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Knots

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Meander BR.png Meander LB.png
Meander BR.png Meander LT & BR.png Meander XH.png Meander LB.png
Meander TR.png Meander XH.png Meander XV.png Meander LT.png
Meander TR.png Meander LT.png
There is 1 prime knot with 3 crossings (the trefoil knot), up to chirality (handedness).
Meander BR.png Meander LB.png
Meander BR.png Meander XH.png Meander LB & TR.png Meander LB.png
Meander TR.png Meander XV.png Meander XH.png Meander LT.png
Meander TR.png Meander LT.png

A knot is a closed non-self-intersecting curve embedded in 3-space.

Knots include:

A086825 Number of knots (trivial, prime or composite) with 
n
crossings, for 
n   ≥   0
. (For 
n = 0
, 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 
n
crossings, for 
n   ≥   0
. (For 
n = 0
, 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 
n
crossings, for 
n   ≥   0
. (For 
n = 0
, we get the trivial knot, i.e. the unknot.)
( 
1, 2, 4, 9, 17, 42
matches A049961
(n), n   ≥   1
: 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.

Prime knots with 
n
crossings

n
Symbol Name
3
 31
trefoil knot
4
 41
figure eight knot
5
 51
Solomon's seal knot
 52
 ?
6
 61
stevedore's knot
 62
Miller Institute knot
 63
 ?
7
 71
 ?
 72
 ?
 73
 ?
 74
 ?
 75
 ?
 76
 ?
 77
 ?

A002863 Number of prime knots with 
n
crossings, for 
n   ≥   1
.
{0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, ...}
A002864 Number of alternating prime knots with 
n
crossings, for 
n   ≥   1
.
{0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, 1288, 4878, 19536, 85263, 379799, 1769979, ...}
A051763 Number of nonalternating prime knots with 
n
crossings, for 
n   ≥   1
. (A002863A002864)
{0, 0, 0, 0, 0, 0, 0, 3, 8, 42, 185, 888, 5110, 27436, 168030, 1008906, ...}
A116584 Number of prime knots with up to 
n
crossings, for 
n   ≥   1
.
{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.

Composite knots with 
n
crossings

n
Symbol Name Properties
6
 31 # 31
granny knot
 31 # 3 *1
square knot
7
 31 # 41
 ? Amphichiral
8
 31 # 51
 ?
 31 # 5 *1
 ?
 31 # 52
 ?
 31 # 5 *2
 ?

A?????? (Is it A227050, or do we have a case of the strong law of small numbers?) Number of composite knots with 
n
crossings, for 
n   ≥   1
.
{0, 0, 0, 0, 0, 2, 1, 4, ...}

Knot chirality

Chiral knots

(...)

Amphichiral knots

(...)

External links