Knot theory

In topology, knot theory is the study of mathematical knots. Real-life knots, like those in shoelaces, inspired this area of math. But a mathematical knot has its ends joined so it can’t be undone. The simplest knot looks like a smooth ring and is called an "unknot". In math, a knot is a circle inside a three-dimensional space.

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

Mathematicians use special numbers and formulas called knot invariants to tell knots apart. Some important tools include knot polynomials, knot groups, and other math ideas.

People began knot theory in the 1800s to make big lists of knots and links, which are several knots tangled together. Since then, mathematicians have listed many knots and links have been tabulated.

To learn more about knots, mathematicians study them in different spaces and shapes. 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 tied knots for a very long time, even before we had writing. Knots were useful for tying things together and also looked nice. They had special meanings in different cultures. For example, Chinese artists made pretty knots hundreds of years ago. Special knots also appear in Tibetan Buddhist art and Celtic designs.

In the 1700s, mathematicians started studying knots more closely. In the 1800s, a scientist thought atoms might be like knots. This led to the first lists of different kinds of knots. In the 1900s, mathematicians used new ideas to understand knots better. Later, they found that knots are connected to other areas of math and science.

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 to make 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 is the same for knots that look the same. If you draw a knot in different ways but it is really the same knot, the invariant will give the same number. Some invariants can tell you if two knots are different.

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.

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. This helps us see patterns and shapes that are useful for studying knots and links.

Higher dimensions

When you think about knots, like tying a shoelace, you can untie them if you move into a fourth dimension. In this way, you can lift one part of the knot out of 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 joined together by cutting them and connecting the ends. This is called the knot sum. Picture each knot drawn on paper without touching. Then, find a way to connect parts of each knot to create a new one. Depending on how you connect them, you might end up with one of two new knots.

When we think of knots as having a direction, joining them follows special rules. Some knots cannot be split into simpler ones and are called prime knots. Others can be made by joining prime knots and are called composite knots. This idea is like how numbers can be split into prime numbers.

Multiplication of knots

In 2020, two mathematicians found a new way to connect knots. They made a rule to join two knots and create a new one. This rule has special patterns that relate to other knot features.

Tabulating knots

See also: List of prime knots and Knot tabulation

Knots are grouped by how many times they cross themselves. This is called the crossing number. Tables of knots usually show the simplest knots, called prime knots. The number of knots with more crossings grows very fast, so it is hard to list them all.

People have listed over 6 billion knots and links.

There are different ways to write down knots to help list 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 shows 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 studies how knots are made from crossings. But some real folded chains have special bonds that traditional methods don't cover. In 2019, researchers expanded knot theory to study these special bonds. They made new ways to understand and sort these complex structures, showing there is more to knots than we usually see.

Applications

Knot theory studies special kinds of loops and 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. Scientists are now using knot theory in new areas like quantum computing and the creation of new materials.