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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Gauss notation</span></span>

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<p>

<b>Gauss notation</b> (also known as a <b>Gauss code or</b> <b>Gauss words</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) is a <a href="Notation" class="mw-redirect" title="Notation">notation</a> for <a href="Mathematical_knot" class="mw-redirect" title="Mathematical knot">mathematical knots</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It is created by enumerating and classifying the crossings of an embedding of the knot in a plane.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It is named after the German mathematician <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (1777–1855).

</p><p>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.<sup id="cite_ref-math.stackexchange.com_6-0" class="reference"><a href="#cite_note-math.stackexchange.com-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>For example, the <a href="Trefoil_knot" title="Trefoil knot">trefoil knot</a> in Gauss code can be given as: 1,−2,3,−1,2,−3.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>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.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p><p>This last issue is often solved by using the <b>extended Gauss code</b>. 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.<sup id="cite_ref-math.stackexchange.com_6-1" class="reference"><a href="#cite_note-math.stackexchange.com-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGibson2011" class="citation journal cs1">Gibson, Andrew (1 April 2011). "Homotopy invariants of Gauss words". <i>Mathematische Annalen</i>. <b>349</b> (4): <span class="nowrap">871–</span>887. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0902.0062">0902.0062</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00208-010-0536-0">10.1007/s00208-010-0536-0</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1432-1807">1432-1807</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14328996">14328996</a>.</cite></span>

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<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFNashRassias2016" class="citation book cs1"><a href="John_Forbes_Nash_Jr." title="John Forbes Nash Jr.">Nash, John F.</a>; Rassias, Michael Th., eds. (5 July 2016). <a href="Open_Problems_in_Mathematics" title="Open Problems in Mathematics"><i>Open Problems in Mathematics</i></a>. Switzerland: Springer. p. 340. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-32162-2</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/953456173">953456173</a>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://knotinfo.math.indiana.edu/descriptions/gauss_notation.html">"Knot Table: Gauss Notation"</a>. <i>knotinfo.math.indiana.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">30 June</span> 2020</span>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.math.toronto.edu/drorbn/Students/GreenJ/gausscode.html">"Gauss Code"</a>. <i>www.math.toronto.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">30 June</span> 2020</span>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFLisitsaPotapovSaleh2009" class="citation book cs1">Lisitsa, Alexei; Potapov, Igor; Saleh, Rafiq (2009). <a rel="nofollow" class="external text" href="https://cgi.csc.liv.ac.uk/~igor/papers/lata2009.pdf">"Automata on Gauss Words"</a> <span class="cs1-format">(PDF)</span>. In Dediu, Adrian Horia; Ionescu, Armand Mihai; Martín-Vide, Carlos (eds.). <i>Language and Automata Theory and Applications</i>. Lecture Notes in Computer Science. Vol. 5457. Berlin, Heidelberg: Springer. pp. <span class="nowrap">505–</span>517. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-00982-2_43">10.1007/978-3-642-00982-2_43</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-00982-2</bdi>.</cite></span>

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<li id="cite_note-math.stackexchange.com-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-math.stackexchange.com_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-math.stackexchange.com_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/2704332/how-to-count-the-crossing-number-of-a-knot-with-5-crossing">"How to count the crossing number of a knot with $5$ crossing?"</a>. <i>Mathematics Stack Exchange</i><span class="reference-accessdate">. Retrieved <span class="nowrap">10 September</span> 2023</span>.</cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://katlas.org/wiki/Gauss_Codes">"Gauss Codes"</a>. <i><a href="Knot_Atlas" class="mw-redirect" title="Knot Atlas">Knot Atlas</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">10 September</span> 2023</span>.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFGouesbetMeunier-Guttin-CluzelLetellier1999" class="citation journal cs1">Gouesbet, G.; Meunier-Guttin-Cluzel, S.; Letellier, C. (1999). "Computer evaluation of Homfly polynomials by using Gauss codes, with a skein-template algorithm". <i>Applied Mathematics and Computation</i>. <b>105</b> (<span class="nowrap">2–</span>3): <span class="nowrap">271–</span>289. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0096-3003%2898%2910106-6">10.1016/S0096-3003(98)10106-6</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1710214">1710214</a>.</cite> See p. 274</span>

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</ol></div></div>

<ul><li><a href="Conway_notation_(knot_theory)" title="Conway notation (knot theory)">Conway notation (knot theory)</a></li>

<li><a href="Dowker%E2%80%93Thistlethwaite_notation" title="Dowker–Thistlethwaite notation">Dowker–Thistlethwaite notation</a></li></ul>

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</style><div id="Carl_Friedrich_Gauss153" style="font-size:114%;margin:0 4em"><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Gauss_composition_law" title="Gauss composition law">Gauss composition law</a></li>

<li><a href="Gauss_map" title="Gauss map">Gauss map</a></li>

<li><a href="Gauss's_method" title="Gauss's method">Gauss's method</a></li>

<li><a href="Gaussian_brackets" title="Gaussian brackets">Gaussian brackets</a></li>

<li><a href="Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a></li>

<li><a href="Gaussian_period" title="Gaussian period">Gaussian period</a></li>

<li><a href="Gaussian_surface" title="Gaussian surface">Gaussian surface</a></li>

<li><a href="Gaussian_units" title="Gaussian units">Gaussian units</a></li>

<li><a href="Gauss's_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a></li>

<li><a href="Gauss's_law" title="Gauss's law">Gauss's law</a></li>

<li><a href="Gauss's_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a></li></ul>

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