Knots and links and the sciences of nature: An introduction. (Nœuds et links et les sciences de la nature: Une introduction.) (French) Zbl 0882.57003

This is an expository article, which aims to describe some of the connections of knot theory with theoretical physics. One of the themes developed is the contrast between local and global information, with the classical Gauss integrals for the linking number of two curves typifying the input of local information which can be integrated to give a global invariant. The alternative simple way to find the linking number from a count based on a diagram of the link gives a basic illustration of the combinatorial approach. This example is discussed initially, followed by a range of more recent ‘quantum’ invariants of knots, which again admit the definition and calculation using skein theory – essentially algebraic combinatorics based on diagrams – or by means of analogues of the Gauss integral techniques using integrals of the Chern-Simons form. The integral approach allows a fairly natural theoretical understanding of the behaviour of invariants when pieces of knot exterior are put together under controlled conditions, and the article goes on to discuss the way in which this leads to ‘quantum field theories’ for 3-manifolds containing a (possibly empty) set of embedded curves. A nice introduction to the theory underlying the ideas of magnetic monopoles and gauge fields is included, which highlights the transition from an abelian setting, in which Maxwell’s equations naturally belong, to the nonabelian contexts of the SU(2) and SU(3) theories where quantum field theory appears. Explicit work with the Jones polynomial and its relations to statistical mechanics models helps to round out the picture of the interconnections and provides a useful bridge to give mathematicians some helpful physical insights in these areas.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M10 Covering spaces and low-dimensional topology
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57R65 Surgery and handlebodies