A knot is called prime if, for any decomposition as a connected sum, one of the factors is unknotted (Livingston 1993, pp. 5 and 78). A knot which is not prime is called a composite knot. It is often possible to combine two prime knots to create two different composite knots, depending on the orientation of the two. 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 prime knots.
In general, it is nontrivial to determine if a given knot is prime or composite (Hoste et al. 1998). However, in the case of alternating knots, Menasco (1984) showed that a reduced alternating diagram represents a prime knot iff the diagram is itself prime ("an alternating knot is prime iff it looks prime"; Hoste et al. 1998).
There is no known formula for giving the number of distinct prime knots as a function of the number of crossings. The numbers of distinct prime
knots having 
,
2, ... crossings are 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, ... (OEIS
A002863). A pictorial enumeration of prime
knots of up to 10 crossings appears in Rolfsen (1976, Appendix C). Note, however,
that in this table, the Perko pair 10-161 and 10-162
are actually identical, and the uppermost crossing in 10-144 should be changed (Jones
1987). The 
th
knot having 
crossings in this (arbitrary) ordering of knots is given the symbol 
. The following table summarizes a number of named prime
knots.
| knot symbol | prime knot |
 | unknot |
 | trefoil knot |
 | figure eight knot |
 | Solomon's seal knot |
 | stevedore's knot |
 | Miller Institute knot |
| -- | Conway's knot |
| -- | Kinoshita-Terasaka knot |
Thistlethwaite has used Dowker notation to enumerate the number of prime knots of up to 13 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).
Let 
be the number of distinct prime knots
with 
crossings, counting chiral
versions of the same knot separately. Then
 |
(Ernst and Summers 1987). Welsh has shown that the number of knots is bounded by an exponential in 
,
and it is also known that
![lim sup[N(n)]^(1/n)<13.5](/images/equations/PrimeKnot/NumberedEquation2.svg) |
(Welsh 1991, Hoste et al. 1998, Thistlethwaite 1998).
Knots." Math. Intell. 20, 33-48,
Fall 1998.Jones, V. F. R. "Hecke Algebra Representations
of Braid Groups and Link Polynomials." Ann. Math. 126, 335-388,
1987.Livingston, C.