# The Distribution of Knots in the Petaluma Model

## TL;DR

This paper studies the distribution of knots in the Petaluma model, showing that as the number of petals increases, the probability of any specific knot type diminishes, and it establishes bounds relating crossing and petal numbers.

## Contribution

It proves that in the n-petal model, the probability of specific knots tends to zero and improves bounds linking crossing and petal numbers, demonstrating the model's capacity to generate many distinct knots.

## Key findings

- Probability of specific knots decays to zero as petals increase
- At least exponentially many distinct knots are represented
- Improved bounds between crossing number and petal number

## Abstract

The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability, rely on the prevalence of localized connect summands as the complexity of the knot increases. However this phenomenon is not clear in other models, including petal diagrams, random grid diagrams, and uniform random polygons. Thus we provide a new approach to investigate this question.