Knot

In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). While in common usage, knots can be tied in string and rope such that one or more strands are left open on either side of the knot, the mathematical theory of knots terms an object of this type a "braid" rather than a knot. To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand.

A knot can be generalized to a link, which is simply a knotted collection of one or more closed strands.

The study of knots and their properties is known as knot theory. Knot theory was given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations (Thompson 1867). P. G. Tait then cataloged possible knots by trial and error. Much progress has been made in the intervening years.

Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed (Livingston 1993, p. 5; Adams 1994, pp. 8-9). Knots that can be so decomposed are then known as composite knots. The total number (prime plus composite) of distinct knots (treating mirror images as equivalent) having , 1, ... crossings are 1, 0, 0, 1, 1, 2, 5, 8, 25, ... (OEIS A086825).

Klein proved that knots cannot exist in an even-dimensional space . It has since been shown that a knot cannot exist in any dimension . Two distinct knots cannot have the same knot complement (Gordon and Luecke 1989), but two links can! (Adams 1994, p. 261).

Knots are most commonly cataloged based on the minimum number of crossings present (the so-called link crossing number). Thistlethwaite has used Dowker notation to enumerate the number of prime knots of up to 13 crossings, and alternating knots up to 14 crossings. In this compilation, mirror images are counted as a single knot type. Hoste et al. (1998) subsequently tabulated all prime knots up to 16 crossings. Hoste and Weeks subsequently began compiling a list of 17-crossing prime knots (Hoste et al. 1998).

Another possible representation for knots uses the braid group. A knot with crossings is a member of the braid group .

There is no general algorithm to determine if a tangled curve is a knot or if two given knots are interlocked. Haken (1961) and Hemion (1979) have given algorithms for rigorously determining if two knots are equivalent, but they are too complex to apply even in simple cases (Hoste et al. 1998).

Broden et al. (2024) proved that all knots can be embedded into the Menger sponge (Barber 2024), proved that every pretzel knot can be embedded into the tetrix, and conjectured that every knot can be embedded into the tetrix.

The following tables give the number of distinct prime, alternating, nonalternating, torus, and satellite knots for to 16 (Hoste et al. 1998).

	prime knots	prime alternating knots	prime nonalternating knots	torus knots	satellite knots
Sloane	A002863	A002864	A051763	A051764	A051765
3	1	1	0	1	0
4	1	1	0	0	0
5	2	2	0	1	0
6	3	3	0	0	0
7	7	7	0	1	0
8	21	18	3	1	0
9	49	41	8	1	0
10	165	123	42	1	0
11	552	367	185	1	0
12	2176	1288	888	0	0
13	9988	4878	5110	1	2
14	46972	19536	27436	1	2
15	253293	85263	168030	2	6
16	1388705	379799	1008906	1	10

The numbers of chiral noninvertible , amphichiral noninvertible, amphichiral noninvertible, chiral invertible , and fully amphichiral and invertible knots are summarized in the following table for to 16 (Hoste et al. 1998).

Sloane	A051766	A051767	A051768	A051769	A052400
3	0	0	0	1	0
4	0	0	0	0	1
5	0	0	0	2	0
6	0	0	0	2	1
7	0	0	0	7	0
8	0	0	1	16	4
9	2	0	0	47	0
10	27	0	6	125	7
11	187	0	0	365	0
12	1103	1	40	1015	17
13	6919	0	0	3069	0
14	37885	6	227	8813	41
15	226580	0	1	26712	0
16	1308449	65	1361	78717	113

If a knot is amphichiral, the "amphichirality" is , otherwise (Jones 1987). Arf invariants are designated . Braid words are denoted (Jones 1987). Conway's knot notation for knots up to 10 crossings is given by Rolfsen (1976). Hyperbolic volumes are given (Adams et al. 1991; Adams 1994). The braid index is given by Jones (1987). Alexander polynomials are given in Rolfsen (1976), but with the polynomials for 10-083 and 10-086 reversed (Jones 1987). The Alexander polynomials are normalized according to Conway, and given in abbreviated form for .

The Jones polynomials for knots of up to 10 crossings are given by Jones (1987), and the Jones polynomials can be either computed from these, or taken from Adams (1994) for knots of up to 9 crossings (although most polynomials are associated with the wrong knot in the first printing). The Jones polynomials can be listed in the abbreviated form for , and correspond either to the knot depicted by Rolfsen or its mirror image, whichever has the lower power of . The HOMFLY polynomial and Kauffman polynomial F(a,x) are given in Lickorish and Millett (1988) for knots of up to 7 crossings. M. B. Thistlethwaite has tabulated the HOMFLY polynomial and Kauffman polynomial F for knots of up to 13 crossings.