Topological Ideas and Fluid Mechanics

TLDR

Topological concepts have long informed physics and fluid mechanics, exemplified by Gauss’s 1833 note linking topology to measurable electric currents and the introduction of the linking number invariant. The authors analyze two closed copper circuits carrying current, examining the magnetic interaction between them. They derive a formula expressing the magnetic action induced by the currents solely in terms of the linking number, independent of geometric details.

Abstract

The use of topological ideas in physics and fluid mechanics dates back to the very origin of topology as an independent science. In a brief note in 1833 Karl Gauss, while lamenting the lack of progress in the “geometry of position” (or Geometria Situs, as topology was then known I, gives a remarkable example of the relationship between topology and measurable physical quantities such as electric currents. He considers two inseparably linked circuits, each of them a copper wire with ends joined, and flowing electric current. Without comment he puts forward a formula that gives the relationship between the magnetic action induced by the currents and a pure number that depends only on the type of link, and not on the geometry. This number is a topological invariant now known as the linking number. The formula, as well as the very first studies in topology done by Johann Benedict Listing in 1847, became known to Kelvin (then William Thomson), James Clerk Maxwell and Peter Guthrie Tait in Britain.