Knot (mathematics)

In mathematics, a knot is a special shape made by looping a circle through three-dimensional space without any ends. Imagine taking a string, forming a loop with no loose ends, and then bending and twisting that loop in space — that's the basic idea of a mathematical knot. These knots are different from the ones we tie in real life because they exist only in perfect, smooth space with no friction or thickness to consider.

Two knots are considered the same, or equivalent, if you can change one into the other by gently moving and stretching the space around it, without cutting or tearing the loop. This idea is called being "ambient isotopic." The study of these special loops is called knot theory, and it connects to many other areas of math, like graph theory.

Knot theory helps us understand many interesting patterns and solves problems in physics, chemistry, and even biology. For example, some molecules and DNA structures form knot-like shapes, and mathematicians use knot theory to study their properties. Knots are fascinating because even simple-looking loops can have very different shapes and behaviors when twisted and turned in space.

Formal definition

A knot is a special way of looping a circle in three-dimensional space. Imagine taking a rubber band and looping it so that it doesn't have any loose ends—it’s all one continuous loop.

We can flatten this loop onto a flat surface, like paper, to make it easier to study. When we do this, most of the loop doesn’t overlap, but there are a few points where the loop crosses over itself. By noting which part of the loop goes over and which goes under at these crossings, we can describe the knot completely. These drawings are called knot diagrams, and they help mathematicians understand and compare different knots.

Types of knots

The simplest knot is called the unknot, which looks like a smooth circle and isn't really knotted at all. Next are slightly more complex knots, such as the trefoil knot, the figure-eight knot, and the cinquefoil knot.

When several knots are linked or tangled together, they are called links. A knot is a special kind of link made from just one piece. Some knots behave in unusual ways, and these are called wild. Most commonly studied knots are "tame," meaning they can be made from straight line segments.

Applications to graph theory

Main article: linkless embedding

In graph theory, we can study knots by looking at special drawings called diagrams. These diagrams help us understand how knots behave and how they can be changed. By coloring parts of these drawings, we can create new graphs that tell us more about the original knot.

In three dimensions, not all graphs can be drawn without parts crossing over each other. Some special graphs, called linkless and knotless embeddings, avoid these crossings in special ways. One important graph, called K₇, always forms a special knot called the trefoil knot, no matter how it is drawn in space.

Generalization

In math, the idea of a knot can be expanded to include more than just circles in space. It can describe situations where one shape can be placed inside another in a special, twisted way that can't be undone easily.

There are special rules about when these shapes can be twisted, and some famous mathematicians have studied these rules. For example, some shapes can be twisted inside others in surprising ways, while others cannot. This helps us understand the different ways shapes can fit and move inside each other.