Table of Contents
50 relations: Alexander polynomial, Alternating knot, Annals of Mathematics, Braid group, Center (group theory), Chirality (mathematics), Cinquefoil knot, Coprime integers, Critical point (mathematics), Crossing number (knot theory), Cylindrical coordinate system, Dunce hat (topology), Euclidean space, Figure-eight knot (mathematics), Genus (mathematics), Genus g surface, Greatest common divisor, Handlebody, Holomorphic function, Hyperbolic link, If and only if, Integer, John Pardon, Jones polynomial, Knot (mathematics), Knot group, Knot tabulation, Knot theory, Linear flow on the torus, Link (knot theory), Mikhael Gromov (mathematician), Morgan Prize, N-sphere, Notices of the American Mathematical Society, Parametrization (geometry), Presentation of a group, Prime knot, Retraction (topology), Rotational symmetry, Satellite knot, Seifert fiber space, Seifert surface, Stretch factor, Topological conjugacy, Topopolis, Torus, Trefoil knot, Unknot, 3-sphere, 71 knot.
- Fibered knots and links
- Torus knots and links
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. Torus knot and Alexander polynomial are knot theory.
See Torus knot 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.
See Torus knot and Alternating knot
Annals of Mathematics
The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
See Torus knot and Annals of Mathematics
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. Torus knot and braid group are knot theory.
See Torus knot and Braid group
Center (group theory)
In abstract algebra, the center of a group is the set of elements that commute with every element of.
See Torus knot and Center (group theory)
Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. Torus knot and chirality (mathematics) are knot theory.
See Torus knot and Chirality (mathematics)
Cinquefoil knot
In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. Torus knot and cinquefoil knot are Fibered knots and links, knot theory and torus knots and links.
See Torus knot and Cinquefoil knot
Coprime integers
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.
See Torus knot and Coprime integers
Critical point (mathematics)
In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below).
See Torus knot and Critical point (mathematics)
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 Torus knot and Crossing number (knot theory)
Cylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane containing the purple section).
See Torus knot and Cylindrical coordinate system
Dunce hat (topology)
In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed. Torus knot and dunce hat (topology) are algebraic topology.
See Torus knot and Dunce hat (topology)
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space.
See Torus knot and Euclidean space
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. Torus knot and figure-eight knot (mathematics) are Fibered knots and links and knot theory.
See Torus knot and Figure-eight knot (mathematics)
Genus (mathematics)
In mathematics, genus (genera) has a few different, but closely related, meanings. Torus knot and genus (mathematics) are algebraic topology.
See Torus knot and Genus (mathematics)
Genus g surface
In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior of a disk is removed from each of g distinct tori and the boundaries of the g many disks are identified (glued together), forming a g-torus.
See Torus knot and Genus g surface
Greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.
See Torus knot and Greatest common divisor
Handlebody
In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces.
Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.
See Torus knot and Holomorphic function
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. Torus knot and hyperbolic link are knot theory.
See Torus knot and Hyperbolic link
If and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.
See Torus knot and If and only if
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.
John Pardon
John Vincent Pardon (born June 1989) is an American mathematician who works on geometry and topology.
See Torus knot and John Pardon
Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Torus knot and Jones polynomial are knot theory.
See Torus knot and Jones polynomial
Knot (mathematics)
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other.
See Torus knot and Knot (mathematics)
Knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space.
Knot tabulation
Ever since Sir William Thomson's vortex theory, mathematicians have tried to classify and tabulate all possible knots. Torus knot and knot tabulation are knot theory.
See Torus knot and Knot tabulation
Knot theory
In topology, knot theory is the study of mathematical knots. Torus knot and knot theory are algebraic topology.
See Torus knot and Knot theory
Linear flow on the torus
In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus \mathbb^n.
See Torus knot and Linear flow on the torus
Link (knot theory)
In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.
See Torus knot and Link (knot theory)
Mikhael Gromov (mathematician)
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory.
See Torus knot and Mikhael Gromov (mathematician)
Morgan Prize
The Morgan Prize (full name Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student) is an annual award given to an undergraduate student in the US, Canada, or Mexico who demonstrates superior mathematics research.
See Torus knot and Morgan Prize
N-sphere
In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer.
Notices of the American Mathematical Society
Notices of the American Mathematical Society is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue.
See Torus knot and Notices of the American Mathematical Society
Parametrization (geometry)
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.
See Torus knot and Parametrization (geometry)
Presentation of a group
In mathematics, a presentation is one method of specifying a group.
See Torus knot and Presentation of a group
Prime knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable.
Retraction (topology)
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace.
See Torus knot and Retraction (topology)
Rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn.
See Torus knot and Rotational symmetry
Satellite knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Torus knot and satellite knot are knot theory.
See Torus knot and Satellite knot
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles.
See Torus knot 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. Torus knot and Seifert surface are knot theory.
See Torus knot and Seifert surface
Stretch factor
The stretch factor (i.e., bilipschitz constant) of an embedding measures the factor by which the embedding distorts distances.
See Torus knot and Stretch factor
Topological conjugacy
In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other.
See Torus knot and Topological conjugacy
Topopolis
A topopolis is a proposed tube-shaped space habitat, rotating to produce artificial gravity via centrifugal force on the inner surface, which is extended into a loop around the local planet or star.
Torus
In geometry, a torus (tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle.
Trefoil knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. Torus knot and trefoil knot are Fibered knots and links, knot theory and torus knots and links.
See Torus knot and Trefoil knot
Unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Torus knot and unknot are Fibered knots and links, knot theory and torus knots and links.
3-sphere
In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. Torus knot and 3-sphere are algebraic topology.
71 knot
In knot theory, the 71 knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number seven. Torus knot and 71 knot are Fibered knots and links, knot theory and torus knots and links.
See also
Fibered knots and links
- (−2,3,7) pretzel knot
- 62 knot
- 63 knot
- 71 knot
- Carrick mat
- Cinquefoil knot
- Fibered knot
- Figure-eight knot (mathematics)
- Hopf link
- Perko pair
- Torus knot
- Trefoil knot
- Unknot
Torus knots and links
- 71 knot
- Cinquefoil knot
- Hopf link
- Torus knot
- Trefoil knot
- Unknot
References
Also known as (3,3,−2) pretzel knot, (3,4)-torus knot, (5,3)-torus knot, (5,3,−2) pretzel knot, (9,2)-torus knot, 10 124 knot, 8 19 knot, 9 1 knot, Double torus knot, Double torus knots, Double torus link, G-torus knot, List of torus knots, Nonafoil knot, Torus link.