A knot is an embedding $S^1 \hookrightarrow \mathbb R^3$ (or $S^3$), considered up to ambient isotopy — continuous deformations not allowed to cut the curve. A link is several disjoint knot components. Trivial cases include the unknot and the unlink.

Knots are studied via knot diagrams: 2D projections with over/under crossings. Two diagrams represent the same knot iff they differ by planar isotopy and the three Reidemeister moves:

• $R_1$: add/remove a twist.
• $R_2$: slide a strand over another.
• $R_3$: move a strand past a crossing.

A knot invariant is a function of knots invariant under Reidemeister moves. Examples: crossing number, tricolorability, Alexander polynomial $\Delta_K(t)$ (1928), Jones polynomial $V_K(t)$ (1984 Fields medal), HOMFLYPT, Khovanov homology.

The trefoil knot $3_1$ has crossing number 3, three Wirtinger generators, and is tricolorable (unknot is not — proves it's nontrivial). The Jones polynomial of the trefoil is $V_{3_1}(t) = -t^{-4} + t^{-3} + t^{-1}$.

Interactive: rotating trefoil