Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots we see every day, like those in shoelaces and rope, a mathematical knot is special because the ends are joined so it cannot be undone. The simplest knot looks like a smooth ring, called an "unknot". In math, a knot is described as a circle that lives inside a three-dimensional space.

Knots can be described in many different ways. For example, a knot can be drawn as a flat picture called a knot diagram, and the same knot can look very different in these pictures. One big question in knot theory is how to tell if two different pictures actually show the same knot.

Mathematicians have found clever ways to tell knots apart using special numbers and formulas called knot invariants. Some important tools include knot polynomials, knot groups, and other math ideas.

People started knot theory in the 1800s to make big lists of knots and links, which are several knots tangled together. Since then, more than six billion knots and links have been tabulated.

To understand knots better, mathematicians also study them in different spaces and with different shapes. For example, they look at knots made from higher-dimensional shapes or study how knots appear in things like proteins and DNA.

History

Main article: History of knot theory

People have been tying knots for a very long time, even before history was written down. Knots were not just useful for tying things together; they were also beautiful and held special meanings in different cultures. For example, Chinese artists made beautiful knots hundreds of years ago, and special knots appear in Tibetan Buddhist art and Celtic designs.

In the 1700s, mathematicians began studying knots more carefully. In the 1800s, a scientist thought atoms might be like knots, which led to the first lists of different kinds of knots. In the 1900s, mathematicians used new ideas to understand knots better. In the late 1900s, amazing discoveries showed how knots are connected to other areas of math and science, like how molecules twist and turn.

Knot equivalence

Knot theory is a part of math where we study knots. Imagine taking a string, tying a knot in it, and then connecting the two ends so it forms a loop that can’t be undone. In math, a knot is this special loop in three-dimensional space.

Two knots are considered the same, or equivalent, if you can move one knot around smoothly in space — without cutting or tearing — to look exactly like the other. This means you can stretch and bend the knot, but not break it, to make it match another knot. This idea helps mathematicians understand and classify different kinds of knots.

Knot diagrams

A good way to see and work with knots is by flattening them onto a flat surface, like a shadow on a wall. When you look at the shadow, sometimes the knot seems to cross over itself at certain points, called crossings. To know which part of the knot is on top and which is underneath at each crossing, we often draw a small break in the line that is underneath.

These drawings are called knot diagrams when they show a single knot and link diagrams when they show more than one knot.

Main article: Reidemeister move

In 1927, researchers found that any two drawings of the same knot can be changed into each other using three simple moves. These moves are called Reidemeister moves. They include: (1) making or undoing a small twist in the knot; (2) two parts of the knot meeting and passing by each other; and (3) three parts of the knot meeting at one point. These moves help us understand how knots can look different but still be the same.

Knot invariants

Main article: Knot invariant

A knot invariant is a special number that stays the same for knots that look the same. For example, if you change how you draw a knot but it is really the same knot, the invariant will give you the same number. Some invariants can tell you if two knots are different, but not all invariants can do this.

Classical knot invariants include the knot group and the Alexander polynomial. Later, other invariants like quantum knot polynomials were found. These are just a few of the many invariants used in knot theory today.

Knot polynomials

Main article: Knot polynomial

A knot polynomial is a special kind of knot invariant that is a math expression. Famous examples are the Jones polynomial, the Alexander polynomial, and the Kauffman polynomial. The Alexander–Conway polynomial is another example that uses a letter, like z, in its calculations.

This polynomial can also work for links, which are several knots tangled together. The rules for calculating it involve changing parts of the link diagram and following specific steps. For example, the Alexander–Conway polynomial for a special knot called the trefoil shows that it is truly knotted and not just a simple loop.

Hyperbolic invariants

Main article: Hyperbolic knot

Many knots have a special property where the space around them can be studied using a type of geometry called hyperbolic geometry. This geometry helps us understand the shape and size of the space inside a knot.

One example is the Borromean rings, a set of three linked rings where removing one ring unlinks the others. By using hyperbolic geometry, we can picture what the inside of these links looks like from the perspective of someone living close to one of the rings. This helps us see patterns and shapes that are useful for studying knots and links.

Higher dimensions

When you think about knots in three dimensions, like tying a shoelace, you can untie them if you move into a fourth dimension. This is because you can lift one part of the knot out of the normal space, move it around, and then put it back so it looks different.

In four dimensions, any loop of string that doesn’t cross itself can be changed to look like a simple circle. This idea helps mathematicians study special types of knots called slice knots and ribbon knots. There is also an interesting question about whether every slice knot is also a ribbon knot.

We can also think about knots made from spheres, not just strings. For example, a two-dimensional sphere, like the surface of a ball, can be placed in four-dimensional space in ways that make it look "knotted." These ideas help mathematicians understand more about shapes and spaces.

Adding knots

Main article: Knot sum

Two knots can be combined by cutting them and joining the ends together. This process is called the knot sum. Imagine drawing each knot on paper without them touching. Then, find a space where you can connect parts of each knot to make a new one. Depending on how you connect them, you might get one of two possible new knots.

When we think of knots as having a direction, combining them follows certain rules. Some knots cannot be broken down into simpler ones and are called prime knots. Others can be made by combining prime knots and are called composite knots. This idea of breaking knots down is similar to how numbers can be broken into prime numbers.

Multiplication of knots

In 2020, two mathematicians introduced a new way to combine knots. They created a special rule to connect two knots together, making a new one. This rule has special patterns and relationships with other knot properties.

Tabulating knots

See also: List of prime knots and Knot tabulation

Knots have been organized based on how many times they cross themselves. This is called the crossing number. Tables of knots usually include only the simplest knots called prime knots. The number of knots with more crossings grows very quickly, making it hard to list them all.

People have managed to list over 6 billion knots and links. The number of prime knots for each crossing number goes like this: 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705...

Different ways to write down knots have been created to help with listing them. Early tables tried to include all knots with up to 10 crossings.

Alexander–Briggs notation

Main article: Dowker–Thistlethwaite notation

The Dowker–Thistlethwaite notation, also called the Dowker notation or code, for a knot is a list of even numbers. These numbers come from following the knot and marking the crossings with numbers. Each crossing is marked twice, so the numbers come in pairs. A sign is used to show which part of the knot passes over and which passes under.

Conway notation

Main article: Conway notation (knot theory)

The Conway notation for knots and links is named after John Horton Conway. It is based on the idea of tangles. This notation shows how to build a picture of the knot or link. It starts with a basic shape and adds parts called tangles.

Gauss code

Main article: Gauss code

Gauss code is another way to write down a knot using numbers. Each crossing is given one number. If the crossing is where the knot goes over, a positive number is used. If it goes under, a negative number is used.

Knots with intra-chain bonds

Classical knot theory looks at how knots are formed by crossings. But some real-life folded chains have special bonds that are not covered by traditional methods. In 2019, researchers expanded knot theory to study these special bonds in chains. They created new ways to understand and classify these complex structures, showing there is more to knots than what we usually see.

Applications

Knot theory, which studies special kinds of loops, is useful in many areas of science. In chemistry, it helps us understand the shape and behavior of tiny parts called molecules. In biology, it helps explain how certain enzymes work with DNA, the material that carries instructions for living things.

In physics, ideas from knot theory help describe unusual shapes and movements in nature. Recently, scientists have started using knot theory in new areas like quantum computing and the creation of new materials.