## Western Kentucky University

# The Development of a Computer Program to Simplify Complex Knot Diagrams using Global Moves

## Institution

Western Kentucky University

## Faculty Advisor/ Mentor

Uta Ziegler

## Abstract

A mathematical knot is similar in concept to the everyday headphone cable, with the ends closed together to form a continuous loop. These knots are the subject of discussion in molecular biology, mathematics, physics, and chemistry. For example, some enzymes are known to interact with DNA by decreasing the self-entanglement of the genomic material (e.g. after replication). An important tool when it comes to the application of knot theory itself is the unraveling, or simplification, of knots. Every knot has many different two-dimensional representations, called diagrams – the entangled headphone cable can be arranged many different ways on a flat surface. Intuitively, the simpler it looks, the easier it is to identify the characteristics of the knot and untangle the cable. For this project, the number of ‘crossings’ in the cable (diagram) was used as the measure of complexity. Prior research used ‘transformations’ to change the appearance of a diagram without changing the knot itself (cutting the cable to get rid of crossings). However, each transformation involved only one to three crossings at a time. Even with a fast computer, using these transformations to reduce the number of crossings in a complicated diagram proved time-consuming. The purpose of this research project was to investigate whether an additional ‘global slide’ transformation, which usually involves a larger number of crossings, would simplify a diagram faster. The research resulted in the implementation of a computer program that can perform the various transformations on a given diagram. Data collected with the program indicated that, in some instances, the ‘global slide’ transformation lead to a further reduction in the number of crossings in diagrams where other programs would become “stuck.'' The results demonstrated the potential of the ‘global’ transformations to simplify knot diagrams and suggested further pursuit of such transformations is justified.

The Development of a Computer Program to Simplify Complex Knot Diagrams using Global Moves

A mathematical knot is similar in concept to the everyday headphone cable, with the ends closed together to form a continuous loop. These knots are the subject of discussion in molecular biology, mathematics, physics, and chemistry. For example, some enzymes are known to interact with DNA by decreasing the self-entanglement of the genomic material (e.g. after replication). An important tool when it comes to the application of knot theory itself is the unraveling, or simplification, of knots. Every knot has many different two-dimensional representations, called diagrams – the entangled headphone cable can be arranged many different ways on a flat surface. Intuitively, the simpler it looks, the easier it is to identify the characteristics of the knot and untangle the cable. For this project, the number of ‘crossings’ in the cable (diagram) was used as the measure of complexity. Prior research used ‘transformations’ to change the appearance of a diagram without changing the knot itself (cutting the cable to get rid of crossings). However, each transformation involved only one to three crossings at a time. Even with a fast computer, using these transformations to reduce the number of crossings in a complicated diagram proved time-consuming. The purpose of this research project was to investigate whether an additional ‘global slide’ transformation, which usually involves a larger number of crossings, would simplify a diagram faster. The research resulted in the implementation of a computer program that can perform the various transformations on a given diagram. Data collected with the program indicated that, in some instances, the ‘global slide’ transformation lead to a further reduction in the number of crossings in diagrams where other programs would become “stuck.'' The results demonstrated the potential of the ‘global’ transformations to simplify knot diagrams and suggested further pursuit of such transformations is justified.