Knot theory

Math can look at knots.

A math knot is a closed loop. The ends are joined together. This means you cannot untie it.

People have used knots for a long time. Some knots are used to show strength. Others are used to make art.

Math helps us study these shapes. We can use math to study tiny things like DNA. This helps us learn how life works.

Imagine a piece of string. You wrap it around and around. Then, you glue the ends together. This creates a math knot. Because the ends are joined, you cannot untie it.

A simple ring is the easiest knot. We call it an unknot.

Maths uses knots to study shapes. One big goal is to see if two knots are the same. You can move a knot around. You can stretch it or bend it. But you cannot cut it. If you can make one knot look like another, they are equivalent.

To help, math uses knot diagrams. These are flat drawings of the knots. They show where one part of the string goes over another.

Scientists also use knot theory for real things. It helps us study DNA. DNA is a tiny string inside our bodies. It can get tangled.

Knowing how knots work helps us understand life. People have made knot art for many years. Some monks used knots in beautiful books. Since the 1800s, math has found billions of different knots.

Imagine a piece of string. You wrap it around itself in many ways. Then, you glue the two ends together. This creates a mathematical knot. Because the ends are joined, you cannot untie it.

A simple ring is the easiest knot. Mathematicians call this an unknot. Knot theory is the study of these shapes. It helps us understand how things can be tangled. It is a special part of a field called topology. Topology looks at how shapes can change without being broken.

To study knots, math uses flat drawings called knot diagrams.

Think of a knot casting a shadow on a wall. The shadow shows where the string crosses over itself. These points are called crossings. In a diagram, we show which part is on top. We do this by leaving a small break in the line that goes underneath.

Two knots are equivalent if you can change one into the other. You can stretch or bend the string smoothly. However, you are never allowed to cut the string. You also cannot pass the string through itself.

Humans have loved knots for a very long time.

Archaeologists found that people tied knots in prehistoric times. Some cultures used knots in beautiful artwork. For example, Celtic monks made intricate patterns in the Book of Kells. In the 1800s, math began to study knots more deeply. Peter Guthrie Tait wanted to make a list of all knots.

He published a table of knots with up to ten crossings in 1885. Since then, people have found more than six billion knots and links.

Mathematicians use special tools called invariants to tell knots apart.

An invariant is a value that stays the same for the same knot. Even if you move the knot, the invariant does not change. One example is the Alexander polynomial. Another is the Jones polynomial, which Vaughan Jones found in 1984. These tools help solve the recognition problem. This is the hard job of deciding if two drawings show the same knot. In 2021, Marc Lackenby shared a new way to help solve this.

Knot theory is not just for math books. It helps scientists study the real world.

For instance, it helps us understand DNA. DNA is a long molecule that can get tangled in our bodies. Scientists use knot theory to study how these tangles work. It can also help in building quantum computers. This field connects tiny molecules to huge ideas about how space works.

Knot theory is a branch of topology that studies mathematical knots. While everyday knots in shoelaces can be untied, a mathematical knot is different. It is created by taking a one-dimensional line segment and wrapping it around itself. The two free ends are then fused together to form a closed loop.

This process creates a simple closed curve in three-dimensional Euclidean space. The simplest possible knot is a single ring, which mathematicians call the "unknot." Knot theory helps us understand the fundamental properties of these entangled shapes.

To study these complex shapes, mathematicians use knot diagrams. A knot diagram is a two-dimensional projection of a knot, much like a shadow cast on a wall.

In these drawings, the points where the string appears to cross itself are called crossings. To make the diagram useful, we must show which part of the strand is the over-strand and which is the under-strand. This is usually done by drawing a small break in the line that passes underneath. A fundamental challenge in the field is the recognition problem. This is the task of determining if two different diagrams actually represent the same knot.

Two knots are considered equivalent if one can be transformed into the other through a process called ambient isotopy. This means you can stretch, bend, or deform the knot smoothly. However, you are strictly forbidden from cutting the string or passing it through itself.

In 1927, mathematicians like Kurt Reidemeister discovered that any two diagrams of the same knot are related by three specific moves. These are known as Reidemeister moves. They include creating or removing a kink, passing two strands through each other, and sliding a strand across a crossing.

These moves allow mathematicians to manipulate diagrams while preserving the knot's identity.

Human interest in knots is ancient and spans many cultures. Archaeologists have found evidence of knot tying dating back to prehistoric times. Knots have been used for practical tasks and for beautiful spiritual symbolism. For example, Celtic monks created intricate knotwork in the 1200-year-old Book of Kells.

In the 19th century, the field began to formalize as a mathematical discipline. Peter Guthrie Tait was a key figure who sought to classify knots.

In 1885, he published a table of knots with up to ten crossings. Since those early beginnings, mathematicians have tabulated more than six billion knots and links.

To distinguish between different knots, researchers use tools called knot invariants. An invariant is a quantity or property that remains exactly the same for all equivalent descriptions of a knot. If two diagrams produce different invariants, they cannot be the same knot.

Some invariants are simple, like tricolorability, while others are highly complex. Classical invariants include the knot group and the Alexander polynomial. In 1984, the discovery of the Jones polynomial by Vaughan Jones changed the field. This discovery revealed deep connections between knot theory and quantum field theory.

Modern knot theory has expanded into many different dimensions and spaces. Mathematicians study higher-dimensional knots, which involve n-dimensional spheres embedded in higher-dimensional space. The field also explores links, which are collections of several knotted components entangled with one another.

Even the way we add knots together is studied, creating new complex structures.

These mathematical advancements allow for a much more precise understanding of how objects can be intertwined in space.

Beyond pure mathematics, knot theory has vital applications in the physical sciences. It is used to study the entanglement of polymers and the structure of DNA. Scientists use it to determine if a molecule is chiral, meaning it has a specific "handedness."

In biology, knot theory helps describe how enzymes like topoisomerase act on DNA strands. Furthermore, the field may play a crucial role in the development of quantum computers through topological quantum computation. This shows how abstract mathematical loops can explain the very building blocks of life and technology.