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:

(For In[1] see Setup)

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.