(M),Λ) for Λ>0. We derive explicit lower bounds for Λin dimensions where previous methods failed, namely for (n,k)\in {(4,1),(5,1),…

The study shows a limit on cosmetic surgeries for certain knots.

It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parame…

Introduces integer-valued Heegaard Floer theory with canonical orientations.