Mathematics › Geometry and Topology

Geometric and Algebraic Topology

Works Count: 65372

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This cluster of papers explores the intersection of symplectic topology, knot invariants, and related areas such as holomorphic disks, Floer homology, contact geometry, group theory, hyperbolic manifolds, quantum topology, braid groups, and the study of protein knotting.

Keywords

Symplectic Topology; Knot Invariants; Holomorphic Disks; Floer Homology; Contact Geometry; Group Theory; Hyperbolic Manifolds; Quantum Topology; Braid Groups; Protein Knotting

An enumeration of knots and links, and some of their algebraic properties

1970-01-01

James Conway

The topology of four-dimensional manifolds

1982-01-01

Michael Freedman

To my teachers and friends 0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's.Of the mysteries still remaining after that period of great success the … To my teachers and friends 0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's.Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four.Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days 1 that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five.There is such a principle.It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary.The main impact, as outlined in §1, is to the classification of 1-connected 4-manifolds and topological end recognition.However, certain applications to nonsimply connected problems such as knot concordance are also obtained.The discovery of this principle was made in three stages.From 1973 to 1975 Andrew Casson developed his theory of "flexible handles" 2 .These are certain pairs having the proper homotopy type of the common place open 2-handle H = (D 2 X D 2 , dD 2 X D 2 ) but "flexible" in the sense that finding imbeddings is rather easy; in fact imbedding is implied by a homotopy theoretic criterion.It was clear to Casson 3 that: (1) no known invariant-link theoretic

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Combinatorial Group Theory

1990-01-01

Roger C. Lyndon , Paul E. Schupp

Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups … Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups 7. Quadratic sets of Word 8. Equations in Free Groups 9. Abstract Lenght Functions 10. Representations of Free Groups 11. Free Pruducts with Amalgamation Chapter II. Generators and Relations 1. Introduction 2. Finite Presentations 3. Fox Calculus, Relation Matrices, Connections with Cohomology 4. The Reidemeister-Schreier Method 5. Groups with a Single Defining Relator 6. Magnus' Treatment of One-Relator Groups Chapter III. Geometric Methods 1. Introduction 2. Complexes 3. Covering Maps 4. Cayley Complexes 5. Planar Caley Complexes 6. F-Groups Continued 7. Fuchsian Complexes 8. Planar Groups with Reflections 9. Singular Subcomplexes 10. Sherical Diagrams 11. Aspherical Groups 12. Coset Diagrams and Permutation Representations 13. Behr Graphs Chpter IV. Free Products and HNN Extensions 1. Free Products 2. Higman-Neumann-Neumann Extensions and Free Products with Amalgamation 3. Some Embedding Theorems 4. Some Decision Problems 5. One-Relator Groups 6. Bipolar Structures 7. The Higman Embedding Theorem 8. Algebraically Closed Groups Chapter V. Small Cancellation Theory 1. Diagrams 2. The Small Cancellation Hypotheses 3. The Basic Formulas 4. Dehn's Algorithm and Greendlinger's Lemma 5. The Conjugacy Problem 6. The Word Problem 7. The Cunjugacy Problme 8. Applications to Knot Groups 9. The Theory over Free Products 10. Small Cancellation Products 11. Small Cancellation Theory over free Products with Amalgamation and HNN Extensions Bibliography Index of Names Subject Index

Lectures on Three-Manifold Topology

1980-12-31

William Jaco

Loop theorem-sphere theorem: The Tower Construction Connected sums 2-manifolds embedded in 3-manifolds Hierarchies Three-manifold groups Seifert fibered manifolds Peripheral structure Essential homotopies (the annulus-torus theorems) Characteristic Seifert pairs Deforming homotopy … Loop theorem-sphere theorem: The Tower Construction Connected sums 2-manifolds embedded in 3-manifolds Hierarchies Three-manifold groups Seifert fibered manifolds Peripheral structure Essential homotopies (the annulus-torus theorems) Characteristic Seifert pairs Deforming homotopy equivalences Bibliography Index.

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Introduction: Motivation

2020-03-06

Chris Wendl

The introduction motivates the remainder of the book via two specific examples of theorems from the early days of symplectic topology in which intersection theory plays a prominent role. We … The introduction motivates the remainder of the book via two specific examples of theorems from the early days of symplectic topology in which intersection theory plays a prominent role. We sketch closely analogous proofs of both theorems, emphasizing the way that intersection theory is used, but point out why the second theorem (on symplectic 4-manifolds that are standard near infinity) requires a nonobvious extension of homological intersection theory to punctured holomorphic curves. We then discuss informally some of the properties this theory will need to have and what kinds of subtle issues may arise.

