# What was the topological constant which Gauss derived in the context of his investigations in electromagnetism and knot theory?

My question refers to the following problem in electromagnetic theory which Gauss was concerned with (see Zur theorie der electrodynamik, volume 5 of Gauss's werke): Determine the work done on a magnetic monopole of strengh m when it is moved along a closed loop L_1 in the magnetic field produced by a closed current loop L_2 with current I. This problem led Gauss to the introduction of the Gauss's linking integral for computing the linking number of two interwinged loops. According to several books, this integral is constant under derformations of the loops as long as their number of interwingings is constant (so it's a measure of the degree of entaglement of the two loops). I'm trying to understand what was his result (what was his topological constant) and also what is his meaning, because according to several books is has an advanced physical meaning (the Gauss's linking integral is an advanced analytic tool).

• So what is your question? You tell us that you are trying to understand the Gauss linking integral. But what is your specific question? Are you asking about references or what? From your message I infer that you know some references. Commented Jun 5, 2016 at 20:28
• My first question is: what is the topological constant that he found? I guess it's in the form: 4n\pi*C, where n is the number of interwingings, and C is a constant that is a function of physical constants (like \epsilon_0) and m and I (strength of the magnetic monopole and the current in the loop). My second question is: what is the meaning of this complicated analytic formulation? and how it's connected to later developements in physics?. Commented Jun 5, 2016 at 21:49
• Perhaps i should clarify why i'm asking that question. With my current level of mathematical knowledge, i'm unable to read and understand all the articles i found (i have references), but i'm very curius about this subject and wants to hear an explanation from an expert or at least from someone with better knowledge than me. I wasn't even able to infer what was the result for the topological constant. Commented Jun 5, 2016 at 22:04

I would suggest the following paper: http://www.maths.ed.ac.uk/~aar/papers/ricca.pdf

it contains a quite detailed analysis of Gauss original works in modern terms. The linking number appears in computation of so-called helicity of vector fields, and in this form it is often applied to magnetic vector field.