A knot is defined as a closed, non-self-intersecting curve embedded in 3-D. A knot is a single component Link. Klein
proved that knots cannot exist in an Even-numbered 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). The Knot Sum of any number of knots cannot be the
Unknot unless each knot in the sum is the Unknot.

Knots can be cataloged based on the minimum number of crossings present. Knots are usually further broken down into Prime Knots. 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.

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. The number of distinct Prime Knots for knots from to 13 crossings are 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988 (Sloane's A002863). Combining Prime Knots gives one additional type of knot for knots of six and seven crossings.

Let be the number of distinct Prime Knots of 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 .

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 . Another possible representation for knots uses the Braid Group. A knot with crossings is a member of the Braid Group . There is no general method known for deciding whether two given knots are equivalent or interlocked. There is no general Algorithm to determine if a tangled curve is a knot. Haken (1961) has given an Algorithm, but it is too complex to apply to even simple cases.

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, Hildebrand, and Weeks 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 are 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 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.

07-001
07-002
07-003
07-004
07-005
07-006
07-007

08-001
08-002
08-003
08-004
08-005
08-006
08-007
08-008
08-009
08-010
08-011
08-012
08-013
08-014
08-015
08-016
08-017
08-018
08-019
08-020
08-021

09-001
09-002
09-003
09-004
09-005
09-006
09-007
09-008
09-009
09-010
09-011
09-012
09-013
09-014
09-015
09-016
09-017
09-018
09-019
09-020
09-021
09-022
09-023
09-024
09-025
09-026
09-027
09-028
09-029
09-030
09-031
09-032
09-033
09-034
09-035
09-036
09-037
09-038
09-039
09-040
09-041
09-042
09-043
09-044
09-045
09-046
09-047
09-048
09-049

10-001
10-002
10-003
10-004
10-005
10-006
10-007
10-008
10-009
10-010
10-011
10-012
10-013
10-014
10-015
10-016
10-017
10-018
10-019
10-020
10-021
10-022
10-023
10-024
10-025
10-026
10-027
10-028
10-029
10-030
10-031
10-032
10-033
10-034
10-035
10-036
10-037
10-038
10-039
10-040
10-041
10-042
10-043
10-044
10-045
10-046
10-047
10-048
10-049
10-050
10-051
10-052
10-053
10-054
10-055
10-056
10-057
10-058
10-059
10-060
10-061
10-062
10-063
10-064
10-065
10-066
10-067
10-068
10-069
10-070
10-071
10-072
10-073
10-074
10-075
10-076
10-077
10-078
10-079
10-080
10-081
10-082
10-083
10-084
10-085
10-086
10-087
10-088
10-089
10-090
10-091
10-092
10-093
10-094
10-095
10-096
10-097
10-098
10-099
10-100
10-101
10-102
10-103
10-104
10-105
10-106
10-107
10-108
10-109
10-110
10-111
10-112
10-113
10-114
10-115
10-116
10-117
10-118
10-119
10-120
10-121
10-122
10-123
10-124
10-125
10-126
10-127
10-128
10-129
10-130
10-131
10-132
10-133
10-134
10-135
10-136
10-137
10-138
10-139
10-140
10-141
10-142
10-143
10-144
10-145
10-146
10-147
10-148
10-149
10-150
10-151
10-152
10-153
10-154
10-155
10-156
10-157
10-158
10-159
10-160
10-161
10-162
10-163
10-164
10-165
10-166

**References**

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.

Anderson, J. ``The Knotting Dictionary of Kännet.'' http://www.netg.se/~jan/knopar/english/index.htm.

Ashley, C. W. *The Ashley Book of Knots.* New York: McGraw-Hill, 1996.

Bogomolny, A. ``Knots....'' http://www.cut-the-knot.com/do_you_know/knots.html.

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.

Eppstein, D. ``Knot Theory.'' http://www.ics.uci.edu/~eppstein/junkyard/knot/.

Erdener, K.; Candy, C.; and Wu, D. ``Verification and Extension of Topological Knot Tables.'' ftp://chs.cusd.claremont.edu/pub/knot/FinalReport.sit.hqx.

Ernst, C. and Sumner, D. W. ``The Growth of the Number of Prime Knots.'' *Proc. Cambridge Phil. Soc.* **102**, 303-315, 1987.

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.

Kauffman, L. *Knots and Applications.* River Edge, NJ: World Scientific, 1995.

Kauffman, L. *Knots and Physics.* Teaneck, NJ: World Scientific, 1991.

Lickorish, W. B. R. and Millett, B. R. ``The New Polynomial Invariants of Knots and Links.'' *Math. Mag.* **61**, 1-23, 1988.

Livingston, C. *Knot Theory.* Washington, DC: Math. Assoc. Amer., 1993.

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.

Rolfsen, D. ``Table of Knots and Links.'' Appendix C in *Knots and Links.* Wilmington, DE: Publish or Perish Press,
pp. 280-287, 1976.

``Ropers Knots Page.'' http://huizen2.dds.nl/~erpprs/kne/kroot.htm.

Sloane, N. J. A. Sequence
A002863/M0851
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and extended entry in
Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

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: 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/.

Thompson, W. T. ``On Vortex Atoms.'' *Philos. Mag.* **34**, 15-24, 1867.

Weisstein, E. W. ``Knots.'' Mathematica notebook Knots.m.

© 1996-9

1999-05-26