#knots

A knot feels ordinary because we meet it in ordinary places: shoelaces, cords, thread, rope, hair, fishing line. It feels like a nuisance rather than a deep fact about space.

But a knot is not just “something tangled.”

A real knot is a kind of geometric trap. It is a structure that cannot be undone by ordinary motion. You can pull, twist, stretch, and slide it around, but unless you cut the strand or let it pass through itself, the knot remains.

That raises a quiet question:

Why is a knot possible at all?

The answer depends on something so familiar that it is easy to overlook: we live in three spatial dimensions.

A knot is not just a drawing

In mathematics, a knot is usually thought of as a closed loop sitting in space.

A loose piece of rope with two free ends is not quite the ideal object, because its ends can often be threaded backward through the tangle. A closed loop has no such escape hatch. It is more like a rubber band, a loop of string with its ends joined, or a tiny circular railway that must keep its track continuous.

One compact way to say this is:

\[ \text{a knot is an embedding of } S^1 \text{ in three-dimensional space.} \]

Here \(S^1\) just means a circle: a one-dimensional closed loop. “Embedded” means the loop is placed into space without cutting through itself.

The important part is not the notation. The important part is this:

A knot is a closed loop arranged in space so that it cannot be untangled without cutting the loop or letting it pass through itself.