Knot

A knot is an embedding of a circle in three-dimensional Euclidean space. The usual theory considers tame knots, which can be represented by polygonal knots or closed smooth curves. The unknot is included, and a nontrivial knot cannot be deformed into a round circle by an ambient isotopy. A physical knot tied in an open rope gives a mathematical knot after the ends are joined to form one closed 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 (Thomson 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 nontrivial tame 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 with two or more nontrivial factors are 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, 26, ... (OEIS A086825).

Every smooth embedding of a circle in with is unknotted. Higher-dimensional spheres can nevertheless be knotted in higher-dimensional spaces. In three dimensions, the knot complement determines a tame knot up to mirror image. If the complements are related by an orientation-preserving homeomorphism, the knots are ambiently isotopic (Gordon and Luecke 1989). Distinct links can have homeomorphic complements (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).

Every tame knot is the closure of a braid (Alexander 1923). The index of the braid group counts strands, rather than crossings.

There are algorithms for deciding whether a finite diagram represents the unknot, whether two tame knots are equivalent, and whether a link is split. Haken (1961) and Hemion (1979) provided foundational algorithms using normal surfaces and three-dimensional topology. Several methods now solve knot and link equivalence (Lackenby 2016).

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, non-alternating, 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.

Precomputed knots and their properties are available in the Wolfram Language through KnotData.