Two knots are equivalent if one can be smoothly transformed into the other without cutting. Knot invariants, like the Jones polynomial or Alexander polynomial, are tools to distinguish between knots.

Common examples include the unknot (a simple loop), the trefoil knot (the simplest nontrivial knot), and the figure-eight knot. More complex knots are categorized by crossings and properties.

Knot theory appears in molecular biology (DNA topology), fluid dynamics (vortex knots), and even quantum field theory and cryptography.