Katana VentraIP

Knot theory

In topology, knot theory is the study 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 so 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, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. 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.[1] In practice, knots are often distinguished 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). For example, a higher-dimensional knot is an n-dimensional sphere embedded in (n+2)-dimensional Euclidean space.

(where is any diagram of the )

unknot

List of knot theory topics

Molecular knot

Circuit topology

Quantum topology

Ribbon theory

Contact geometry#Legendrian submanifolds and knots

Knots and graphs

Necktie § Knots

Lamp cord trick

(2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1

Adams, Colin

Adams, Colin; Crawford, Thomas; DeMeo, Benjamin; Landry, Michael; Lin, Alex Tong; Montee, MurphyKate; Park, Seojung; Venkatesh, Saraswathi; Yhee, Farrah (2015), "Knot projections with a single multi-crossing", Journal of Knot Theory and Its Ramifications, 24 (3): 1550011, 30, :1208.5742, doi:10.1142/S021821651550011X, MR 3342136, S2CID 119320887

arXiv

Adams, Colin; Hildebrand, Martin; (1991), "Hyperbolic invariants of knots and links", Transactions of the American Mathematical Society, 326 (1): 1–56, doi:10.1090/s0002-9947-1991-0994161-2, JSTOR 2001854

Weeks, Jeffrey

; King, Henry C. (1981), "All knots are algebraic", Comment. Math. Helv., 56 (3): 339–351, doi:10.1007/BF02566217, S2CID 120218312

Akbulut, Selman

(1995), "On the Vassiliev knot invariants", Topology, 34 (2): 423–472, doi:10.1016/0040-9383(95)93237-2

Bar-Natan, Dror

Burton, Benjamin A. (2020). . 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz Int. Proc. Inform. Vol. 164. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. pp. 25:1–25:17. doi:10.4230/LIPIcs.SoCG.2020.25.

"The Next 350 Million Knots"

Collins, Graham (April 2006), "Computing with Quantum Knots", , 294 (4): 56–63, Bibcode:2006SciAm.294d..56C, doi:10.1038/scientificamerican0406-56, PMID 16596880

Scientific American

(1914), "Die beiden Kleeblattschlingen", Mathematische Annalen, 75 (3): 402–413, doi:10.1007/BF01563732, S2CID 120452571

Dehn, Max

(1970), "An enumeration of knots and links, and some of their algebraic properties", Computational Problems in Abstract Algebra, Pergamon, pp. 329–358, doi:10.1016/B978-0-08-012975-4.50034-5, ISBN 978-0-08-012975-4

Conway, John H.

Doll, Helmut; Hoste, Jim (1991), "A tabulation of oriented links. With microfiche supplement", Math. Comp., 57 (196): 747–761, :1991MaCom..57..747D, doi:10.1090/S0025-5718-1991-1094946-4

Bibcode

(1962), "Knotted (4k âˆ’ 1)-spheres in 6k-space", Annals of Mathematics, Second Series, 75 (3): 452–466, doi:10.2307/1970208, JSTOR 1970208

Haefliger, André

(1962), "Ãœber das Homöomorphieproblem der 3-Mannigfaltigkeiten. I", Mathematische Zeitschrift, 80: 89–120, doi:10.1007/BF01162369, ISSN 0025-5874, MR 0160196

Haken, Wolfgang

(1998), "Algorithms for recognizing knots and 3-manifolds", Chaos, Solitons and Fractals, 9 (4–5): 569–581, arXiv:math/9712269, Bibcode:1998CSF.....9..569H, doi:10.1016/S0960-0779(97)00109-4, S2CID 7381505

Hass, Joel

Hoste, Jim; ; Weeks, Jeffrey (1998), "The First 1,701,935 Knots", Math. Intelligencer, 20 (4): 33–48, doi:10.1007/BF03025227, S2CID 18027155

Thistlethwaite, Morwen

Hoste, Jim (2005). "The Enumeration and Classification of Knots and Links". Handbook of Knot Theory. pp. 209–232. :10.1016/B978-044451452-3/50006-X. ISBN 978-0-444-51452-3.

doi

(1965), "A classification of differentiable knots", Annals of Mathematics, Second Series, 1982 (1): 15–50, doi:10.2307/1970561, JSTOR 1970561

Levine, Jerome

Kontsevich, M. (1993). "Vassiliev's knot invariants". I. M. Gelfand Seminar. ADVSOV. Vol. 16. pp. 137–150. :10.1090/advsov/016.2/04. ISBN 978-0-8218-4117-4.

doi

(1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, doi:10.1007/978-1-4612-0691-0, ISBN 978-0-387-98254-0, S2CID 122824389

Lickorish, W. B. Raymond

Perko, Kenneth (1974), "On the classification of knots", , 45 (2): 262–6, doi:10.2307/2040074, JSTOR 2040074

Proceedings of the American Mathematical Society

Rolfsen, Dale (1976), , Mathematics Lecture Series, vol. 7, Berkeley, California: Publish or Perish, ISBN 978-0-914098-16-4, MR 0515288

Knots and Links

Schubert, Horst (1949). Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. :10.1007/978-3-642-45813-2. ISBN 978-3-540-01419-5.

doi

Silver, Daniel (2006). "Knot Theory's Odd Origins". American Scientist. 94 (2): 158. :10.1511/2006.2.158.

doi

Simon, Jonathan (1986), "Topological chirality of certain molecules", Topology, 25 (2): 229–235, :10.1016/0040-9383(86)90041-8

doi

Sossinsky, Alexei (2002), Knots, mathematics with a twist, Harvard University Press,  978-0-674-00944-8

ISBN

Turaev, Vladimir G. (2016). Quantum Invariants of Knots and 3-Manifolds. :10.1515/9783110435221. ISBN 978-3-11-043522-1. S2CID 118682559.

doi

(2013). "Reduced Knot Diagram". MathWorld. Wolfram. Retrieved 8 May 2013.

Weisstein, Eric W.

Weisstein, Eric W. (2013a). . MathWorld. Wolfram. Retrieved 8 May 2013.

"Reducible Crossing"

(1989), "Quantum field theory and the Jones polynomial", Comm. Math. Phys., 121 (3): 351–399, Bibcode:1989CMaPh.121..351W, doi:10.1007/BF01217730, S2CID 14951363

Witten, Edward

(1963), "Unknotting combinatorial balls", Annals of Mathematics, Second Series, 78 (3): 501–526, doi:10.2307/1970538, JSTOR 1970538

Zeeman, Erik C.

Further reading[edit]

Introductory textbooks[edit]

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is (Rolfsen 1976). Other good texts from the references are (Adams 2004) and (Lickorish 1997). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics. (Cromwell 2004) is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required.

This is an online version of an exhibition developed for the 1989 Royal Society "PopMath RoadShow". Its aim was to use knots to present methods of mathematics to the general public.

"Mathematics and Knots"