The geometry and topology of three-manifolds

1979-01-01

William P. Thurston

4-Manifolds and Kirby Calculus

1999-08-31

Robert E. Gompf , András I. Stipsicz

4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic … 4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$s Symplectic 4-manifolds Stein surfaces Appendices: Solutions Notation, important figures Bibliography Index.

Foliations and the topology of 3-manifolds

1983-01-01

David Gabai

Corollary 6.5.A nontήvial link Lin S 3 is nonsplit if and only ifL is the set of cores of Reeb components of some foliation ΦofS 3 .The => direction follows … Corollary 6.5.A nontήvial link Lin S 3 is nonsplit if and only ifL is the set of cores of Reeb components of some foliation ΦofS 3 .The => direction follows from Theorem 5.5.Novikov [21] proved the converse in 1965.We therefore answer the so-called "Reeb placement problem" of Laudenbach and Roussarie [16] who asked which links could be realized as cores of Reeb components of foliations of S 3 .The holonomy of our foliations along the toral leaves is in general C°.The C 00 problem is open although it can be solved for the alternating knots, fibred knots, many other knots, and certain "sums" of such knots using the constructions [6]-[8] Corollary 6.7.Let R t be a Seifert surface for the oriented link L, C S 3 for i -1,2, and R be any Murasugi sum (or generalized plumbing) of R { and R 2 with L -dR.Then R is a minimal genus surface for the oriented link L if and only if each /^ is a minimal genus surface for the oriented link L z .This generalizes the classical result due to Seifert in the 1930's that the connected sum of minimal genus surfaces is a surface of minimal genus.Corollary 6.9.Let M be a compact connected irreducible oriented 3-manifold whose boundary ΘM is a (possibly empty) union of incompressible tori, and H 2 (M,dM) is not generated by tori and annuli.Then there exists a C 00 transversely oriented foliation ^ on M such that ^ ίίl ΘM, < $\ ΘM has no Reeb components, and no leaf of ^ is compact.In particular we have Corollary 6.11.Let M be either a compact connected oriented 3-manifold whose interior has a complete hyperbolic metric and H 2 (M, ΘM) Φ 0, or M -S 3 -N(L) where L is a nonsplit nontrivial link in S 3 .Then there exists a C°°t ransversely oriented foliation ¥ of M such that 3F has no compact leaves, ®ί ffl ΘM, and < $\ ΘM has no Reeb components.The conditions that ΘM be a union of incompressible tori and M be irreducible are necessary by Novikov's work.The question of whether a manifold possesses a C°° codimension-1 foliation without compact leaves has been precisely answered by the work of Thurston [31], Levitt [18], Wood [34], and Milnor [19] for circle bundles over surfaces and for most Seifert fibred spaces by [4]; see also [5].The 2-dimensional homology of these spaces (except for trivial cases) is generated by tori and annuli.It would be interesting to

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Introduction to symplectic topology

2006-02-06

Dusa McDuff

Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and … Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of <italic>Introduction to Symplectic Topology</italic> was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. In 1998, a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to improve the exposition, new material has been added in many places, and various proofs have been tightened up. Copious new references to key papers have been added to the bibliography. In particular, the material on contact geometry has been significantly expanded, many more details on linear complex structures and on the symplectic blowup and blowdown have been added, the section on <italic>J</italic>-holomorphic curves in Chapter 4 has been thoroughly revised, there are new sections on GIT and on the topology of symplectomorphism groups, and the section on Floer homology has been revised and updated. Chapter 13 has been completely rewritten and has a new title (Questions of Existence and Uniqueness). It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of Taubes–Seiberg–Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds. Chapter 14 on open problems has been added.

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Spaces of constant curvature

2010-12-01

Joseph A. Wolf

This sixth edition illustrates the high degree of interplay between group theory and geometry. The reader will benefit from the very concise treatments of riemannian and pseudo-riemannian manifolds and their … This sixth edition illustrates the high degree of interplay between group theory and geometry. The reader will benefit from the very concise treatments of riemannian and pseudo-riemannian manifolds and their curvatures, of the representation theory of finite groups, and of indications of recent progress in discrete subgroups of Lie groups.

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Notes on Cobordism Theory

1969-12-31

R. E. Stong

These notes contain the first complete treatment of cobordism, a topic that has become increasingly important in the past ten years. The subject is fully developed and the latest theories … These notes contain the first complete treatment of cobordism, a topic that has become increasingly important in the past ten years. The subject is fully developed and the latest theories are treated. Originally published in 1968. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

A Course in Metric Geometry

2001-06-12

Dmitri Burago , Yuri Burago , Sergei Ivanov

Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded Above Spaces of Curvature … Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded Above Spaces of Curvature Bounded Below Bibliography Index.

