“Knot”的概念、定义、习题解题方法及翻译-数学辞典

单词

Knot

释义

Knot

A knot is defined as a closed, non-self-intersecting curve embedded in 3-D. A knot is a single component Link. Kleinproved that knots cannot exist in an Even-numbered dimensional space . It has since been shown that a knot cannotexist in any dimension . Two distinct knots cannot have the same Knot Complement (Gordon and Luecke1989), but two Links can! (Adams 1994, p. 261). The Knot Sum of any number of knots cannot be theUnknot 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 intoPrime Knots. Knot theory was given its first impetus when Lord Kelvin proposed a theory that atomswere 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 13crossings, and Alternating Knots up to 14 crossings. In this compilation, MirrorImages are counted as a single knot type. The number of distinct Prime Knots forknots from to 13 crossings are 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988 (Sloane's A002863). Combining PrimeKnots gives one additional type of knot for knots of six and seven crossings.

Let be the number of distinct Prime Knots of crossings, counting Chiralversions 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 10-161and 10-162are actually identical,and the uppermost crossing in 10-144should 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 BraidGroup. A knot with crossings is a member of the Braid Group . There is no general method known fordeciding whether two given knots are equivalent or interlocked. There is no general Algorithm to determine if atangled 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 KnotNotation for knots up to 10 crossings is given by Rolfsen (1976). Hyperbolic volumes are given (Adams, Hildebrand, andWeeks 1991; Adams 1994). The Braid Index is given by Jones (1987). Alexander Polynomials are given in Rolfsen (1976), but with the Polynomials for 10-083and10-086reversed (Jones 1987). The Alexander Polynomials are normalized accordingto Conway, and given in abbreviated form for .

The Jones Polynomials for knots of up to 10 crossings are given by Jones (1987), and theJones Polynomials can be either computed from these, or taken from Adams (1994) for knots ofup 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 thelower 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 Ffor Knots of up to 13 crossings.

03-001

04-001

05-00105-002

06-00106-00206-003

07-00107-00207-00307-00407-00507-00607-007

08-00108-00208-00308-00408-00508-00608-00708-00808-00908-01008-01108-01208-01308-01408-01508-01608-01708-01808-01908-02008-021

09-00109-00209-00309-00409-00509-00609-00709-00809-00909-01009-01109-01209-01309-01409-01509-01609-01709-01809-01909-02009-02109-02209-02309-02409-02509-02609-02709-02809-02909-03009-03109-03209-03309-03409-03509-03609-03709-03809-03909-04009-04109-04209-04309-04409-04509-04609-04709-04809-049

10-00110-00210-00310-00410-00510-00610-00710-00810-00910-01010-01110-01210-01310-01410-01510-01610-01710-01810-01910-02010-02110-02210-02310-02410-02510-02610-02710-02810-02910-03010-03110-03210-03310-03410-03510-03610-03710-03810-03910-04010-04110-04210-04310-04410-04510-04610-04710-04810-04910-05010-05110-05210-05310-05410-05510-05610-05710-05810-05910-06010-06110-06210-06310-06410-06510-06610-06710-06810-06910-07010-07110-07210-07310-07410-07510-07610-07710-07810-07910-08010-08110-08210-08310-08410-08510-08610-08710-08810-08910-09010-09110-09210-09310-09410-09510-09610-09710-09810-09910-10010-10110-10210-10310-10410-10510-10610-10710-10810-10910-11010-11110-11210-11310-11410-11510-11610-11710-11810-11910-12010-12110-12210-12310-12410-12510-12610-12710-12810-12910-13010-13110-13210-13310-13410-13510-13610-13710-13810-13910-14010-14110-14210-14310-14410-14510-14610-14710-14810-14910-15010-15110-15210-15310-15410-15510-15610-15710-15810-15910-16010-16110-16210-16310-16410-16510-166

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 KnotConjectures, 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. SequenceA002863/M0851in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and extended entry inSloane, 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.

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