Gauss notation

Gauss notation (also known as a Gauss code or Gauss words{{Cite journal |last=Gibson |first=Andrew |date=1 April 2011 |title=Homotopy invariants of Gauss words |journal=Mathematische Annalen |language=en |volume=349 |issue=4 |pages=871–887 |doi=10.1007/s00208-010-0536-0 |arxiv=0902.0062 |s2cid=14328996 |issn=1432-1807}}) is a notation for mathematical knots.{{Cite book |title=Open Problems in Mathematics |title-link=Open Problems in Mathematics |editor-last1=Nash |editor-first1=John F. |editor-link1=John Forbes Nash Jr. |editor-last2=Rassias |editor-first2=Michael Th. |date=5 July 2016 |isbn=978-3-319-32162-2 |publisher=Springer |location=Switzerland |page=340 |oclc=953456173}}{{Cite web |title=Knot Table: Gauss Notation |url=https://knotinfo.math.indiana.edu/descriptions/gauss_notation.html |access-date=30 June 2020 |website=knotinfo.math.indiana.edu}} It is created by enumerating and classifying the crossings of an embedding of the knot in a plane.{{Cite web |title=Gauss Code |url=https://www.math.toronto.edu/drorbn/Students/GreenJ/gausscode.html |access-date=30 June 2020 |website=www.math.toronto.edu}}{{Cite book |last1=Lisitsa |first1=Alexei |last2=Potapov |first2=Igor |last3=Saleh |first3=Rafiq |title=Language and Automata Theory and Applications |chapter=Automata on Gauss Words |date=2009 |editor-last=Dediu |editor-first=Adrian Horia |editor2-last=Ionescu |editor2-first=Armand Mihai |editor3-last=Martín-Vide |editor3-first=Carlos |series=Lecture Notes in Computer Science |volume=5457 |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=505–517 |doi=10.1007/978-3-642-00982-2_43 |isbn=978-3-642-00982-2 |chapter-url=https://cgi.csc.liv.ac.uk/~igor/papers/lata2009.pdf}} 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.{{Cite web |title=How to count the crossing number of a knot with $5$ crossing? |url=https://math.stackexchange.com/questions/2704332/how-to-count-the-crossing-number-of-a-knot-with-5-crossing |access-date=10 September 2023 |website=Mathematics Stack Exchange |language=en}}

For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3.{{Cite web |title=Gauss Codes |url=http://katlas.org/wiki/Gauss_Codes |access-date=10 September 2023 |website=Knot Atlas}}

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.{{cite journal |last1=Gouesbet |first1=G. |last2=Meunier-Guttin-Cluzel |first2=S. |last3=Letellier |first3=C. |doi=10.1016/S0096-3003(98)10106-6 |issue=2–3 |journal=Applied Mathematics and Computation |mr=1710214 |pages=271–289 |title=Computer evaluation of Homfly polynomials by using Gauss codes, with a skein-template algorithm |volume=105 |year=1999}} See p. 274

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.

Gauss notation

References

See also