Typical knots: size, link component count, and writhe

TL;DR

This paper models the typical behavior of knots and links using grid diagrams, analyzing invariants like size, component count, and writhe, revealing their distributions and growth patterns as grid size increases.

Contribution

It introduces numerical analysis and generating functions for knot invariants, providing new insights into their distributions and asymptotic behaviors.

Findings

Knot size is uniformly distributed and linearly related to grid size.

Number of link components grows logarithmically with grid size.

Variance of writhe grows quadratically with grid size.

Abstract

We model the typical behavior of knots and links using grid diagrams. Links are ubiquitous in the sciences, and their "normal" or "typical" behavior is of significant importance in understanding situations such as the topological state of DNA or the statistical mechanics of ring polymers. We examine three invariants: the expected size of a random knot; the expected number of components of a random link; and the expected writhe of a random knot. We investigate the first two numerically and produce generating functions which codify the observed patterns: knot size is uniformly distributed and linearly dependent upon grid size, and the number of components follows a distribution whose mean and variance grow with log_2 of grid size; in particular, for any fixed k, the k-component links grow vanishingly rare as grid size increases. Finally, we observe that the odd moments of writhe vanish, and we perform an exploratory data analysis to discover that variance grows with the square of grid size and kurtosis is constant at approximately 3.5. We continue this project in a future work, where we investigate genus and the effects of crossing change on it.