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, ... (Sloane's 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).
The following tables give the number of distinct prime, alternating, nonalternating, torus,
and satellite knots for  to 16 (Hoste et al. 1998).
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.
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory
of Knots. New York: W. H. Freeman, pp. 280-286, 1994.
Adams, C.; Hildebrand, M.; and Weeks, J. "Hyperbolic Invariants of Knots and
Links." Trans. Amer. Math. Soc. 1, 1-56, 1991.
Alexander, J. W. and Briggs, G. B. "On Types of Knotted Curves."
Ann. Math. 28, 562-586, 1927.
Aneziris, C. N. The Mystery of Knots: Computer Programming for Knot Tabulation.
Singapore: World Scientific, 1999.
Ashley, C. W. The Ashley Book of Knots. New York: McGraw-Hill, 1996.
Bar-Natan, D. "The Mathematica Package KnotTheory.m."
http://www.math.toronto.edu/~drorbn/KAtlas/Manual/.
Bar-Natan, D. "The Rolfsen Knot Table." http://www.math.toronto.edu/~drorbn/KAtlas/Knots/.
Bogomolny, A. "Knots...." http://www.cut-the-knot.org/do_you_know/knots.shtml.
Bruzelius, L. "Knots and Splices." http://pc-78-120.udac.se:8001/WWW/Nautica/Bibliography/Knots&Splices.html.
Caudron, A. "Classification des noeuds et des enlacements." Prepublication Math. d'Orsay. Orsay, France: Université Paris-Sud, 1980.
Cerf, C. "Atlas of Oriented Knots and Links." Topology Atlas Invited Contributions 3, No. 2, 1-32, 1998. http://at.yorku.ca/t/a/i/c/31.htm.
Cha, J. C. and Livingston, C. "KnotInfo: Table of Knot Invariants."
http://www.indiana.edu/~knotinfo/.
Cha, J. C. and Livingston, C. "Unknown Values in the Table of Knots."
2008 May 16. http://arxiv.org/abs/math.GT/0503125/.
Conway, J. H. "An Enumeration of Knots and Links." In Computational Problems in Abstract Algebra (Ed. J. Leech). Oxford, England: Pergamon Press,
pp. 329-358, 1970.
Eppstein, D. "Knot Theory." http://www.ics.uci.edu/~eppstein/junkyard/knot.html.
Erdener, K.; Candy, C.; and Wu, D. "Verification and Extension of Topological Knot Tables." ftp://chs.cusd.claremont.edu/pub/knot/FinalReport.sit.hqx.
ERP Productions. "Ropers Knot Pages, Real Knots: Knotting, Bends and Hitches."
http://www.realknots.com/.
GANG. "GANG Knot Library." http://www.gang.umass.edu/library/knots/.
Gordon, C. and Luecke, J. "Knots are Determined by their Complements."
J. Amer. Math. Soc. 2, 371-415, 1989.
Haken, W. "Theorie der Normalflachen." Acta Math. 105, 245-375,
1961.
Hemion, G. "On the Classification of Homeomorphisms of 2-Manifolds and the Classification
of 3-Manifolds." Acta Math. 142, 123-155, 1979.
Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots."
Math. Intell. 20, 33-48, Fall 1998.
Jones, V. "Hecke Algebra Representations of Braid Groups and Link Polynomials."
Ann. Math. 126, 335-388, 1987.
Kauffman, L. Knots and Applications. River Edge, NJ: World Scientific,
1995.
Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, 1991.
Kirkman, T. P. "The Enumeration, Description, and Construction of Knots Fewer than Ten Crossings." Trans. Roy. Soc. Edinburgh 32, 1885,
281-309.
Kirkman, T. P. "The 634 Unifilar Knots of Ten Crossings Enumerated and
Defined." Trans. Roy. Soc. Edinburgh 32, 483-506, 1885.
Korpegård, J. "The Knotting Dictionary of Kännet." http://www.korpegard.nu/knot/.
Lickorish, W. B. R. and Millett, B. R. "The New Polynomial Invariants
of Knots and Links." Math. Mag. 61, 1-23, 1988.
Listing, J. B. "Vorstudien zur Topologie." Göttingen Studien, University of Göttingen, Germany, 1848.
Little, C. N. "On Knots, with a Census of Order Ten." Trans. Connecticut
Acad. Sci. 18, 374-378, 1885.
Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.
Murasugi, K. and Kurpita, B. I. A Study of Braids. Dordrecht, Netherlands: Kluwer, 1999.
Neuwirth, L. "The Theory of Knots." Sci. Amer. 140, 84-96,
Jun. 1979.
Perko, K. "Invariants of 11-Crossing Knots." Prepublications Math. d'Orsay. Orsay, France: Université Paris-Sub, 1980.
Perko, K. "Primality of Certain Knots." Topology Proc. 7,
109-118, 1982.
Praslov, V. V. and Sossinsky, A. B. Knots, Links, Braids and 3-Manifolds: An Introduction to the New
Invariants in Low-Dimensional Topology. Providence, RI: Amer. Math. Soc.,
1996.
Przytycki, J. "A History of Knot Theory from Vandermonde to Jones." Proc.
Mexican Nat. Congress Math., Nov. 1991.
Reidemeister, K. Knotentheorie. Berlin: Springer-Verlag, 1932.
Rolfsen, D. "Table of Knots and Links." Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press,
pp. 280-287, 1976.
Schubert, H. Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturwiss. Klasse, 3rd
Abhandlung. 1949.
Sloane, N. J. A. Sequences A002863/M0851 and A086825 in "The On-Line Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Plouffe, S. Figure M0851 in The Encyclopedia of Integer Sequences. San Diego: Academic
Press, 1995.
Stoimenow, A. "Polynomials of Knots with Up to 10 Crossings." Rev. March
9, 2002. http://www.math.toronto.edu/stoimeno/poly.ps.gz.
Suber, O. "Knots on the Web." http://www.earlham.edu/~peters/knotlink.htm.
Tait, P. G. "On Knots I, II, and III." Scientific Papers, Vol. 1.
Cambridge, England: University Press, pp. 273-347, 1898.
Thistlethwaite, M. B. "Knot Tabulations and Related Topics." In Aspects
of Topology in Memory of Hugh Dowker 1912-1982 (Ed. I. M. James
and E. H. Kronheimer). Cambridge, England: Cambridge University Press,
pp. 2-76, 1985.
Thistlethwaite, M. B. ftp://chs.cusd.claremont.edu/pub/knot/Thistlethwaite_Tables/.
Thistlethwaite, M. B. "Morwen's Home Page." http://www.math.utk.edu/~morwen/.
Thompson, W. T. "On Vortex Atoms." Philos. Mag. 34,
15-24, 1867.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, pp. 132-135, 1991.
Weisstein, E. W. "Books about Knot Theory." http://www.ericweisstein.com/encyclopedias/books/KnotTheory.html.
 Zia. "Zia Knotting."
http://204.30.30.10/hac/knots/
|
Contact the MathWorld Team
