Teorija čvorova

Examples of different knots including the trivial knot (top left) and (below it) the trefoil knot

A knot diagram of the trefoil knot, the simplest non-trivial knot

U topologiji, teorija čvorova je studija of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R³ (in topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R³ upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.

A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.

The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.

Istorija

Glavni članak: Istorija teorije čvorova

Intricate Celtic knotwork in the 1200-year-old Book of Kells

Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (see Chinese knotting). The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.

The first knot tabulator, Peter Guthrie Tait

A mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Carl Friedrich Gauss, who defined the linking integral Silver 2006. In the 1860s, Lord Kelvin's theory that atoms were knots in the aether led to Peter Guthrie Tait's creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as the Tait conjectures. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology.

These topologists in the early part of the 20th century—Max Dehn, J. W. Alexander, and others—studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial. This would be the main approach to knot theory until a series of breakthroughs transformed the subject.

In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in 1984 Sossinsky 2002, стр. 71–89, and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology.

In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not Simon 1986. Tangles, strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA Flapan 2000. Knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation Collins 2006.

Literatura

• Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1 
• Adams, Colin; Hildebrand, Martin; Weeks, Jeffrey (1991), „Hyperbolic invariants of knots and links”, Transactions of the American Mathematical Society, 326 (1): 1—56, JSTOR 2001854, doi:10.1090/s0002-9947-1991-0994161-2 
• Akbulut, Selman; King, Henry C. (1981), „All knots are algebraic”, Comm. Math. Helv., 56 (3): 339—351, doi:10.1007/BF02566217 
• Bar-Natan, Dror (1995), „On the Vassiliev knot invariants”, Topology, 34 (2): 423—472, doi:10.1016/0040-9383(95)93237-2 
• Collins, Graham (април 2006), „Computing with Quantum Knots”, Scientific American, 294 (4), стр. 56—63, Bibcode:2006SciAm.294d..56C, doi:10.1038/scientificamerican0406-56 CS1 одржавање: Формат датума (веза)
• Dehn, Max (1914), „Die beiden Kleeblattschlingen”, Mathematische Annalen, 75: 402—413 
• Conway, John Horton (1970), „An enumeration of knots and links, and some of their algebraic properties”, Computational Problems in Abstract Algebra, Pergamon, стр. 329—358, ISBN 978-0080129754, OCLC 322649 
• Doll, Helmut; Hoste, Jim (1991), „A tabulation of oriented links. With microfiche supplement”, Math. Comp., 57 (196): 747—761, Bibcode:1991MaCom..57..747D, doi:10.1090/S0025-5718-1991-1094946-4 
• Flapan, Erica (2000), „When topology meets chemistry: A topological look at molecular chirality”, Outlooks, Cambridge University Press, ISBN 978-0-521-66254-3 
• Haefliger, André (1962), „Knotted (4k − 1)-spheres in 6k-space”, Annals of Mathematics, Second Series, 75 (3): 452—466, JSTOR 1970208, doi:10.2307/1970208 
• Hass, Joel (1998), „Algorithms for recognizing knots and 3-manifolds”, Chaos, Solitons and Fractals, 9 (4–5): 569—581, Bibcode:1998CSF.....9..569H, arXiv:math/9712269 , doi:10.1016/S0960-0779(97)00109-4 
• Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeffrey (1998), „The First 1,701,935 Knots”, Math. Intelligencer, 20 (4): 33—48, doi:10.1007/BF03025227 