Morse theory for Lagrangian intersections

1988-01-01

Andreas Floer

Let P be a compact symplectic manifold and let L C P be a Lagrangian submanifold with π2{P,L) = 0.For any exact diffeomorphism φ of P with the property that … Let P be a compact symplectic manifold and let L C P be a Lagrangian submanifold with π2{P,L) = 0.For any exact diffeomorphism φ of P with the property that φ(L) intersects L transverally, we prove a Morse inequality relating the set φ(L) Π L to the cohomology of L. As a consequence, we prove a special case of the Arnold conjecture: If τr 2 (P) = 0 and φ is an exact diffeomorphism all of whose fixed points are nondegenerate, then the number of fixed points is greater than or equal to the sum over the Z2-Betti numbers of P.

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Geometry of the complex of curves I: Hyperbolicity

1999-10-01

Howard Masur , Yair N. Minsky

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Holomorphic disks and knot invariants

2003-11-15

Peter Ozsváth , Zoltán Szabó

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A classifying invariant of knots, the knot quandle

1982-01-01

David Joyce

The geometry of tensor calculus, I

1991-07-01

André Joyal , Ross Street

Quantum field theory and the Jones polynomial

1989-09-01

Edward Witten

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Convex polytopes, Coxeter orbifolds and torus actions

1991-03-01

Michael W. Davis , Tadeusz Januszkiewicz

Symplectic invariants and Hamiltonian dynamics

1995-01-01

Helmut Hofer , Eduard Zehnder

There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of Hamiltonian dynamics. One of the links is provided by a special class of symplectic … There is a mysterious relation between rigidity phenomena of symplectic geometry and global periodic solutions of Hamiltonian dynamics. One of the links is provided by a special class of symplectic invariants discovered by I. Ekeland and H. Hofer in [2], [3] called symplectic capacities. We first recall this concept in a more general setting from [26] and consider the class of all symplectic manifolds (M, ω) possibly with boundary, but of fixed dimension 2n. Here ω is a symplectic structure, i.e. a two-form on M which is closed and nondegenerate.

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Three dimensional manifolds, Kleinian groups and hyperbolic geometry

1982-05-01

William P. Thurston

1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: … 1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the

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Varieties of Group Representations and Splittings of 3-Manifolds

1983-01-01

Marc Culler , Peter B. Shalen

Holomorphic disks and topological invariants for closed three-manifolds

2004-05-01

Peter Ozsváth , Zoltán Szabó

The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure.Given a Heegaard splitting of Y = U … The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure.Given a Heegaard splitting of Y = U 0 ∪ Σ U 1 , these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U 0 and U 1 .

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Introduction to Knot Theory.

1964-12-01

H. F. Trotter , Richard H. Crowell , Ralph H. Fox

Prerequisites.- I * Knots and Knot Types.- 1. Definition of a knot.- 2. Tame versus wild knots.- 3. Knot projections.- 4. Isotopy type, amphicheiral and invertible knots.- II * The … Prerequisites.- I * Knots and Knot Types.- 1. Definition of a knot.- 2. Tame versus wild knots.- 3. Knot projections.- 4. Isotopy type, amphicheiral and invertible knots.- II * The Fundamental Group.- 1. Paths and loops.- 2. Classes of paths and loops.- 3. Change of basepoint.- 4. Induced homomorphisms of fundamental groups.- 5. Fundamental group of the circle.- III * The Free Groups.- 1. The free group F[A].- 2. Reduced words.- 3. Free groups.- IV * Presentation of Groups.- 1. Development of the presentation concept.- 2. Presentations and presentation types.- 3. The Tietze theorem.- 4. Word subgroups and the associated homomorphisms.- 5. Free abelian groups.- V * Calculation of Fundamental Groups.- 1. Retractions and deformations.- 2. Homotopy type.- 3. The van Kampen theorem.- VI * Presentation of a Knot Group.- 1. The over and under presentations.- 2. The over and under presentations, continued.- 3. The Wirtinger presentation.- 4. Examples of presentations.- 5. Existence of nontrivial knot types.- VII * The Free Calculus and the Elementary Ideals.- 1. The group ring.- 2. The free calculus.- 3. The Alexander matrix.- 4. The elementary ideals.- VIII * The Knot Polynomials.- 1. The abelianized knot group.- 2. The group ring of an infinite cyclic group.- 3. The knot polynomials.- 4. Knot types and knot polynomials.- IX * Characteristic Properties of the Knot Polynomials.- 1. Operation of the trivialize.- 2. Conjugation.- 3. Dual presentations.- Appendix I. Differentiable Knots are Tame.- Appendix II. Categories and groupoids.- Appendix III. Proof of the van Kampen theorem.- Guide to the Literature.

