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Figure-eight knot (mathematics)

Index Figure-eight knot (mathematics)

In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. [1]

Table of Contents

  1. 42 relations: Alexander polynomial, Alternating knot, Braid group, Cameron Gordon (mathematician), Chiral knot, Clausen function, Computer-assisted proof, Covering space, Crossing number (knot theory), Dehn surgery, Fibered knot, Figure-eight knot, Geometrization conjecture, Geometry & Topology, Gieseking manifold, Haken manifold, Hyperbolic geometry, Hyperbolic link, Hyperbolic volume, Ian Agol, Ideal point, Inventiones Mathematicae, John Milnor, John R. Stallings, Jones polynomial, Knot complement, Knot theory, Manifold decomposition, Marc Lackenby, Milnor map, Prime decomposition of 3-manifolds, Prime knot, Robion Kirby, Seifert fiber space, Seifert surface, Tetrahedron, Trefoil knot, Unknot, William Thurston, (−2,3,7) pretzel knot, 2-bridge knot, 3-manifold.

  2. Alternating knots and links
  3. Double torus knots and links
  4. Fibered knots and links
  5. Hyperbolic knots and links
  6. Non-tricolorable knots and links
  7. Prime knots and links
  8. Twist knots

Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. Figure-eight knot (mathematics) and Alexander polynomial are knot theory.

See Figure-eight knot (mathematics) and Alexander polynomial

Alternating knot

In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. Figure-eight knot (mathematics) and alternating knot are alternating knots and links.

See Figure-eight knot (mathematics) and Alternating knot

Braid group

In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see); and in monodromy invariants of algebraic geometry. Figure-eight knot (mathematics) and braid group are knot theory.

See Figure-eight knot (mathematics) and Braid group

Cameron Gordon (mathematician)

Cameron Gordon (born 1945) is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of Mathematics at the University of Texas at Austin, known for his work in knot theory.

See Figure-eight knot (mathematics) and Cameron Gordon (mathematician)

Chiral knot

In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed).

See Figure-eight knot (mathematics) and Chiral knot

Clausen function

In mathematics, the Clausen function, introduced by, is a transcendental, special function of a single variable.

See Figure-eight knot (mathematics) and Clausen function

Computer-assisted proof

A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.

See Figure-eight knot (mathematics) and Computer-assisted proof

Covering space

In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself.

See Figure-eight knot (mathematics) and Covering space

Crossing number (knot theory)

In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot.

See Figure-eight knot (mathematics) and Crossing number (knot theory)

Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. Figure-eight knot (mathematics) and Dehn surgery are 3-manifolds.

See Figure-eight knot (mathematics) and Dehn surgery

Fibered knot

In knot theory, a branch of mathematics, a knot or link K in the 3-dimensional sphere S^3 is called fibered or fibred (sometimes Neuwirth knot in older texts, after Lee Neuwirth) if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t then the intersection of F_s and F_t is exactly K. Figure-eight knot (mathematics) and fibered knot are fibered knots and links.

See Figure-eight knot (mathematics) and Fibered knot

Figure-eight knot

The figure-eight knot or figure-of-eight knot is a type of stopper knot.

See Figure-eight knot (mathematics) and Figure-eight knot

Geometrization conjecture

In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. Figure-eight knot (mathematics) and geometrization conjecture are 3-manifolds.

See Figure-eight knot (mathematics) and Geometrization conjecture

Geometry & Topology

Geometry & Topology is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications.

See Figure-eight knot (mathematics) and Geometry & Topology

Gieseking manifold

In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. Figure-eight knot (mathematics) and Gieseking manifold are 3-manifolds.

See Figure-eight knot (mathematics) and Gieseking 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. Figure-eight knot (mathematics) and Haken manifold are 3-manifolds.

See Figure-eight knot (mathematics) and Haken manifold

Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry.

See Figure-eight knot (mathematics) and Hyperbolic geometry

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. Figure-eight knot (mathematics) and hyperbolic link are 3-manifolds, hyperbolic knots and links and knot theory.

