Folding, Meanders and Arches

Abstract

The concept of folding has an important place in polymer physics. Typically one considers the statistical model of a polymer chain made of say n identical constituents, and which may be folded onto itself. The entropy of such a system is obtained by counting the number of inequivalent ways of folding the chain. The combinatorial problem of enumerating all the compact foldings of a closed polymer chain happens to be equivalent to another geometrical problem, that of enumerating meanders [1], i.e. configurations of a closed road crossing a river through n bridges. To our knowledge, this is still an open problem, which indeed has been adressed by very few authors. Apart from the above folding interpretation, the meander problem arises in such various domains as the classification of 3-manifolds [3], computer science [4] and fine arts [5].