Nullification of knots and links

TL;DR

This paper investigates various mathematical nullification numbers for knots and links, analyzing their properties, differences, and biological relevance, especially in DNA topology, and shows that certain nullification classes are abundant.

Contribution

It introduces and compares multiple nullification numbers, analyzes their properties for specific knots, and demonstrates the abundance of links with nullification number one.

Findings

Nullification numbers differ depending on their definitions.

Detailed analysis of nullification numbers for 2-bridge knots and links.

Links with nullification number one are exponentially numerous with respect to crossing number.

Abstract

In this paper, we study a geometric/topological measure of knots and links called the nullification number. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, one can intuitively regard it as a way to measure how easily a knotted circular DNA can unknot itself through recombination of its DNA strands. It turns out that there are several different ways to define such a number. These definitions lead to nullification numbers that are related, but different. Our aim is to explore the mathematical properties of these nullification numbers. First, we give specific examples to show that the nullification numbers we defined are different. We provide detailed analysis of the nullification numbers for the well known 2-bridge knots and links. We also explore the relationships among the three nullification numbers, as well as their relationships with other knot invariants. Finally, we study a special class of links, namely those links whose general nullification number equals one. We show that such links exist in abundance. In fact, the number of such links with crossing number less than or equal to n grows exponentially with respect to n.