# Knot Fertility and Lineage

## TL;DR

This paper introduces the descendant relation among knots, explores properties of fertile knots with many descendants, and provides computational data to understand knot fertility better.

## Contribution

It defines the descendant relation for knots, studies fertile knots with numerous descendants, and offers computational insights and open questions in knot theory.

## Key findings

- Identification of fertile knots with many descendants
- Properties of the descendant relation among knots
- Computational data on knot fertility

## Abstract

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot $H$ is a descendant of another knot $K$ if $H$ can be obtained from a minimal crossing diagram of $K$ by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration.