Unknotted cycles

TL;DR

This paper explores the topological properties of permutation cycle diagrams, showing that permutations representing unknots are counted by Schr"{o}der numbers, and employs Bennequin's inequality for proof.

Contribution

It establishes a novel connection between permutation cycle diagrams and knot theory, specifically characterizing unknots and unlinks through enumeration.

Findings

Permutations corresponding to unknots are enumerated by Schr"{o}der numbers.

Permutations corresponding to unlinks are also enumerated.

The proof utilizes Bennequin's inequality.

Abstract

Noting that cycle diagrams of permutations visually resemble grid diagrams used to depict knots and links in topology, we consider the knot (or link) obtained from the cycle diagram of a permutation. We show that the permutations which correspond in this way to an unknot are enumerated by the Schr\"{o}der numbers, and also enumerate the permutations corresponding to an unlink. The proof uses Bennequin's inequality.