# Explanation of a passage from Gauss's writings on knot theory.

Gauss's writings on knot theory include five lists of knots (see Geometria Situs - Gauss's werke: volume 8, p.282-285: http://gdz.sub.uni-goettingen.de/pdfcache/PPN236010751/PPN236010751___LOG_0082.pdf): the first for knots with one cross (with only one knot on the list), the second for knots with two crossings (with three knots on the list),the third for knots with three crossings (with 15 knots on the list) the fourth for knots with four crossings (with 105 knots on the list), but the fifth list is with unclear meaning (it contains 120 knots and it has no title). I guess Gauss made these tables as a first step toward knot tabulation, but i'm not sure about it; in particular i want to know whether the lists he made for each number of crossings are complete, what's the meaning of the six drawings on page 284, and what's the idea behind the fifth list.

These are lists of what are now known as Gauss words. A generic immersion of a circle in a plane can be (mostly) recorded by listing off the double points (what he calls Knotenpunkten) in a traversal of the circle, starting at a particular point and going in a particular direction. For example, $aabb$ represents visiting vertex $a$ twice and then vertex $b$ twice, for an immersion of a circle with two double points.

He performs an exhaustive study of all possible Gauss words with up to five double points. Anything with stars is a Gauss word that does not come from an immersed circle, for example $abab$. The number of stars indicates some information about the way in which it is impossible, since Gauss was interested in a necessary and sufficient condition to determine which words could be realized as immersed circles.

A systematic enumeration of potential words is a straightforward combinatorics problem. For example, with four double points, we are looking for all ways of choosing a set of four pairs of positions in a string of length eight. That is $\frac{1}{4!}\binom{8}{2,2,2,2}=105$, just as in Gauss's table.

However, cyclic shifts of a realizable Gauss word correspond to the same immersed circle (for example, $aabb$ and $baab$). For the words with four double points that are realizable as immersed curves, he finds there are five unique ones. The six pictures represent the five possible immersed curves, with the middle two being equivalent (in that they have equivalent Gauss words).

For the list for five double points, since there would be $945$ words, he cut some corners to reduce that number to $120$. The table is organized by five-letter prefixes, which for each sequence with the same prefix, only the first has the prefix printed. It looks like he skipped prefixes that would result in unrealizable words.

This sort of tabulation was carried out on a large scale by Dowker and Thistlethwaite using a computer, as a first step to tabulate knots (where a knot also has over- and under-crossing information). They describe a necessary and sufficient condition for realizability that they found they could implement on a computer.

Dowker, C. H.; Thistlethwaite, Morwen B., Classification of knot projections, Topology Appl. 16, 19-31 (1983). ZBL0516.57002.

This is one of very many papers on necessary and sufficient conditions. For example, there is even an $O(n)$ algorithm that can give a realization if and only if one exists.

Rosenstiehl, Pierre; Tarjan, Robert E., Gauss codes, planar Hamiltonian graphs, and stack-sortable permutations, J. Algorithms 5, 375-390 (1984). ZBL0588.68034.