Knot theory

In mathematics, knot theory is the study of the geometric properties of knots, or simple closed curves. It is a branch of topology, which studies how one topological space is placed and preserved into another.

Knot theory looks at how a one-dimensional string lies in three-dimensional space. Types of knots include the trivial knot (or unknot), the trefoil knot, and the figure-eight knot.

The purpose of knot theory is to determine when two given knots are the same. Two knots may be considered equivalent if the first knot can be slowly altered and deformed into the second knot through a series of moves.

Knot theory has important applications in various scientific fields, including physics, molecular biology, chemistry, and cryptography. Examples of its real-life uses include understanding the process of DNA replication and improving data encryption in security systems.

Background

Knots have appeared in architecture and jewelry throughout history. The decorative Celtic knot is one of the most widely recognized knots. The aesthetic of the knot inspired scientists. During the late nineteenth century, knot theory grew out of scientists' quest to understand matter. British physicist Lord Kelvin theorized that atoms were knots in the ether, the mysterious substance that made up space. He believed that different elements were different knots. Scottish physicist Peter Guthrie Tait then spent years developing a meticulous table of knots, aiming to construct a periodic table of elements. In doing so, Tait sought to distinguish two knots from one another.

Later experiments showed Kelvin's theory was wrong. However, the misguided model for the atom—and the beauty of Tait's tables—captured the attention of mathematicians.

The study of knot theory in mathematics focuses on the shape of knots. A knot is typically thought of as a single string with loose ends. Mathematical knots are closed loops, with no starting or ending point. A knot can cross over itself multiple times, but cannot intersect itself. It does not matter how long or thick the knot is.

Knots exist in Euclidean space, or ordinary three-dimensional space. Knots can be untied or tied. The points where a knot crosses itself are called crossings. There are an infinite number of knots as the number of crossings increase. As knots become more complex, they are organized through knot tables.

The most basic knot is an untied circle, which looks like a rubber band. The unknotted circle is known as a trivial knot, or an unknot. The simplest tied knot is the trefoil knot, which looks like a pretzel. A trefoil knot crosses itself in three places. The popular figure-eight knot crosses itself in four places. Knots that cannot be broken down into two or more simpler knots are called prime knots. Trefoil knots and figure-eight knots are prime knots.

Overview

Knot theory seeks to prove whether two knots are the same. It operates under the conditions that a knot cannot be cut or intersect itself. Knots can be deformed to determine if two knots are equivalent to each other without changing the knots themselves.

Although knots exist in three-dimensional space, they are drawn in two-dimensional space. These projections of knots are called knot diagrams. Knot diagrams enable mathematicians to study knots and their crossings. While knots are embedded circles, they can also be considered entwined polygons, a plane with sides made up of a finite number of segmented straight lines. This allows the shape of the knot to be altered.

The deleting or adding of points results in replacements called elementary knot moves. Elementary knot moves can be used to determine if two knots are equivalent.

There are four kinds of elementary knot moves:

• An edge in the space of the knot may be divided by a point, replaced by two new edges.
• A point may be removed from two edges, replacing it with a single edge.
• If a point exists in the space outside a given knot, the point may form a triangle with the edge. If the triangle does not intersect the knot, then the original edge may be removed and the two new adjacent edges added.
• If the triangle exists in space and does not intersect the knot, then the two adjacent edges may be removed and replaced by one edge.

By applying a finite amount of elementary knot moves, a knot can be deformed into another knot. Two knots are considered equivalent if the first can be changed continuously into the second without intersecting itself.

Another way to determine if two knots are equivalent is by mapping the first knot to the second by using its inverse form, or mirror image. The mirror image of a knot is called its homeomorphism.

For this process, a knot must have orientation, which is the direction a knot flows in. A knot can be assigned a left or right orientation. Additionally, the homeomorphism must preserve the knot's orientation. If a mirror image of the first knot—the homeomorphism—preserves its orientation and can be mapped to the second knot without altering its orientation, then the two knots are equivalent.

Performing an elementary knot move does not change the knot, but it may change the knot diagram, offering two distinct projections of the same knot. Reidemeister moves, developed by German mathematician Kurt Reidemeister, provide a way to get from one projection to the second without changing the knot itself. They are made in relation to elementary knot moves.

There are three types of Reidemeister moves:

• The first removes or adds a crossing in the knot.
• The second adds or takes out two crossings.
• The third relocates a strand of the knot from one side of the crossing to the other.

Knot theory plays a vital role in real-life applications of science. In physics, it is used in the equilibrium statistical mechanics of mechanical systems and quantum field theory for elementary particles.

In molecular biology, knot theory is used to understand the replication process of DNA. DNA is a double helix structure. During cell reproduction, DNA must unfold itself, make a second copy that is embedded with the original, and then separate itself. Enzymes aid the complicated process of knotting and unknotting strands of DNA.

In cryptography, stored information is transformed into a protected, or encrypted, form. Knot theory enhances security protocols by turning mathematical data from knots into encryption keys. This allows security systems to read coded information in employee badges. On a consumer level, knot theory enables cell phone devices to be used for payment through apps.

Bibliography

Adams, Colin C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society, 2000.

Kawauchi, Akio. A Survey of Knot Theory. Birkhauser, 1996.

"Knot Theory." Cornell University, www.math.cornell.edu/~mec/2008-2009/HoHonLeung/page3‗knots.htm. Accessed 27 June 2017.

"Knot Theory." Knot Plot, knotplot.com/knot-theory. Accessed 26 June 2017.

Lickorish, W.B. Raymond. An Introduction to Knot Theory. Springer, 1997.

Murasugi, Kunio. Knot Theory and Its Applications. Springer, 1996.

Nelson, Sam. "A Revolution in Knot Theory." Phys.org, 10 Nov. 2011, phys.org/news/2011-11-revolution-theory.html. Accessed 26 June 2017.

"What Is Knot Theory?" Osaka City University, www.sci.osaka-cu.ac.jp/~kawauchi/WhatIsKnotTheory.pdf. Accessed 26 June 2017.

"You've Heard of String Theory. What about Knot Theory?" Science Daily, 10 Feb. 2016, www.sciencedaily.com/releases/2016/02/160210170411.htm. Accessed 26 June 2017.