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 for exclusively prime knots (sequence A002863 in the OEIS) and for prime or composite knots (sequence A086825 in the OEIS) are given in the following table.
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Number of prime knots
with n crossings0 0 1 1 2 3 7 21 49 165 552 2176 9988 46972 253293 1388705 8053393 48266466 294130458 Composite knots 0 0 0 0 0 2 1 5 ... ... ... ... ... ... Total 0 0 1 1 2 5 8 26 ... ... ... ... ... ...
Enantiomorphs are counted only once in this table and the following chart (i.e. a knot and its mirror image are considered equivalent).
Schubert's theorem
A theorem due to Horst Schubert (1919–2001) states that every knot can be uniquely expressed as a connected sum of prime knots.
See also
- List of prime knots
References
- Schubert, H. "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57–104.
External links
- Weisstein, Eric W. "Prime Knot". MathWorld.
- "Prime Links with a Non-Prime Component", The Knot Atlas.
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 Simplest prime link 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 for exclusively prime knots sequence A002863 in the OEIS and for prime or composite knots sequence A086825 in the OEIS are given in the following table n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19Number of prime knots with n crossings 0 0 1 1 2 3 7 21 49 165 552 2176 9988 46972 253293 1388705 8053393 48266466 294130458Composite knots 0 0 0 0 0 2 1 5 Total 0 0 1 1 2 5 8 26 Enantiomorphs are counted only once in this table and the following chart i e a knot and its mirror image are considered equivalent A chart of all prime knots with seven or fewer crossings not including mirror images plus the unknot which is not considered prime Schubert s theoremA theorem due to Horst Schubert 1919 2001 states that every knot can be uniquely expressed as a connected sum of prime knots See alsoList of prime knotsReferencesSchubert H Die eindeutige Zerlegbarkeit eines Knotens in Primknoten S B Heidelberger Akad Wiss Math Nat Kl 1949 1949 57 104 External linksWeisstein Eric W Prime Knot MathWorld Prime Links with a Non Prime Component The Knot Atlas
