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Three-twist knot | |
---|---|

Common name | [[Figure-of-nine knot^{ [1] }]] |

Arf invariant | 0 |

Braid length | 6 |

Braid no. | 3 |

Bridge no. | 2 |

Crosscap no. | 2 |

Crossing no. | 5 |

Genus | 1 |

Hyperbolic volume | 2.82812 |

Stick no. | 8 |

Unknotting no. | 1 |

Conway notation | [32] |

A-B notation | 5_{2} |

Dowker notation | 4, 8, 10, 2, 6 |

Last /Next | 5_{1} / 6_{1} |

Other | |

alternating, hyperbolic, prime, reversible, twist |

In knot theory, the **three-twist knot** is the twist knot with three-half twists. It is listed as the **5 _{2} knot** in the Alexander-Briggs notation, and is one of two knots with crossing number five, the other being the cinquefoil knot.

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 together so that 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, **R**^{3}. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of **R**^{3} upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

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.

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. It is a knot invariant.

The three-twist knot is a prime knot, and it is invertible but not amphichiral. Its Alexander polynomial is

In knot theory, a **prime knot** or **prime link** is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be **composite knots** or **composite links**. It can be a nontrivial problem to determine whether a given knot is prime or not.

In mathematics, especially in the area of topology known as knot theory, an **invertible knot** is a knot that can be continuously deformed to itself, but with its orientation reversed. A **non-invertible knot** is any knot which does not have this property. The **invertibility** of a knot is a knot invariant. An **invertible link** is the link equivalent of an invertible knot.

In mathematics, the **Alexander polynomial** is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the **Alexander–Conway polynomial**, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.

its Conway polynomial is

and its Jones polynomial is

^{ [2] }

Because the Alexander polynomial is not monic, the three-twist knot is not fibered.

In algebra, a **monic polynomial** is a single-variable polynomial in which the leading coefficient is equal to 1. Therefore, a monic polynomial has the form

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 .

The three-twist knot is a hyperbolic knot, with its complement having a volume of approximately 2.82812.

If the fibre of the knot in the initial image of this page were cut at the bottom right of the image, and the ends were pulled apart, it would result in a single-stranded figure-of-nine knot (not the figure-of-nine loop).

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.

In knot theory, a branch of mathematics, the **trefoil knot** is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.

In the mathematical field of knot theory, the **HOMFLY polynomial**, sometimes called the **HOMFLY-PT** polynomial or the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables *m* and *l*.

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 knot theory, a **torus knot** is a special kind of knot that lies on the surface of an unknotted torus in **R**^{3}. 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.

In mathematics, a **Seifert surface** is a surface whose boundary is a given knot or link.

In knot theory, a **Lissajous knot** is a knot defined by parametric equations of the form

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. It is listed as the **5 _{1} knot** in the Alexander-Briggs notation, and can also be described as the (5,2)-torus knot. The cinquefoil is the closed version of the double overhand knot.

In knot theory, the **stevedore knot** is one of three prime knots with crossing number six, the others being the 6_{2} knot and the 6_{3} knot. The stevedore knot is listed as the **6 _{1} knot** in the Alexander–Briggs notation, and it can also be described as a twist knot with four twists, or as the (5,−1,−1) pretzel knot.

In knot theory, the **7 _{1} knot**, also known as the

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.

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.

In knot theory, the **6 _{2} knot** is one of three prime knots with crossing number six, the others being the stevedore knot and the 6

In knot theory, the **6 _{3} knot** is one of three prime knots with crossing number six, the others being the stevedore knot and the 6

In the mathematical theory of knots, **L10a140** is the name in the Thistlewaite link table of a link of three loops, which has ten crossings between the loops when presented in its simplest visual form. It is of interest because it is presumably the simplest link which possesses the Brunnian property — a link of connected components that, when one component is removed, becomes entirely unconnected — other than the six-crossing Borromean rings.

- ↑ Pinsky, Tali (1 September 2017). "On the topology of the Lorenz system".
*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*. The Royal Society.**473**(2205): 20170374. doi:10.1098/rspa.2017.0374. PMC 5627380 . PMID 28989313 . Retrieved 26 August 2018.(b) the knot with three half-twists, called the 52 knot.

- ↑ " 5_2 ",
*The Knot Atlas*.

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Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.