![<span class="mw-page-title-main">Figure-eight knot (mathematics)</span> Unique knot with a crossing number of four](https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Blue_Figure-Eight_Knot.png/320px-Blue_Figure-Eight_Knot.png)
In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.
![<span class="mw-page-title-main">Torus</span> Doughnut-shaped surface of revolution](https://upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tesseract_torus.png/320px-Tesseract_torus.png)
In geometry, a torus 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. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
![<span class="mw-page-title-main">Knot theory</span> Study of mathematical knots](https://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg/320px-Tabela_de_n%C3%B3s_matem%C3%A1ticos_01%2C_crop.jpg)
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
![<span class="mw-page-title-main">Parametric equation</span> Representation of a curve by a function of a parameter](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Butterfly_transcendental_curve.svg/320px-Butterfly_transcendental_curve.svg.png)
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called a parametric curve and parametric surface, respectively. In such cases, the equations are collectively called a parametric representation, or parametric system, or parameterization of the object.
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients.
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,
![<span class="mw-page-title-main">Torus knot</span> Knot which lies on the surface of a torus in 3-dimensional space](https://upload.wikimedia.org/wikipedia/commons/thumb/0/07/TorusKnot3D.png/320px-TorusKnot3D.png)
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 are not coprime. A torus knot is trivial if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.
![<span class="mw-page-title-main">Seifert surface</span> Orientable surface whose boundary is a knot or link](https://upload.wikimedia.org/wikipedia/commons/e/e4/Borromean_Seifert_surface.png)
In mathematics, a Seifert surface is an orientable surface whose boundary is a given knot or link.
![<span class="mw-page-title-main">Fibered knot</span> Mathematical knot](https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Blue_Figure-Eight_Knot.png/320px-Blue_Figure-Eight_Knot.png)
In knot theory, a branch of mathematics, a knot or link in the 3-dimensional sphere is called fibered or fibred if there is a 1-parameter family of Seifert surfaces for , where the parameter runs through the points of the unit circle , such that if is not equal to then the intersection of and is exactly .
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
![<span class="mw-page-title-main">Hopf bifurcation</span> Critical point where a periodic solution arises](https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Hopfeigenvalues.png/320px-Hopfeigenvalues.png)
In the mathematical theory of bifurcations, a Hopfbifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis as a parameter crosses a threshold value. Under reasonably generic assumptions about the dynamical system, the fixed point becomes a small-amplitude limit cycle as the parameter changes.
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:
![<span class="mw-page-title-main">Lissajous knot</span> Knot defined by parametric equations defining Lissajous curves](https://upload.wikimedia.org/wikipedia/commons/thumb/8/88/Lissajous_8_21_Knot.png/320px-Lissajous_8_21_Knot.png)
In knot theory, a Lissajous knot is a knot defined by parametric equations of the form
![7<sub>1</sub> knot Mathematical knot with crossing number 7](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Blue_7_1_Knot.png/320px-Blue_7_1_Knot.png)
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. It is the simplest torus knot after the trefoil and cinquefoil.
![<span class="mw-page-title-main">Square knot (mathematics)</span> Connected sum of two trefoil knots with opposite chirality](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Blue_Square_Knot.png/320px-Blue_Square_Knot.png)
In knot theory, the square knot is a composite knot obtained by taking the connected sum of a trefoil knot with its reflection. It is closely related to the granny knot, which is also a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the square knot and the granny knot are the simplest of all composite knots.
![<span class="mw-page-title-main">Granny knot (mathematics)</span> Connected sum of two trefoil knots with same chirality](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Math-granny-knot-6crossings.svg/320px-Math-granny-knot-6crossings.svg.png)
In knot theory, the granny knot is a composite knot obtained by taking the connected sum of two identical trefoil knots. It is closely related to the square knot, which can also be described as a connected sum of two trefoils. Because the trefoil knot is the simplest nontrivial knot, the granny knot and the square knot are the simplest of all composite knots.
![<span class="mw-page-title-main">Twist knot</span> Family of mathematical knots](https://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Blue_8_1_Knot.png/320px-Blue_8_1_Knot.png)
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots.
![<span class="mw-page-title-main">Unknotting number</span> Minimum number of times a specific knot must be passed through itself to become untied](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Unknotting_trefoil.svg/320px-Unknotting_trefoil.svg.png)
In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself to untie it. If a knot has unknotting number , then there exists a diagram of the knot which can be changed to unknot by switching crossings. The unknotting number of a knot is always less than half of its crossing number. This invariant was first defined by Hilmar Wendt in 1936.