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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. [1]

## 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.

## Braid group

In mathematics, the braid group on strands, denoted by, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group.

## 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 G, denoted Z(G),The notation Z is from German Zentrum, meaning "center".

## 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 a and b are said to be relatively prime, mutually prime, or coprime (also spelled co-prime) if the only positive integer that evenly 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 or undefined.

## 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.

## Deformation retract

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

## 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.

## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, when at least one of them is not zero, is the largest positive integer that divides the numbers without a remainder.

## Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis.

## 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").

## Irrational winding of a torus

In topology, a branch of mathematics, an irrational winding of a torus is a continuous injection of a line into a torus that is used to set up several counterexamples.

## 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 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 theory

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

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

## Parametrization

Parametrization (or parameterization; also parameterisation, parametrisation) is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object.

## 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.

## Rotational symmetry

Generally, an object with 'rotational symmetry' also known in biological contexts as 'radial symmetry', is an object that looks the same after a certain amount of rotation.

## 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.

## 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 (also called a 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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