• Hoste, Jim (2005), „The enumeration and classification of knots and links”, Handbook of Knot Theory (PDF), Amsterdam: Elsevier 
• Levine, Jerome (1965), „A classification of differentiable knots”, Annals of Mathematics, Second Series, 1982 (1): 15—50, JSTOR 1970561, doi:10.2307/1970561 
• Kontsevich, Maxim (1993), „Vassiliev's knot invariants”, I. M. Gelfand Seminar, Adv. Soviet Math., 2, Providence, RI: American Mathematical Society, 16: 137—150, ISBN 9780821841174, doi:10.1090/advsov/016.2/04 
• Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, Springer-Verlag, ISBN 978-0-387-98254-0 
• Perko, Kenneth (1974), „On the classification of knots”, Proceedings of the American Mathematical Society, 45 (2): 262—6, JSTOR 2040074, doi:10.2307/2040074 
• Rolfsen, Dale (1976), Knots and Links, Mathematics Lecture Series, 7, Berkeley, California: Publish or Perish, ISBN 978-0-914098-16-4, MR 0515288 
• Schubert, Horst (1949), „Die eindeutige Zerlegbarkeit eines Knotens in Primknoten”, Heidelberger Akad. Wiss. Math.-Nat. Kl. (3): 57—104 
• Silver, Dan (2006), „Knot theory's odd origins” (PDF), American Scientist, 94 (2), стр. 158—165, doi:10.1511/2006.2.158 
• Simon, Jonathan (1986), „Topological chirality of certain molecules”, Topology, 25 (2): 229—235, doi:10.1016/0040-9383(86)90041-8 
• Sossinsky, Alexei (2002), Knots, mathematics with a twist, Harvard University Press, ISBN 978-0-674-00944-8 
• Turaev, V. G. (1994), „Quantum invariants of knots and 3-manifolds”, De Gruyter Studies in Mathematics, Berlin: Walter de Gruyter & Co., 18, ISBN 978-3-11-013704-0, arXiv:hep-th/9409028  
• Weisstein, Eric W. „Reduced Knot Diagram”. MathWorld. Wolfram. Приступљено 8. 5. 2013. CS1 одржавање: Формат датума (веза)
• Weisstein, Eric W. „Reducible Crossing”. MathWorld. Wolfram. Приступљено 8. 5. 2013. CS1 одржавање: Формат датума (веза)
• Witten, Edward (1989), „Quantum field theory and the Jones polynomial”, Comm. Math. Phys., 121 (3): 351—399, Bibcode:1989CMaPh.121..351W, doi:10.1007/BF01217730 
• Zeeman, E. C. (1963), „Unknotting combinatorial balls”, Annals of Mathematics, Second Series, 78 (3): 501—526, JSTOR 1970538, doi:10.2307/1970538 
• Burde, Gerhard; Zieschang, Heiner (1985), Knots, De Gruyter Studies in Mathematics, 5, Walter de Gruyter, ISBN 978-3-11-008675-1 
• Crowell, Richard H.; Fox, Ralph (1977). Introduction to Knot Theory. ISBN 978-0-387-90272-2. 
• Kauffman, Louis H. (1987), On Knots, ISBN 978-0-691-08435-0 
• Kauffman, Louis H. (2013), Knots and Physics (4th изд.), World Scientific, ISBN 978-981-4383-00-4 
• Menasco, William W.; Thistlethwaite, Morwen, ур. (2005), Handbook of Knot Theory, Elsevier, ISBN 978-0-444-51452-3 
• Menasco and Thistlethwaite's handbook surveys a mix of topics relevant to current research trends in a manner accessible to advanced undergraduates but of interest to professional researchers.
• Livio, Mario (2009), „Ch. 8: Unreasonable Effectiveness?”, Is God a Mathematician?, Simon & Schuster, стр. 203—218, ISBN 978-0-7432-9405-8 

Spoljašnje veze

• "Mathematics and Knots"

Istorija

• Thomson, Sir William (1867), „On Vortex Atoms”, Proceedings of the Royal Society of Edinburgh, VI: 94—105 
• Silliman, Robert H. (децембар 1963), „William Thomson: Smoke Rings and Nineteenth-Century Atomism”, Isis, 54 (4): 461—474, JSTOR 228151, doi:10.1086/349764 CS1 одржавање: Формат датума (веза)
• Movie of a modern recreation of Tait's smoke ring experiment
• History of knot theory (on the home page of Andrew Ranicki)

Tabele čvorova i softver

• KnotInfo: Table of Knot Invariants and Knot Theory Resources
• "Main Page", The Knot Atlas. — detailed info on individual knots in knot tables
• KnotPlot — software to investigate geometric properties of knots