Characterization of Knots and Links Arising From Site-specific Recombination on Twist Knots

TL;DR

This paper presents a comprehensive model for predicting all possible knots and links resulting from site-specific recombination on twist knots, revealing their classification, growth rate, and specific product types, aiding in experimental data interpretation.

Contribution

It extends previous models by fully characterizing all potential recombination products from twist knots and analyzing their growth and classification.

Findings

All product knots and links fall into three known families.

Number of product knots/links with minimal crossing number n grows as n^5.

Product types are tightly constrained when minimal crossing number increases by one.

Abstract

We develop a model characterizing all possible knots and links arising from recombination starting with a twist knot substrate, extending previous work of Buck and Flapan. We show that all knot or link products fall into three well-understood families of knots and links, and prove that given a positive integer $n$, the number of product knots and links with minimal crossing number equal to $n$ grows proportionally to $n^5$. In the (common) case of twist knot substrates whose products have minimal crossing number one more than the substrate, we prove that the types of products are tightly prescribed. Finally, we give two simple examples to illustrate how this model can help determine previously uncharacterized experimental data.