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# 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. 

## Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type.

## 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.

## Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

## Braid group

In mathematics, the braid group on strands (denoted), 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.

## Braid theory

In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations.

## Center (group theory)

In abstract algebra, the center of a group,, is the set of elements that commute with every element of.

## 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.

## 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.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Critical point (mathematics)

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0.

## 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.

## 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, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.

## 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.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## 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.

## Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## John Pardon

John Vincent Pardon (born June 1989) is an American mathematician who works on geometry and topology.

## Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984.

## Knot (mathematics)

In mathematics, a knot is an embedding of a circle S^1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies).

## 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.

## Knot theory

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

## 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 which is represented by the following differential equations with respect to the standard angular coordinates (θ1, θ2,..., θn): The solution of these equations can explicitly be expressed as If we represent the torus as \mathbb.

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together.

## Mikhail Leonidovich Gromov

Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Михаи́л Леони́дович Гро́мов; born 23 December 1943), is a French-Russian mathematician known for work in geometry, analysis and group theory.

## Morgan Prize

The Morgan Prize (actually 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.

## 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.

## Parametrization

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.

## Presentation of a group

In mathematics, one method of defining a group is by a presentation.

## Prime knot

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable.

## Retract

In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace which preserves the position of all points in that subspace.

## Rotational symmetry

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn.

## Seifert fiber space

A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles.

## Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is a surface whose boundary is a given knot or link.

## Stretch factor

In mathematics, the stretch factor of an embedding measures the factor by which the embedding distorts distances.

## Topological conjugacy

In mathematics, two functions are said to be topologically conjugate to one another if there exists a homeomorphism that will conjugate the one into the other.

## Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

## Trefoil knot

In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot.

## Unknot

The unknot arises in the mathematical theory of knots.

## 3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.

## 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.

## References

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