# 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". Among the findings of this article is a new historical thesis of the role which the linking integral served in Gauss's ideas on topology, and an interpretation of it as a computational tool.

This interpretation of Gauss linking integral, as a computation rather then a definition, seems to be the missing block in the reconstruction of Gauss's thoughts on topological issues, as the many indirect reports (by Mobius, Listing, etc) document. I always had a strong intuitive feeling that Gauss's introduction of the linking number cannot be just a definition, since one doesn't need an integral in order to define the linking number (one can do it simply by projecting the link into a plane and then counting negative and positive crossings).

Since viewing braids as mathematical objects seem to inspire Gauss's "meditations" on topology, i'd like to understand more about the issues discussed in Epple's article.

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 $$\frac {{1}}{{2}}$$. If Gauss did apply a kind of linking integral for braids (how one can define linking number for a braid with more then two strands?), a question arises - are there any known definitions of a "linking number" for general braids?

• It's tempting to connect the 1847 letter from Mobius to Gauss, which mentions a treatise on all possible configurations of a "thread" (which seems to mean a "braid") as an introduction for a paper on electricity and magnetism, with this fragment, and imagine it as a detailed presentation of his thoughts on braids. However, it seems to me completely unclear how such classical physics issues are related to braids - and therefore i'd like to know if anyone is aware of connections between electromagnetism and braids and can tell something about it.

• 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). Oct 30, 2018 at 13:10
• This really belongs on Math OF or Math.SE as you are asking about the mathematical details rather than focusing on the history, pe se. Feb 5, 2021 at 16:48

I found a useful reference that might help answering this question (this doesn't constitute an answer to the questions in post). The reference is from p. 46-47 of the book "From Riemann to Differential Geometry and Relativity" - in those pages the author cites a 1863 letter from Enrico Betti to another mathematician:

What gave Riemann the idea of cuts was that Gauss defined them to him, talking about other matters, in a private conversation. In his writings one finds that analysis situs, that is, the consideration of quantities independently from their measure, is important; in the last years of his life he has been much concerned with a problem in analysis situs, namely: given a winding thread and knowing, at every one of its self-intersections, which part is above and which below, to find whether it can be unwound without making knots; this problem he didn't succeed in solving except in special cases...

I guess the "thread problem" which is mentioned in Betti's letter is a topological problem (on braids) in the modern sense. So understanding the meaning of the problem which Gauss tried to solve might shed some light about the note mentioned in Epple's article - in which Gauss made the first mathematical discussion of a certain braid.

In addition, there is a letter from Mobius to Gauss on February 2, 1847:

As i heard from Wilhelm Weber, you found it nessecary, a few years ago, as an introduction or preperation for the theory of electrical or magnetic currents, to write a treatise on all possible wraps (configurations) of a thread. Is it not to be hoped that this essay will appear soon? The fulfillment of this hope will be very much appreciated by me and certainly also by many others.