Prime Link

A prime link is a link that cannot be represented as a knot sum of other links. Doll and Hoste (1991) list polynomials for oriented links of nine or fewer crossings, and Rolfsen (1976) gives a table of links with small numbers of components and crossings.

The following table summarizes the number of distinct prime -components links having specified crossing numbers. The

components	OEIS	prime -component links with 1, 2, ... crossings
1	A002863	0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, ...
2	A048952	0, 1, 0, 1, 1, 3, 8, 16, 61, 185, 638, ...
3	A048953	0, 0, 0, 0, 0, 3, 1, 10, 21, 74, 329, ...
4	A087071	0, 0, 0, 0, 0, 0, 0, 3, 1, 15, 39, ...
5		0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, ...
total	A086771	0, 1, 1, 2, 3, 9, 16, 50, 132, 442, 1559, ...

The following table lists some named links. The notation and ordering follows that of Rolfsen (1976), where  denotes the th -component link with crossing number .

link number	name
	unlink
	Hopf link
	Whitehead link
	Borromean rings

A listing of the first few simple links follows, arranged by link crossing number.

01

01

01

01

01 02 03

01 02 03 04 05 06 07 08

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

01 02 03

01

01 02 03 04 05 06 07 08 09 10

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21

01 02 03

01

REFERENCES:

Bar-Natan, D. "The Hoste-Thistlethwaite Link Table." http://www.math.toronto.edu/~drorbn/KAtlas/Links/.

Doll, H. and Hoste, J. "A Tabulation of Oriented Links." Math. Comput. 57, 747-761, 1991.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976.

Scharein, R. G. "Number of Prime Links." http://www.pims.math.ca/knotplot/manual/PrimeLinks.html.

Sloane, N. J. A. Sequences A002863/M0851, A048952, A048953, A086771, and A087071 in "The On-Line Encyclopedia of Integer Sequences."

Welsh, D. J. A. "On the Number of Knots and Links." Colloq. Math. Soc. J. Bolyai 60, 713-718, 1991.