Knot

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 $\geq 4$. It has since been shown that a knot cannot exist in any dimension $\geq 4$. 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 $N(n)$ for knots from $n=3$ 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 $C(n)$ be the number of distinct Prime Knots of $n$ crossings, counting Chiral versions of the same knot separately. Then

$${\textstyle{1\over 3}}(2^{n-2}-1)\leq N(n) \mathrel{\hbox{\h... ...o 0pt{% \lower.5ex\hbox{$\sim$}\hss}\raise.4ex\hbox{$<$}}} e^n$$

(Ernst and Summers 1987). Welsh has shown that the number of knots is bounded by an exponential in $n$.

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 $k$th knot having $n$ crossings in this (arbitrary) ordering of knots is given the symbol $n_k$. Another possible representation for knots uses the Braid Group. A knot with $n+1$ crossings is a member of the Braid Group $n$. 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 $A=1$, otherwise $A=0$ (Jones 1987). Arf Invariants are designated $a$. Braid Words are denoted $b$ (Jones 1987). Conway's Knot Notation $C$ 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 $i$ is given by Jones (1987). Alexander Polynomials $\Delta$ 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 $[a_1, a_2, \ldots$ for $a_1+a_2(x^{-1}+x)+\ldots$.

The Jones Polynomials $W$ for knots of up to 10 crossings are given by Jones (1987), and the Jones Polynomials $V$ 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 $\{n\}\, a_0\, a_1\, \ldots$ for $t^{-n}(a_0+a_1t+\ldots)$, and correspond either to the knot depicted by Rolfsen or its Mirror Image, whichever has the lower Power of $t^{-1}$. The HOMFLY Polynomial $P(\ell,m)$ 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.

03-001

04-001

05-001 05-002

06-001 06-002 06-003

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

See also Alexander Polynomial, Alexander's Horned Sphere, Ambient Isotopy, Amphichiral, Antoine's Necklace, Bend (Knot), Bennequin's Conjecture, Borromean Rings, Braid Group, Brunnian Link, Burau Representation, Chefalo Knot, Clove Hitch, Colorable, Conway's Knot, Crookedness, Dehn's Lemma, Dowker Notation, Figure-of-Eight Knot, Granny Knot, Hitch, Invertible Knot, Jones Polynomial, Kinoshita-Terasaka Knot, Knot Polynomial, Knot Sum, Linking Number, Loop (Knot), Markov's Theorem, Menasco's Theorem, Milnor's Conjecture, Nasty Knot, Pretzel Knot, Prime Knot, Reidemeister Moves, Ribbon Knot, Running Knot, Schönflies Theorem, Shortening, Signature (Knot), Skein Relationship, Slice Knot, Slip Knot, Smith Conjecture, Solomon's Seal Knot, Span (Link), Splitting, Square Knot, Stevedore's Knot, Stick Number, Stopper Knot, Tait's Knot Conjectures, Tame Knot, Tangle, Torsion Number, Trefoil Knot, Unknot, Unknotting Number, Vassiliev Polynomial, Whitehead Link

References

Knot Theory

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.