• topology glossary
  Topology glossary
  This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology...
  
  
• List of topology topics
• List of general topology topics
• List of algebraic topology topics
• Publications in topology

Knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

• Knot (mathematics)
  Knot (mathematics)
  In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
  
  
• Link (knot theory)
  Link (knot theory)
  In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...
  
  
• Wild knot
  Wild knot
  In the mathematical theory of knots, a knot is tame if it can be "thickened up", that is, if there exists an extension to an embedding of the solid torus S 1 × D 2 into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain...
  
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• Examples of knots
  • Unknot
    Unknot
    The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
    
    
  • Trefoil knot
    Trefoil knot
    In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...
    
    
  • Figure-eight knot (mathematics)
    Figure-eight knot (mathematics)
    In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
    
    
  • Borromean rings
    Borromean rings
    In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.- Mathematical properties :...
    
    
• Types of knots
  • Torus knot
    Torus knot
    In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...
    
    
  • Prime knot
    Prime knot
    In knot theory, a prime knot is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite. It can be a nontrivial problem to determine whether a...
    
    
  • Alternating knot
    Alternating knot
    In knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link. A link is alternating if it has an alternating diagram....
    
    
  • Hyperbolic link
    Hyperbolic link
    In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry...
    
    
• Knot invariant
  Knot invariant
  In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...
  
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  • Crossing number
    Crossing number (knot theory)
    In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....
    
    
  • Linking number
    Linking number
    In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other...
    
    
  • Skein relation
    Skein relation
    A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial which is an invariant of the knot. If two diagrams have different polynomials, they represent different knots. The reverse may not...
    
    
  • Knot polynomial
    Knot polynomial
    In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.-History:The first knot polynomial, the Alexander polynomial, was introduced by J. W...
    
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    • Alexander polynomial
      Alexander polynomial
      In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
      
      
    • Jones polynomial
  • Knot group
    Knot group
    In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,\pi_1....
    
    
• Writhe
  Writhe
  In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings....
  
  
• Quandle
• Seifert surface
  Seifert surface
  In mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
  
  
• Braids
  • Braid theory
    Braid theory
    In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a...
    
    
  • Braid group
    Braid group
    In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
    
    
• Kirby calculus
  Kirby calculus
  In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves...
  
  

Surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

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• Genus (mathematics)
  Genus (mathematics)
  In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
  
  
• Examples
  • Positive Euler characteristic
    Euler characteristic
    In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
    
    
    • 2-disk
      Disk (mathematics)
      In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
      
      
    • Sphere
      Sphere
      A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
      
      
    • Real projective plane
      Real projective plane
      In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
      
      
  • Zero Euler characteristic
    • Annulus
      Annulus (mathematics)
      In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...
      
      
    • Möbius strip
      Möbius strip
      The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
      
      
    • Torus
      Torus
      In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
      
      
    • Klein bottle
      Klein bottle
      In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
      
      
  • Negative Euler characteristic
    • The boundary of the pretzel
      Pretzel
      A pretzel is a type of baked food made from dough in soft and hard varieties and savory or sweet flavors in a unique knot-like shape, originating in Europe...
      
       is a genus three surface
  • Embedded
    Embedding
    In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
    
    /Immersed in Euclidean space
    • Cross-cap
      Cross-cap
      In mathematics, a cross-cap is a two-dimensional surface that is a model of a Möbius strip with a single self intersection. This self intersection precludes the cross-cap from being topologically equivalent to a Möbius strip...
      
      
    • Boy's surface
      Boy's surface
      In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901...
      
      
    • Roman surface
      Roman surface
      The Roman surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry...
      
      
    • Steiner surface
      Steiner surface
      In geometry, a branch of mathematics, the Steiner surfaces, discovered by Jakob Steiner, are mappings of the real projective plane into three-dimensional real projective space...
      
      
    • Alexander horned sphere
      Alexander horned sphere
      The Alexander horned sphere is a wild embedding of a sphere into space, discovered by . It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:...
      
      
    • Klein bottle
      Klein bottle
      In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
      
      
• Mapping class group
  Mapping class group
  In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
  
  
  • Dehn twist
    Dehn twist
    In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface .-Definition:...
    
    
  • Nielsen–Thurston classification

Three-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

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• Moise's Theorem
  Moise's theorem
  In geometric topology, a branch of mathematics, Moise's theorem, proved by , states that any topological 3-manifold has an essentially unique piecewise-linear structure and smooth structure....
  
   (see also Hauptvermutung
  Hauptvermutung
  The Hauptvermutung of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze.This conjecture is now known to be false...
  