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The Geometries of 3-Manifolds

1983-09-01

Peter Scott

structures on 2-dimensional orbifolds . . . . . . . .421 §3.The basic theory of Seifert fibre spaces 428 §4. structures on 2-dimensional orbifolds . . . . . . . .421 §3.The basic theory of Seifert fibre spaces 428 §4.

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A New Construction of Symplectic Manifolds

1995-11-01

Robert E. Gompf

Invariants of 3-manifolds via link polynomials and quantum groups

1991-12-01

Nicolai Reshetikhin , Vladimir G. Turaev

A new polynomial invariant of knots and links

1985-01-01

Peter Freyd , David N. Yetter , Jim Hoste +3 more

The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space.We represent links by plane projections, using the customary conventions … The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space.We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines.Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1.Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation A L+ (t) -A L _(t) + (t 1 ' 2 -r 1 /2)A Lo (t) -0.FIGURE l

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On the Vassiliev knot invariants

1995-04-01

Dror Bar-Natan

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Ribbon graphs and their invaraints derived from quantum groups

1990-01-01

Nicolai Reshetikhin , Vladimir G. Turaev

Topological invariants of knots and links

1928-01-01

James Alexander

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Homological Algebra of Mirror Symmetry

1995-01-01

Maxim Kontsevich

Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without … Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has $$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$ .

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State models and the jones polynomial

1987-01-01

Louis H. Kauffman

A categorification of the Jones polynomial

2000-02-15

Mikhail Khovanov

Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial. Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.

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Chern-Simons gauge theory as a string theory

1995-01-01

Edward Witten

Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these … Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given space-time interpretations. For instance, three-dimensional Chern-Simons gauge theory can arise as a string theory. The world-sheet model in this case involves a limiting case of Floer/Gromov theory of symplectic manifolds. The instantons usually considered in Floer theory give rise to Wilson line insertions in the space-time Chern-Simons theory. A certain holomorphic analog of Chern-Simons theory can also arise as a string theory.

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Virtual Knot Theory

1999-10-01

Louis H. Kauffman

An Introduction to Knot Theory

1997-01-01

W. B. R. Lickorish

Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un

1989-01-01

Pierre Pansu

We exhibit a rigidity property of the simple groups Sp(n, 1) and F7-20 which implies Mostow rigidity. This property does not extend to O(n, 1) and U(n, 1). The proof … We exhibit a rigidity property of the simple groups Sp(n, 1) and F7-20 which implies Mostow rigidity. This property does not extend to O(n, 1) and U(n, 1). The proof relies on quasiconformal theory applied in the CR setting. Extensions are given to a class of solvable Lie groups. As a byproduct, a result on quasiisometries of infinite nilpotent groups is obtained. Dans cet article, on etablit une propriete de rigidite' des groupes simples de rang un Sp(n, 1), n 2 2 et Fqui entraine la rigidite de Mostow: 1. THEOREME. Toute quasiisornwtrie de 1'espace hyperbolique quaternionien HHn, n > 2, (resp. du plan hyperbolique de Cayley CaH2) est d distance bornee d'une isometrie, i.e., differe d'une isomnetrie par une application qui diplace les points d 'une distance bornee. Une application f entre espaces metriques est une quasiisometrie s'il existe des constantes L et C telles que l'image de f soit C-dense et que, pour tous X y, C + -d(x, y) < d(fx, fy) < Ld(x, y) + C. L Une quasiisometrie entre des G et G' est une sorte d'isomorphisme virtuel dans la categorie topologique (en effet, cela correspond 'a une action de G sur un fibre principal C' de groupe G' sur une base compacte; cf. [Ra]). Un isomorphisme entre sous-groupes cocompacts (covolume fini suffit pour les groupes simples de rang un) de groupes de Lie s'etend en une quasiisometrie des espaces symetriques ou des groupes de Lie. Si celle-ci est proche d'une isometrie des espaces symetriques (resp. un isomorphisme des groupes de Lie), les sous-groupes sont conjugues, c'est la rigidite de Mostow [M2]. La propriete ci-dessus ne s'etend pas aux groupes O(n, 1) et U(n, 1). En effet, (paragraphe 11.7) ceux-ci ont beaucoup de quasiisometries, au moins This content downloaded from 207.46.13.92 on Sun, 20 Nov 2016 04:26:46 UTC All use subject to http://about.jstor.org/terms

