Fáry–Milnor theorem

The Fáry–Milnor theorem states that every closed curve forming a nontrivial knot has total curvature at least 4π. The bound is sharp as an infimum for nontrivial knot types.

Connect

Total curvature: The accumulated turning of a curve, measured by integrating curvature along its length. The theorem's 4π threshold is a lower bound on this quantity.
Knot theory: The study of embeddings of circles in three-dimensional space and their equivalence. The theorem is a geometric constraint on knots within this broader subject.
István Fáry: A Hungarian mathematician whose work included a proof of the total-curvature bound for knotted curves. His independent proof gave the theorem one of its names.
Trefoil knot: The simplest nontrivial knot, with three crossings in its standard diagram. It is a basic test case for the lower bound on knotted curves.
Knot invariant: A quantity or structure unchanged under equivalence of knots. The theorem turns a geometric measurement into a constraint determined by knot type.
Fenchel's theorem: Every closed space curve has total curvature at least 2π. It supplies the baseline bound that the knot theorem strengthens.
Knot (mathematics): An embedding of a circle in three-dimensional space, considered up to continuous deformation without cutting. The theorem distinguishes knotted embeddings from the unknot.
John Milnor: An American mathematician known for foundational contributions to topology, geometry, and differential topology. His independent proof established the other eponym and connected curvature with knot complexity.
Torus knot: A knot that lies on the surface of an unknotted solid torus and winds around it in two directions. Explicit torus-knot curves illustrate how knot type constrains possible bending.
Total curvature of knots: The study of lower bounds and extremal values for total curvature among curves of fixed knot type. The theorem is a foundational lower bound in this area.