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• Poincaré conjecture
  Poincaré conjecture
  In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
  
  
  • Thurston elliptization conjecture
    Thurston elliptization conjecture
    William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature. A 3-manifold with such a metric is covered by the 3-sphere, moreover the group of covering transformations...
    
    
• Thurston's geometrization conjecture
  • Hyperbolic 3-manifold
    Hyperbolic 3-manifold
    A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
    
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  • Spherical 3-manifold
    Spherical 3-manifold
    In mathematics, a spherical 3-manifold M is a 3-manifold of the formM=S^3/\Gammawhere \Gamma is a finite subgroup of SO acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed...
    
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  • Euclidean 3-manifolds, Bieberbach Theorem, Flat manifold
    Flat manifold
    In mathematics, a Riemannian manifold is said to be flat if its curvature is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°....
    
    s, Crystallographic groups
  • Seifert fiber space
    Seifert fiber space
    A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...
    
    
• Heegaard splitting
  Heegaard splitting
  In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.-Definitions:...
  
  
  • Waldhausen conjecture
  • Compression body
    Compression body
    In the theory of 3-manifolds, a compression body is a kind of generalized handlebody.A compression body is either a handlebody or the result of the following construction:Compression bodies often arise when manipulating Heegaard splittings....
    
    
  • Handlebody
    Handlebody
    In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds...
    
    
• Incompressible surface
  Incompressible surface
  In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
  
  
  • Dehn's lemma
    Dehn's lemma
    In mathematics Dehn's lemmaasserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disc's interior, implies the existence of another piecewise-linear map of the disc which is an embedding and is identical to the original on the boundary of the...
    
    
  • Loop theorem
    Loop theorem
    In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem....
    
     (aka the Disk theorem)
  • Sphere theorem
    Sphere theorem
    In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows...
    
    
• Haken manifold
  Haken manifold
  In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
  
  
• JSJ decomposition
  JSJ decomposition
  In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson...
  
  
• Branched surface
  Branched surface
  In mathematics, a branched surface is type of topological space. A small piece of an surface looks topologically like ℝ². A small piece of a branched surface, on the other hand, might look like either of the following:*ℝ²;...
  
  
• Lamination
  Train track
  In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions:#The curves meet at a finite set of vertices called switches....
  
  
• Examples
  • 3-sphere
    3-sphere
    In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
    
    
  • Torus bundle
    Torus bundle
    In mathematics, in the sub-field of geometric topology, a torus bundle is a kind of surface bundle over the circle, which in turn are a class of three-manifolds.-Construction:To obtain a torus bundle: let f be an...
    
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  • Surface bundles over the circle
  • Graph manifold
    Graph manifold
    In topology, a graph manifold is a 3-manifold which is obtained by gluing some circle bundles. They were invented and classified by the German topologist Friedhelm Waldhausen in 1967...
    
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  • Knot complement
    Knot complement
    In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...
    
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  • Whitehead manifold
    Whitehead manifold
    In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture....
    
    
• Invariants
  Invariant (mathematics)
  In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
  
  
  • Fundamental group
    Fundamental group
    In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
    
    
  • Heegaard genus
  • tri-genus
    Trigenus
    In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple . It is obtained by minimizing the genera of three orientable handle bodies — with no intersection between their interiors— which decompose the manifold as far as the...
    
    
  • Analytic torsion
    Analytic torsion
    In mathematics, Reidemeister torsion is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and ....
    
    

Manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s in general

• Orientable manifold
• Connected sum
  Connected sum
  In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...
  
  
• Jordan-Schönflies theorem
• Signature (topology)
  Signature (topology)
  In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four ....
  
  
• Handle decomposition
• Handlebody
  Handlebody
  In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds...
  
  
• h-cobordism theorem
• s-cobordism theorem
• Manifold decomposition
  Manifold decomposition
  In topology, a branch of mathematics, a manifold M may be decomposed or split by writing M as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form M....
  
  
• Hilbert-Smith conjecture
  Hilbert-Smith conjecture
  In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively on a manifold M...
  
  
• Mapping class group
  Mapping class group
  In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
  
  
• Orbifold
  Orbifold
  In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
  
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• Examples
  • Exotic sphere
    Exotic sphere
    In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
    
    
  • Homology sphere
    Homology sphere
    In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1. That is,andTherefore X is a connected space, with one non-zero higher Betti number: bn...
    
    
  • Lens space
    Lens space
    A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions....
    
    
  • I-bundle
    I-bundle
    In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber....