# Several questions about Gauss's mathematical conception of braids

I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits of asteroids, a braid, and the first link invariant".

Some basic questions

• Epple begins his exposition of Gauss's fragment with the observation that "Gauss thought of the braid as being devided into six segments, extending from one crossing to the next". Even this basic fact is not clear to me, since the crossings devide the four strands to more than six segments.

• What are the "half-twists" mentioned in the following paragraphs?

More difficult questions

• Why did Gauss used coordinates that are based on complex integers in order to encode a braid? Epple says "That Gauss used complex integers to encode the composition of the braid might be motivated by his well known fascination with these numbers; it might, however, also have a more serious reason (see below)". Unfortunately i coudn't infer from the rest of the text the explanation for why complex numbers are important here.
• Epple suggests to connect the drawing of the braid found in Gauss's notebook with his later definition of the linking integral; according to his interpretation, the linking integral represents a computation rather then a definition - in the case of two strand braid, this analytic approach (the integral) gives rise to a linking number which equals to an integer multiple of $$\frac {{1}}{{2}}$$. Until i read this article, i thought the linking integral always equals an integer, so can someone tell more about such "fractional" linking numbers?
• As is being said in my posted "answer" (which is just intended to give additional information), the indirect reference in Betti's letter states that Gauss was occupied with a certain braid problem that he didn't succeed in solving except in special cases. Is there any term from modern braid theory for the topological problem described in Betti's letter?

General question

One thing that i noticed, which seems very interesting to me, is that several results or drawings which some authors refer to, don't appear in any of the 12 volumes of the published version of Gauss's werke. Just as one example, in several articles on the early history of knot theory, the authors mention a collection of 13 knots made by Gauss. So my question is: is there any website which gives access to those obscure writings of Gauss?

• I looked into Q1 some years ago and I believe the answer is: nobody knows. It's a one-off note by Gauss without any supporting context and nobody has explained it successfully (Epple is perhaps the only one who has made an attempt, but his interpretation requires the highly dubious assumption that Gauss made mistakes in the table, if I recall correctly). – Viktor Blasjo Oct 30 '18 at 13:10