Knot (mathematics)

In mathematics, a knot is a special shape made by looping a circle through space without any ends. Imagine taking a string, forming a loop with no loose ends, and then bending and twisting that loop — 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 space with no friction or thickness.

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 interesting patterns and solves problems in physics, chemistry, and 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 loops can have very different shapes when twisted and turned in space.

Formal definition

A knot is a special way of looping a circle in three-dimensional space. Picture a rubber band looped so that it has no loose ends—it’s one continuous loop.

We can flatten this loop onto paper to study it. Most of the loop doesn’t overlap, but a few points cross over themselves. By noting which part 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. It 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. 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 study knots using special drawings called diagrams. These diagrams help us learn about how knots work and how they can change. By coloring parts of these drawings, we can make new graphs that give us more information 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 include more than just circles in space. It can describe situations where one shape is 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. Some famous mathematicians have studied these rules. Some shapes can be twisted inside others in surprising ways, while others cannot. This helps us understand how shapes can fit and move inside each other.