The Writhe of Permutations and Random Framed Knots

TL;DR

This paper introduces the writhe of permutations as a new circular permutation statistic, analyzes its asymptotic behavior, and connects it to a refined model of random framed knots, revealing a non-Gaussian limit distribution.

Contribution

It defines the permutation writhe, explores its properties, and links it to a novel random framed knot model, extending understanding of permutation statistics and knot theory.

Findings

Writhe of a permutation has interesting properties and interpretations.

Asymptotic distribution of the writhe is non-Gaussian.

Distribution of framing in the knot model matches the permutation writhe.

Abstract

We introduce and study the writhe of a permutation, a circular variant of the well-known inversion number. This simple permutation statistics has several interpretations, which lead to some interesting properties. For a permutation sampled uniformly at random, we study the asymptotics of the writhe, and obtain a non-Gaussian limit distribution. This work is motivated by the study of random knots. A model for random framed knots is described, which refines the Petaluma model. The distribution of the framing in this model is equivalent to the writhe of random permutations.