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Gauss notation

Gauss notation (also known as a Gauss code or Gauss words[1]) is a notation for mathematical knots.[2][3] It is created by enumerating and classifying the crossings of an embedding of the knot in a plane.[2][4][5] It is named after the German mathematician Carl Friedrich Gauss (1777–1855).

Gauss code represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labelled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number.[6]

For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3.[7]

Gauss code is limited in its ability to identify knots by a few problems. The starting point on the knot at which to begin tracing the crossings is arbitrary, and there is no way to determine which direction to trace in. Also, the Gauss code is unable to indicate the handedness of each crossing, which is necessary to identify a knot versus its mirror. For example, the Gauss code for the trefoil knot does not specify if it is the right-handed or left-handed trefoil.[8]

This last issue is often solved by using the extended Gauss code. In this modification, the positive/negative sign on the second instance of every number is chosen to represent the handedness of that crossing, rather than the over/under sign of the crossing, which is made clear in the first instance of the number. A right-handed crossing is given a positive number, and a left handed crossing is given a negative number.[6]

References

  1. ^ Gibson, Andrew (1 April 2011). "Homotopy invariants of Gauss words". Mathematische Annalen. 349 (4): 871–887. arXiv:0902.0062. doi:10.1007/s00208-010-0536-0. ISSN 1432-1807. S2CID 14328996.
  2. ^ a b Nash, John F.; Rassias, Michael Th., eds. (5 July 2016). Open Problems in Mathematics. Switzerland: Springer. p. 340. ISBN 978-3-319-32162-2. OCLC 953456173.
  3. ^ "Knot Table: Gauss Notation". knotinfo.math.indiana.edu. Retrieved 30 June 2020.
  4. ^ "Gauss Code". www.math.toronto.edu. Retrieved 30 June 2020.
  5. ^ Lisitsa, Alexei; Potapov, Igor; Saleh, Rafiq (2009). "Automata on Gauss Words" (PDF). In Dediu, Adrian Horia; Ionescu, Armand Mihai; Martín-Vide, Carlos (eds.). Language and Automata Theory and Applications. Lecture Notes in Computer Science. Vol. 5457. Berlin, Heidelberg: Springer. pp. 505–517. doi:10.1007/978-3-642-00982-2_43. ISBN 978-3-642-00982-2.
  6. ^ a b "How to count the crossing number of a knot with $5$ crossing?". Mathematics Stack Exchange. Retrieved 10 September 2023.
  7. ^ "Gauss Codes". Knot Atlas. Retrieved 10 September 2023.
  8. ^ Gouesbet, G.; Meunier-Guttin-Cluzel, S.; Letellier, C. (1999). "Computer evaluation of Homfly polynomials by using Gauss codes, with a skein-template algorithm". Applied Mathematics and Computation. 105 (2–3): 271–289. doi:10.1016/S0096-3003(98)10106-6. MR 1710214. See p. 274

See also

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