Quadrisecant
In the plane, a generic curve can be crossed arbitrarily many times by a line; for instance, small generic perturbations of the sine curve are crossed infinitely often by the horizontal axis.
Nevertheless, any quadrisecants of the original space curve will remain present nearby in its perturbation.
[3] For generic space curves, the quadrisecants form a discrete set of lines.
In contrast, when trisecants occur, they form continuous families of lines.
Pairs of strands of the curve may appear to cross from all of these points of view, or from a two-dimensional subset of them.
Three strands will form a triple crossing when the point of view lies on a trisecant, and four strands will form a quadruple crossing from a point of view on a quadrisecant.
Each constraint that the crossing of a pair of strands lies on another strand reduces the number of degrees of freedom by one (for a generic curve), so the points of view on trisecants form a one-dimensional (continuously infinite) subset of the sphere, while the points of view on quadrisecants form a zero-dimensional (discrete) subset.
C. T. C. Wall writes that the fact that generic space curves are crossed at most four times by lines is "one of the simplest theorems of the kind", a model case for analogous theorems on higher-dimensional transversals.
[3] Depending on the properties of the curve, it may have no quadrisecants, finitely many, or infinitely many.
These considerations make it of interest to determine conditions for the existence of quadrisecants, or to find bounds on their number in various special cases, such as knotted curves,[5][6] algebraic curves,[7] or arrangements of lines.
[8] In three-dimensional Euclidean space, every nontrivial tame knot or link has a quadrisecant.
[9] Pannwitz proved more strongly that, for a locally flat disk having the knot as its boundary, the number of singularities of the disk can be used to construct a lower bound on the number of distinct quadrisecants.
[5][10] Morton & Mond (1982) conjectured that the number of distinct quadrisecants of a given knot is always at least
[11] 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.
can be approximated with an equivalent polygonal knot with its vertices at the points where the quadrisecants intersect
However, their conjecture is false: in fact, 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.
[9] 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.
This formula assumes that the given curve is non-singular; adjustments may be necessary if it has singular points.
[15][16] In three-dimensional Euclidean space, every set of four skew lines in general position has either two quadrisecants (also in this context called transversals) or none.
If the fourth of the given lines pierces this surface, it has two points of intersection, because the hyperboloid is defined by a quadratic equation.
The two trisecants of the ruled surface, through these two points, form two quadrisecants of the given four lines.
On the other hand, if the fourth line is disjoint from the hyperboloid, then there are no quadrisecants.
complex lines with a given number of pairwise intersections and otherwise skew may be interpreted as an algebraic curve with degree
and with genus determined from its number of intersections, and Cayley's aforementioned formula used to count its quadrisecants.