Theory of Braids

1947-01-01

Emil Artin

A theory of braids leading to a classification was given in my paper Theorie der Zopfe in vol. 4 of the Hamburger Abhandlungen (quoted as Z). Most of the proofs … A theory of braids leading to a classification was given in my paper Theorie der Zopfe in vol. 4 of the Hamburger Abhandlungen (quoted as Z). Most of the proofs are entirely intuitive. That of the main theorem in ?7 is not even convincing. It is possible to correct the proofs. The difficulties that one encounters if one tries to do so come from the fact that projection of the braid, which is an excellent tool for intuitive investigations, is a very clumsy one for rigorous proofs. This has lead me to abandon projections altogether. We shall use the more powerful tool of braid coordinates and obtain thereby farther reaching results of greater generality. A few words about the initial definitions. The fact that we assume of a braid string that it ends in a straight line is of course unimportant. It could be replaced by limit assumptions or introduction of infinite points. The present definition was selected because it makes some of the discussions easier and may be replaced any time by another one. I also wish to stress the fact that the definition of s-isotopy is of a provisional character only and is replaced later (Definition 3) by a general notion of isotopy. More than half of the paper is of a geometric nature. In this part we develop some results that may escape an intuitive investigation (Theorem 7 to 10). We do not prove (as has been done in Z) that the relations (18) (19) are defining relations for the braid group. We refer the reader to a paper by F. Bohnenblust1 where a proof of this fact and of many of our results is given by purely group theoretical methods. Later the proofs become more algebraic. With the developed tools we are able to give a unique normal form for every braid2 (Theorem 17, fig. 4 and remark following Theorem 18). In Theorem 19 we determine the center of the braid group and finally we give a characterisation of braids of braids. I would like to mention in this introduction a few of the more important of the unsolved problems: 1) Assume that two braids can be deformed into each other by a deformation of the most general nature including self intersection of each string but avoiding intersection of two different strings. Are they isotopic? One would be inclined to doubt it. Theorem 8 solves, however, a special case of this problem. 2) In Definition 3, we introduce a notion of isotopy that is already very general. What conditions must be put on a many to many mapping so that the result of Theorem 9 still holds?

Dehn Surgery on Knots

1987-03-01

Marc Culler , C. McA. Gordon , John Luecke +1 more

In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In … In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that

Hyperbolic Groups

1987-01-01

M. Gromov

A Primer on Mapping Class Groups

2011-12-31

Benson Farb , Dan Margalit

The study of the mapping class group Mod( S ) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. … The study of the mapping class group Mod( S ) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod( S ), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod( S ) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

A Primer on Mapping Class Groups (PMS-49)

2017-10-19

Benson Farb , Dan Margalit

The study of the mapping class group Mod(<italic>S</italic>) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book … The study of the mapping class group Mod(<italic>S</italic>) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(<italic>S</italic>), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(<italic>S</italic>) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

The Geometry of Four-Manifolds

1990-09-13

Simon Donaldson , P. B. Kronheimer

Abstract The last ten years have seen rapid advances in the understanding of differentiable four-manifolds not least of which has been the discovery of new 'exotic' manifolds. These results have … Abstract The last ten years have seen rapid advances in the understanding of differentiable four-manifolds not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics and have proved to be a mainspring of current mathematical research. This book provides a lucid and accessible account to the modern study of the geometry of four-manifolds. Consequently, it will form required reading for all those mathematicians and theoretical physicists whose research touches on this topic. Prerequisites are a firm grounding in differential geometry and topology as might be gained from the first year of a graduate course. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology - notably the definition of new invariants of four-manifolds - and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proved in detail. On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles. At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topology and which are close to the frontiers of current research.

Ultraknots and limit knots

2025-06-25

Benjamin Bode | Journal of the Mathematical Society of Japan

Fano’s research trajectory on birational contact transformations of the plane

2025-06-24

Elena Scalambro | Bollettino dell Unione Matematica Italiana

Symplectic embeddings into cylinders for certain symplectic manifolds

2025-06-24

Nil İpek Şirikçi | Hacettepe Journal of Mathematics and Statistics

We present a proof of a nonexistence result for symplectic embeddings of symplectic manifolds satisfying certain conditions into the symplectic cylinder. The proof utilizes an inequality between the displacement energy … We present a proof of a nonexistence result for symplectic embeddings of symplectic manifolds satisfying certain conditions into the symplectic cylinder. The proof utilizes an inequality between the displacement energy and the cylindrical capacity for subsets of $\mathbb{R}^{2n}$ to obtain an inequality for bounded subsets of the symplectic manifold. We also state a corollary which utilizes other results on nondisplaceable Lagrangians.

