In the mathematical theory of knots, the Fáry–Milnor theorem, named after

István Fáry István Fáry (30 June 1922 – 2 November 1984) was a Hungarian-born mathematician known for his work in geometry and algebraic topology.. He proved Fáry's theorem that every planar graph has a straight-line embedding in 1948, and the Fáry� ...

and

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...

, states that three-dimensional smooth curves with small total curvature must be

unknot In the knot theory, mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a Knot (mathematics), knot tied into it, unknotted. To a knot ...

ted. The theorem was proved independently by Fáry in 1949 and Milnor in 1950. It was later shown to follow from the existence of

quadrisecant In geometry, a quadrisecant or quadrisecant line of a space curve is a line that passes through four points of the curve. This is the largest possible number of intersections that a generic space curve can have with a line, and for such curves ...

s .

Statement

If ''K'' is any closed

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

that is sufficiently smooth to define the

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

κ at each of its points, and if the total absolute curvature is less than or equal to 4π, then ''K'' is an

, i.e.: :

\text\ \oint_K , \kappa(s),  \, \mathrms \le 4 \pi,\ \text\ K\ \text.

The

contrapositive In logic and mathematics, contraposition, or ''transposition'', refers to the inference of going from a Conditional sentence, conditional statement into its logically equivalent contrapositive, and an associated proof method known as . The contrap ...

tells us that if ''K'' is not an unknot, i.e. ''K'' is not isotopic to the circle, then the total curvature will be strictly greater than 4π. Notice that having the total curvature less than or equal to 4 is merely a

sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

for ''K'' to be an unknot; it is not a

necessary condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

. In other words, although all knots with total curvature less than or equal to 4π are the unknot, there exist unknots with curvature strictly greater than 4π.

Generalizations to non-smooth curves

For closed polygonal chains the same result holds with the integral of curvature replaced by the sum of angles between adjacent segments of the chain. By approximating arbitrary curves by polygonal chains, one may extend the definition of total curvature to larger classes of curves, within which the Fáry–Milnor theorem also holds (, ).

References

*. *. *. *.

External links

*. Fenner describes a geometric proof of the theorem, and of the related theorem that any smooth closed curve has total curvature at least 2π. {{DEFAULTSORT:Fary-Milnor theorem Knot theory Theorems in topology