There are no approved revisions of this page, so it may
not have been
reviewed.
This article page is a stub, please help by expanding it.
A knot is a closed non-self-intersecting curve embedded in 3-space.
Knots include:
A086825 Number of knots (trivial, prime or composite) with
crossings, for
.
(For , we get the trivial knot, i.e. the unknot.)
-
{1, 0, 0, 1, 1, 2, 5, 8, 25, ...} |
A?????? Number of knots (trivial, prime or composite) with up to
crossings, for
.
(For , we get the trivial knot, i.e. the unknot.)
-
{1, 1, 1, 2, 3, 5, 10, 18, 43, ...} |
A?????? Number of knots (prime or composite, but not trivial) with up to
crossings, for
.
(For , we get the trivial knot, i.e. the unknot.) (
matches
A049961 : coincidence?)
-
{0, 0, 0, 1, 2, 4, 9, 17, 42, ...} |
Prime knots
Prime knots have at least 3 crossings, since 0, 1 or 2 crossings yield the unknot.
A002863 Number of prime knots with
crossings, for
.
-
{0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, ...} |
A002864 Number of alternating prime knots with
crossings, for
.
-
{0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, 1288, 4878, 19536, 85263, 379799, 1769979, ...} |
A051763 Number of nonalternating prime knots with
crossings, for
. (
A002863 −
A002864)
-
{0, 0, 0, 0, 0, 0, 0, 3, 8, 42, 185, 888, 5110, 27436, 168030, 1008906, ...} |
A116584 Number of prime knots with up to
crossings, for
.
-
{0, 0, 1, 2, 4, 7, 14, 35, 84, 249, 801, 2977, 12965, 59937, 313230, 1701935, ...} |
Knot sum
Every composite knot is a unique knot sum of prime knots (called factor knots), where different choices of chiralities may yield distinct knots.
Composite knots
Composite knots have at least 6 crossings, since there are no prime knots with fewer than 3 crossings.
A?????? (Is it
A227050, or do we have a case of the
strong law of small numbers?) Number of composite knots with
crossings, for
.
-
{0, 0, 0, 0, 0, 2, 1, 4, ...} |
Knot chirality
Chiral knots
(...)
Amphichiral knots
(...)
External links
- The Knot Atlas—Bar-Natan, Dror and Morrison, Scott and et al.
- Knotilus
- The Rolfsen Knot Table, from The Knot Atlas. (Table of the unknot and prime knots with 3, 4, 5, 6, 7, 8, 9, or 10 crossings.)
- Weisstein, Eric W., Prime Knot, from MathWorld—A Wolfram Web Resource.
- Weisstein, Eric W., Composite Knot, from MathWorld—A Wolfram Web Resource.
- Weisstein, Eric W., Knot Sum, from MathWorld—A Wolfram Web Resource.
- Weisstein, Eric W., Factor Knots, from MathWorld—A Wolfram Web Resource.
- Morinigo, Marcos A. Prime Factorization of Knots
- Adams, Colin C. (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society. pp. 306. ISBN 9780821836781.