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4.2 Gauss Codes

The Gauss Code of an $n$-crossing knot or link $L$ is obtained as follows:

• Number the crossings of $L$ from 1 to $n$ in an arbitrary manner.
• Order the components of $L$ is some arbitrary manner.
• Start ``walking'' along the first component of $L$, taking note of the numbers of the crossings you've gone through. If in a given crossing crossing you cross on the ``over'' strand, write down the number of that crossing. If you cross on the ``under'' strand, write down the negative of the number of that crossing.
• Do the same for all other components of $L$ (if any).

The resulting list of signed integers (in the case of a knot) or list of lists of signed integers (in the case of a link) is called the Gauss Code of $L$. KnotTheory` has some rudimentary support for Gauss codes:

In[1]

In[2]:= ?GaussCode
GaussCode[i1, i2, ...] represents a knot via its Gauss Code following the conventions used by the knotilus website, http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html. Likewise GaussCode[l1, l2, ...] represents a link, where each of l1, l2,... is a list describing the code read along one component of the link. GaussCode also acts as a "type caster", so for example, GaussCode[K] where K is is a named knot (or link) returns the Gauss code of that knot.

Thus for example, the Gauss codes for the trefoil knot and the Borromean link are:

In[3]:=	GaussCode /@ {Knot[3, 1], Link[6, Alternating, 4]}
Out[3]=	{GaussCode[-1, 3, -2, 1, -3, 2], > GaussCode[{1, -6, 5, -3}, {4, -1, 2, -5}, {6, -4, 3, -2}]}

Ralph Furmaniak, working under the guidance of Stuart Rankin and Ortho Flint at the University of Western Ontario, wrote a web-based server called ``Knotilus'' that takes Gauss codes and outputs pictures of the desired knots and links in several standard image formats.

In[4]:= ?KnotilusURL
KnotilusURL[K_] returns the URL of the knot/link K on the knotilus website,
http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html.

Thus,

In[5]:=	KnotilusURL /@ {Knot[3, 1], Link[6, Alternating, 4]}
Out[5]=	{http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,3,-2,1,-3,2/goTop.html, > http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,5,-3:4,-1,2,-5:6,-4,\ > 3,-2/goTop.html}

Click to get there! http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,3,-2,1,-3,2/goTop.html and http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-6,5,-3:4,-1,2,-5:6,-4,3,-2/goTop.html.