Knot Theory Intro

Knot Theory, a branch of topology, studies mathematical knots—closed loops in 3D space—and their properties under deformation.

Definition

A knot is an embedding of the circle \(S^1\) into \(\mathbb{R}^3\), considered up to ambient isotopy (continuous deformation).

Examples

The unknot (a simple loop) vs. the trefoil knot (three crossings) are distinct; they’re not deformable into each other.

Applications

Used in DNA modeling (tangled strands), physics (quantum field theory), and cryptography.