Steiner Coset Partitions of groups

2025-06-24

Füsun Akman , Papa A. Sissokho | Canadian Mathematical Bulletin

Topological monodromy kernels for fundamental groups of discriminant complements

2025-06-21

Nick Salter | Journal of Topology and Analysis

A linear system on a smooth complex algebraic surface gives rise to a family of smooth curves in the surface. Such a family has a topological monodromy representation valued in … A linear system on a smooth complex algebraic surface gives rise to a family of smooth curves in the surface. Such a family has a topological monodromy representation valued in the mapping class group of a fiber. Extending arguments of Kuno, we show that if the image of this representation is of finite index, then the kernel is infinite. This applies in particular to linear systems on smooth toric surfaces and on smooth complete intersections. In the case of plane curves, we extend the techniques of Carlson–Toledo to show that the kernel is quite rich (e.g. it contains a nonabelian free group).

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A-polynomials, Ptolemy equations and Dehn filling

2025-06-20

Joshua Howie , Daniel V. Mathews , Jessica S. Purcell | Algebraic & Geometric Topology

Real algebraic overtwisted contact structures on 3-spheres

2025-06-20

Şeyma Karadereli , Ferı̇t Öztürk | Algebraic & Geometric Topology

Hamiltonian classification of toric fibres and symmetric probes

2025-06-20

Joé Brendel | Algebraic & Geometric Topology

Fibered 3-manifolds and Veech groups

2025-06-20

Christopher J. Leininger , Kasra Rafi , Nicholas Rouse +2 more | Algebraic & Geometric Topology

Random Artin groups

2025-06-20

Antoine Goldsborough , Nicolas Vaskou | Algebraic & Geometric Topology

Bridge trisections and Seifert solids

2025-06-20

Jason Joseph , Jeffrey Meier , Maggie Miller +1 more | Algebraic & Geometric Topology

Elasticity of Free Type III Actions of Free Groups

2025-06-20

Antoine Poulin | Journal of Topology and Analysis

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A deformation of Asaeda–Przytycki–Sikora homology

2025-06-20

Zhenkun Li , Yi Xie , Boyu Zhang | Algebraic & Geometric Topology

Symplectization of certain Hamiltonian systems in fibered almost-symplectic manifolds

2025-06-20

Francesco Fassò , Nicola Sansonetto | Deleted Journal

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A recipe for exotic 2-links in closed 4-manifolds whose components are topological unknots

2025-06-20

Valentina Bais , Younes Benyahia , Oliviero Malech +1 more | Pacific Journal of Mathematics

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Virtual domination of 3-manifolds, III

2025-06-20

Hongbin Sun | Algebraic & Geometric Topology

Solvability of concordance groups and Milnor invariants

2025-06-20

Alessio Di Prisa , Giovanni Framba | Revista Matemática Iberoamericana

Using Milnor invariants, we prove that the concordance group \mathcal{C}(2) of 2 -string links is not solvable. As a consequence, we prove that the equivariant concordance group of strongly invertible … Using Milnor invariants, we prove that the concordance group \mathcal{C}(2) of 2 -string links is not solvable. As a consequence, we prove that the equivariant concordance group of strongly invertible knots is also not solvable, and we answer a conjecture by Kuzbary (2023).

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The Kakimizu complex for genus one hyperbolic knots in the 3-sphere

2025-06-20

Luis G. Valdez-Sánchez | Algebraic & Geometric Topology

An example of higher-dimensional Heegaard Floer homology

2025-06-20

Tian Yin , Tianyu Yuan | Algebraic & Geometric Topology

Mathematical Preliminaries: 4-Manifolds and Exotic Smoothness Structures

2025-06-19

| WORLD SCIENTIFIC eBooks

Dihedral and Cyclic Covers of a Class of Maps on Surfaces

2025-06-19

Marbarisha M. Kharkongor , Debashis Bhowmik , Dipendu Maity | Proceedings of the National Academy of Sciences India Section A Physical Sciences

4-Manifolds and Non-Commutative Geometry — Exotic Smoothness as Quantized Geometry

2025-06-19

| WORLD SCIENTIFIC eBooks

An algorithm to compute the Hausdorff dimension of regular branch groups

2025-06-19

Jorge Fariña-Asategui | Journal of Computational Algebra

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Rigidity criteria for chainmail consisting of tessellations of torus knots

2025-06-18

NULL AUTHOR_ID | Physical Review Letters

Essential tori in 3–manifolds not detected in any characteristic

2025-06-18

Grace S. Garden , Benjamin Martin , Stephan Tillmann | Geometriae Dedicata

None

2025-06-17

Efstratia Kalfagianni , Joseph M. Melby | Annales de l’institut Fourier

We use Dehn surgery methods to construct infinite families of hyperbolic knots in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>-sphere satisfying a weak form of the Turaev–Viro invariants volume conjecture. The results have applications … We use Dehn surgery methods to construct infinite families of hyperbolic knots in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>-sphere satisfying a weak form of the Turaev–Viro invariants volume conjecture. The results have applications to a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. We obtain an explicit family of pseudo-Anosov mapping classes acting on surfaces of any genus and with one boundary component that satisfy the conjecture.

