Введение
Нетривиальный узел, который нельзя представить как сумму узлов двух нетривиальных узлов.
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. A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers. Knots are characterized by their crossing numbers. The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3) torus knot. The figure eight knot, with four crossings, is the simplest non torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table. {| class="wikitable" style="text align:right;"
|
! n
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|
! Number of prime knotswith n crossings
| 0 || 0 || 1 || 1 || 2 || 3 || 7 || 21 || 49 || 165 || 552 ||2176 || 9988 || 46972 || 253293 || 1388705
|
! Composite knots
| 0 || 0 || 0 || 0 || 0 || 2 || 1 || 4 || || || || || || || ||
|
! Total
| 0 || 0 || || || || || || || || || || || || || ||
|}
Enantiomorphs are counted only once in this table and the following chart (i. e. a knot and its mirror image are considered equivalent). NOTOC
В теории узлов, простой узел или простое звено – это узел, который в определенном смысле является неделимым. В частности, это нетривиальный узел, который нельзя представить как сумму узлов двух нетривиальных узлов. Узлы, не являющиеся простыми, называются составными узлами или составными звеньями. Определение того, является ли данный узел простым или нет, может быть нетривиальной задачей. Примером простых узлов служат тороидальные узлы. Они образуются путем обвития окружности вокруг тора p раз в одном направлении и q раз в другом, где p и q – взаимно простые целые числа. Узлы характеризуются числом пересечений. Самый простой простой узел – это трилистник с тремя пересечениями. Трилистник фактически является тороидальным узлом (2, 3). Восьмерка с четырьмя пересечениями – самый простой узел, не являющийся тороидальным. Для любого положительного целого числа n существует конечное число простых узлов с n пересечениями. Первые несколько значений приведены в следующей таблице.
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. A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers. Knots are characterized by their crossing numbers. The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3) torus knot. The figure eight knot, with four crossings, is the simplest non torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table. {| class="wikitable" style="text align:right;"
|
! n
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|
! Number of prime knotswith n crossings
| 0 || 0 || 1 || 1 || 2 || 3 || 7 || 21 || 49 || 165 || 552 ||2176 || 9988 || 46972 || 253293 || 1388705
|
! Composite knots
| 0 || 0 || 0 || 0 || 0 || 2 || 1 || 4 || || || || || || || ||
|
! Total
| 0 || 0 || || || || || || || || || || || || || ||
|}
Enantiomorphs are counted only once in this table and the following chart (i. e. a knot and its mirror image are considered equivalent). NOTOC
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Число простых узлов с n пересечениями | 0 | 0 | 1 | 1 | 2 | 3 | 7 | 21 | 49 | 165 | 552 | 2176 | 9988 | 46972 | 253293 | 1388705 |
| Составные узлы | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 4 | | | | | | | | |
| Всего | 0 | 0 | | | | | | | | | | | | | | |
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. A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers. Knots are characterized by their crossing numbers. The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3) torus knot. The figure eight knot, with four crossings, is the simplest non torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table. {| class="wikitable" style="text align:right;"
|
! n
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|
! Number of prime knotswith n crossings
| 0 || 0 || 1 || 1 || 2 || 3 || 7 || 21 || 49 || 165 || 552 ||2176 || 9988 || 46972 || 253293 || 1388705
|
! Composite knots
| 0 || 0 || 0 || 0 || 0 || 2 || 1 || 4 || || || || || || || ||
|
! Total
| 0 || 0 || || || || || || || || || || || || || ||
|}
Enantiomorphs are counted only once in this table and the following chart (i. e. a knot and its mirror image are considered equivalent). NOTOC
Энантиоморфы в этой таблице и следующей диаграмме учитываются только один раз (то есть узел и его зеркальное отражение считаются эквивалентными). NOTOC
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. A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers. Knots are characterized by their crossing numbers. The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3) torus knot. The figure eight knot, with four crossings, is the simplest non torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values are given in the following table. {| class="wikitable" style="text align:right;"
|
! n
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|
! Number of prime knotswith n crossings
| 0 || 0 || 1 || 1 || 2 || 3 || 7 || 21 || 49 || 165 || 552 ||2176 || 9988 || 46972 || 253293 || 1388705
|
! Composite knots
| 0 || 0 || 0 || 0 || 0 || 2 || 1 || 4 || || || || || || || ||
|
! Total
| 0 || 0 || || || || || || || || || || || || || ||
|}
Enantiomorphs are counted only once in this table and the following chart (i. e. a knot and its mirror image are considered equivalent). NOTOC
Теорема Шуберта
Теорема, установленная Хорстом Шубертом (1919–2001), утверждает, что любой узел может быть единственным образом представлен в виде связной суммы простых узлов.