See Figure-eight knot (mathematics) and Hyperbolic link

Hyperbolic volume

In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. Figure-eight knot (mathematics) and hyperbolic volume are knot theory.

See Figure-eight knot (mathematics) and Hyperbolic volume

Ian Agol

Ian Agol (born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds.

See Figure-eight knot (mathematics) and Ian Agol

Ideal point

In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.

See Figure-eight knot (mathematics) and Ideal point

Inventiones Mathematicae

Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.

See Figure-eight knot (mathematics) and Inventiones Mathematicae

John Milnor

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems.

See Figure-eight knot (mathematics) and John Milnor

John R. Stallings

John Robert Stallings Jr. (July 22, 1935 – November 24, 2008) was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology.

See Figure-eight knot (mathematics) and John R. Stallings

Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Figure-eight knot (mathematics) and Jones polynomial are knot theory.

See Figure-eight knot (mathematics) and Jones polynomial

Knot complement

In mathematics, the knot complement of a tame knot K is the space where the knot is not. Figure-eight knot (mathematics) and knot complement are knot theory.

See Figure-eight knot (mathematics) and Knot complement

Knot theory

In topology, knot theory is the study of mathematical knots.

See Figure-eight knot (mathematics) and Knot theory

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.

See Figure-eight knot (mathematics) and Manifold decomposition

Marc Lackenby

Marc Lackenby is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory.

See Figure-eight knot (mathematics) and Marc Lackenby

Milnor map

In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. Figure-eight knot (mathematics) and Milnor map are knot theory.

See Figure-eight knot (mathematics) and Milnor map

Prime decomposition of 3-manifolds

In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds. Figure-eight knot (mathematics) and prime decomposition of 3-manifolds are 3-manifolds.

See Figure-eight knot (mathematics) and Prime decomposition of 3-manifolds

Prime knot

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Figure-eight knot (mathematics) and prime knot are prime knots and links.

See Figure-eight knot (mathematics) and Prime knot

Robion Kirby

Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology.

See Figure-eight knot (mathematics) and Robion Kirby

Seifert fiber space

A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. Figure-eight knot (mathematics) and Seifert fiber space are 3-manifolds.

See Figure-eight knot (mathematics) and Seifert fiber space

Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Figure-eight knot (mathematics) and Seifert surface are knot theory.

See Figure-eight knot (mathematics) and Seifert surface

Tetrahedron

In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.

See Figure-eight knot (mathematics) and Tetrahedron

Trefoil knot

In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. Figure-eight knot (mathematics) and trefoil knot are alternating knots and links, fibered knots and links, knot theory, prime knots and links and Twist knots.

See Figure-eight knot (mathematics) and Trefoil knot

Unknot

In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Figure-eight knot (mathematics) and unknot are fibered knots and links, knot theory, non-tricolorable knots and links and prime knots and links.

See Figure-eight knot (mathematics) and Unknot

William Thurston

William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician.

See Figure-eight knot (mathematics) and William Thurston

(−2,3,7) pretzel knot

In geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery constructions. Figure-eight knot (mathematics) and (−2,3,7) pretzel knot are 3-manifolds, fibered knots and links, hyperbolic knots and links, knot theory and non-tricolorable knots and links.

See Figure-eight knot (mathematics) and (−2,3,7) pretzel knot

2-bridge knot

In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points. Figure-eight knot (mathematics) and 2-bridge knot are knot theory.

See Figure-eight knot (mathematics) and 2-bridge knot

3-manifold

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. Figure-eight knot (mathematics) and 3-manifold are 3-manifolds.

See Figure-eight knot (mathematics) and 3-manifold

See also

Alternating knots and links

Double torus knots and links

Fibered knots and links

Hyperbolic knots and links

Non-tricolorable knots and links

Prime knots and links

Twist knots

References

[1] https://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)

Also known as 4 1 knot, 4₁ knot, Figure eight knot (mathematics), Listing knot, Listing's knot.