None

2025-06-17

Corentin Le Bars | Annales de l’institut Fourier

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> be a group acting on a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">CAT</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> space with contracting isometries. We study the random walk generated by an admissible measure on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>. We … Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> be a group acting on a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">CAT</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> space with contracting isometries. We study the random walk generated by an admissible measure on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>. We prove that if the action is non-elementary and under optimal moment assumptions on the measure, the random walk satisfies a central limit theorem. The general approach is inspired from the cocycle argument of Y. Benoist and J-F. Quint, and our strategy relies on the use of hyperbolic models introduced by H. Petyt, A. Zalloum and D. Spriano, which are analogues of the contact graph for the class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">CAT</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> spaces. As a side result, we prove that the probability that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>th-step the random walk acts as a contracting isometry goes to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> goes to infinity.

Progress around the Boone–Higman conjecture

2025-06-16

James Belk , Collin Bleak , Francesco Matucci +1 more | EMS Surveys in Mathematical Sciences

A conjecture of Boone and Higman from the 1970’s asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a … A conjecture of Boone and Higman from the 1970’s asserts that a finitely generated group G has solvable word problem if and only if G can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

Free circle actions on (n−1)-connected (2n+1)-manifolds

2025-06-14

Yi Jiang , Yang Su | Pacific Journal of Mathematics

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A classification of semi-equivelar gems of PL d-manifolds on the surface with Euler characteristic -1

2025-06-14

A.S. Agarwal , Biplab Basak | Topological Methods in Nonlinear Analysis

A semi-equivelar gem of a PL $d$-manifold is a regular colored graph that represents the PL $d$-manifold and regularly embeds on a surface, with the property that the cyclic sequence … A semi-equivelar gem of a PL $d$-manifold is a regular colored graph that represents the PL $d$-manifold and regularly embeds on a surface, with the property that the cyclic sequence of lengths of faces in the embedding around each vertex is identical. In \cite{bb24}, the authors classified semi-equivelar gems of PL $d$-manifolds embedded on surfaces with Euler characteristics greater than or equal to zero. In this article, we focus on classifying semi-equivelar gems of PL $d$-manifolds embedded on the surface with Euler characteristic $-1$. We prove that if a semi-equivelar gem embeds regularly on the surface with Euler characteristic $-1$, then it belongs to one of the following types: $(8^3)$, $(6^2,8)$, $(6^2,12)$, $(10^2,4)$, $(12^2,4)$, $ (4,6,14)$, $(4,6,16)$, $(4,6,18)$, $(4,6,24)$, $(4,8,10)$, $(4,8,12)$ and $(4,8,16)$. Furthermore, we provide constructions that demonstrate the existence of such gems for each of the aforementioned types.

On the volume conjecture for hyperbolic Dehn-filled 3-manifolds along the figure-eight knot

2025-06-13

K. H. Yau Wong , Tian Yang | Quantum Topology

Using Ohtsuki’s method, we prove the asymptotic expansion conjecture and the volume conjecture of the Reshetikhin–Turaev and the Turaev–Viro invariants for all hyperbolic 3-manifolds obtained by doing a Dehn-surgery along … Using Ohtsuki’s method, we prove the asymptotic expansion conjecture and the volume conjecture of the Reshetikhin–Turaev and the Turaev–Viro invariants for all hyperbolic 3-manifolds obtained by doing a Dehn-surgery along the figure-8 knot.

Generalized knots-quivers correspondence

2025-06-13

Marko Stošić | Journal of Knot Theory and Its Ramifications

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Chip-firing on graphs of groups

2025-06-12

Margaret Meyer , Dmitry Zakharov | Discrete Mathematics

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BPS invariants from framed links

2025-06-12

Kai Wang , Shengmao Zhu | Journal of High Energy Physics

A bstract In this article, we investigate the BPS invariants associated with framed links. We extend the relationship between the algebraic curve (i.e. dual A -polynomial) and the BPS invariants … A bstract In this article, we investigate the BPS invariants associated with framed links. We extend the relationship between the algebraic curve (i.e. dual A -polynomial) and the BPS invariants of a knot investigated in [14] to the case of a framed knot. With the help of the framing change formula for the dual A -polynomial of a framed knot, we give several explicit formulas for the extremal A -polynomials and the BPS invariants of framed knots. As to the framed links, we present several numerical calculations for the Ooguri-Vafa invariants and BPS invariants for framed Whitehead links and Borromean rings and verify the integrality property for them.

