Quadrisecant

In geometry, a quadrisecant or quadrisecant line of a curve is a line that passes through four points of the curve. Every knotted curve in three-dimensional Euclidean space has a quadrisecant. The number of quadrisecants of an algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley. Quadrisecants of skew lines are also associated with ruled surfaces and the Schläfli double six configuration.
In knot theory.
In three-dimensional Euclidean space, every non-trivial tame knot or link has a quadrisecant. Originally established in the case of knotted polygons and smooth knots by Erika Pannwitz,
this result was extended to knots in suitably general position and links with nonzero linking number,
and later to all nontrivial tame knots and links.
Pannwitz proved more strongly that the number of distinct quadrisecants is lower bounded by a function of the minimum number of boundary singularities in a locally-flat open disk bounded by the knot. conjectured that the number of distinct quadrisecants of a given knot is always at least formula_1, where formula_2 is the crossing number of the knot. However, counterexamples to this conjecture have since been discovered.
Two-component links have quadrisecants in which the points on the quadrisecant appear in alternating order between the two components, and nontrivial knots have quadrisecants in which the four points, ordered cyclically as formula_3 on the knot, appear in order formula_4 along the quadrisecant. The existence of these alternating quadrisecants can be used to derive the Fáry–Milnor theorem, a lower bound on the total curvature of a nontrivial knot. Quadrisecants have also been used to find lower bounds on the ropelength of knots.
G. T. Jin and H. S. Kim conjectured that, when a knotted curve formula_5 has finitely many quadrisecants, formula_5 can be approximated with an equivalent polygonal knot with its vertices at the points where the quadrisecants meet formula_5, in the same order as they appear on formula_5. However, this is false: for every knot type, there is a realization for which this construction leads to a self-intersecting polygon, and another realization where this construction produces a knot of a different type.
In algebraic geometry.
Arthur Cayley derived a formula for the number of quadrisecants of an algebraic curve in three-dimensional complex projective space, as a function of its degree and genus. For a curve of degree formula_9 and genus formula_10, the number of quadrisecants is
formula_11
This formula assumes that the given curve is non-singular; adjustments may be necessary if it has singular points.
Of skew lines.
In three-dimensional Euclidean space, every set of four skew lines in general position either has two quadrisecants (also called in this context transversals) or none. Any three of the four lines determine a doubly ruled surface, in which one of the two sets of ruled lines contains the three given lines, and the other ruling consists of trisecants to the given lines. If the fourth of the given lines pierces this surface, its two points of intersection lie on the two quadrisecants; if it is disjoint from the surface, then there are no quadrisecants.
The quadrisecants of sets of lines play an important role in the construction of the Schläfli double six, a configuration of twelve lines intersecting each other in 30 crossings. If five lines are given in a three-dimensional space, such that all five are intersected by a common line formula_12 but are otherwise in general position, then each of the five quadruples of the lines formula_13 has a second quadrisecant formula_14, and the five lines formula_14 formed in this way are all intersected by a common line formula_16. These twelve lines and the 30 intersection points formula_17 form the double six.