Completely (iso-)split scale-invariant Coulomb branch geometries are isotrivial

2025-06-11

Philip C. Argyres , Robert Moscrop , Souradeep Thakur +1 more | Journal of High Energy Physics

A bstract We show that scale-invariant special Kähler geometries whose generic r dim ℂ abelian variety fiber is isomorphic (completely split) or isogenous (completely iso-split) as a complex torus to … A bstract We show that scale-invariant special Kähler geometries whose generic r dim ℂ abelian variety fiber is isomorphic (completely split) or isogenous (completely iso-split) as a complex torus to the product of r one-dimensional complex tori have constant τ ij modulus on the Coulomb branch, i.e. are isotrivial. These simple results are useful in organizing the classification of scale-invariant special Kähler geometries, which, in turn, is relevant to the classification of possible 4-dimensional $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 supersymmetric superconformal field theories.

Classification of aut-fixed subgroups in free-abelian times surface groups

2025-06-10

J.J. Lei , Peng Wang , Qiang Zhang | Journal of Pure and Applied Algebra

Alternative Proof of the Ribbonness on Classical Link

2025-06-10

Akio Kawauchi | International Journal of Physics Research and Applications

Alternative proof is given for an earlier presented result that if a link in 3-space bounds a compact oriented proper surface (without closed component) in the upper half 4-space, then … Alternative proof is given for an earlier presented result that if a link in 3-space bounds a compact oriented proper surface (without closed component) in the upper half 4-space, then the link bounds a ribbon surface in the upper half 4-space which is a boundary-relative renewal embedding of the original surface. 2020 Mathematics Subject Classification: Primary 57K45; Secondary 57K40

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Cubulating a free-product-by-cyclic group

2023-06-06

François Dahmani , Suraj Krishna M S

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Band diagrams of immersed surfaces in 4-manifolds

2021-08-29

Mark C. Hughes , Seungwon Kim , Maggie Miller

We study immersed surfaces in smooth 4-manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4-manifold, … We study immersed surfaces in smooth 4-manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4-manifold, representing a level set of the surface with respect to an associated Morse function. We show that every self-transverse immersed surface in a smooth, orientable, closed 4-manifold can be represented by a singular banded unlink diagram, and that such representations are uniquely determined by the ambient isotopy or equivalence class of the surface up to a set of singular band moves which we define explicitly. By introducing additional finger, Whitney, and cusp diagrammatic moves, we can use these singular band moves to describe homotopies or regular homotopies as well. Using these techniques, we introduce bridge trisections of immersed surfaces in arbitrary trisected 4-manifolds and prove that such bridge trisections exist and are unique up to simple perturbation moves. We additionally give some examples of how singular banded unlink diagrams may be used to perform computations or produce explicit homotopies of surfaces.

The homology of permutation racks

2020-10-27

Victoria Lebed , Markus Szymik

Despite a blossoming of research activity on racks and their homology for over two decades, with a record of diverse applications to central parts of contemporary mathematics, there are still … Despite a blossoming of research activity on racks and their homology for over two decades, with a record of diverse applications to central parts of contemporary mathematics, there are still very few examples of racks whose homology has been fully calculated. In this paper, we compute the entire integral homology of all permutation racks. Our method of choice involves homotopical algebra, which was brought to bear on the homology of racks only recently. For our main result, we establish a spectral sequence, which reduces the problem to one in equivariant homology, and for which we show that it always degenerates. The blueprint given in this paper demonstrates the high potential for further exploitation of these techniques.

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Fully Augmented Links in the Thickened Torus

2020-07-24

Alice Kwon

In this paper we study the geometry of fully augmented link complements in the thickened torus and describe their geometric properties, generalizing the study of fully augmented links in $S^3$. … In this paper we study the geometry of fully augmented link complements in the thickened torus and describe their geometric properties, generalizing the study of fully augmented links in $S^3$. We classify which fully augmented links in the thickened torus are hyperbolic, show that their complements in the thickened torus decompose into ideal right-angled torihedra, and that the edges of this decomposition are canonical. We also study volume density of fully augmented links in $S^3$, defined to be the ratio of its volume and the number of augmentations. We prove the Volume Density Conjecture for fully augmented links which states that the volume density of a sequence of fully augmented links in $S^3$ which diagrammatically converge to a biperiodic link, converges to the volume density of that biperiodic link.

A combinatorial proof of invariance of double-point enhanced grid homology

2018-10-07

Timothy Ratigan , Joshua Wang , Luya Wang

We prove that the minus version of Lipshitz's double-point enhanced grid homology is a knot invariant through purely combinatorial means. We prove that the minus version of Lipshitz's double-point enhanced grid homology is a knot invariant through purely combinatorial means.