Quadrisecant

In geometry, a quadrisecant or quadrisecant line of a space curve or system of curves is a line that passes through four points of the curve or curves. This is the largest possible number of intersections that a generic space curve can have with a line, and for such curves the quadrisecants form a discrete set of lines. Quadrisecants have been studied for curves of several different types:

• Knots and links in knot theory, when nontrivial, always have quadrisecants, and the existence and number of quadrisecants has been studied in connection with knot invariants including the minimum total curvature and the ropelength of a knot.
• The number of quadrisecants of a non-singular algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley.
• Quadrisecants of skew lines must touch each of four lines once. They are associated with ruled surfaces and the Schläfli double six configuration.

Definition and motivation[]

A quadrisecant is just a line that intersects a curve, surface, or other set in four distinct points, analogously to a secant line, a line that intersects a curve or surface in two points, and a trisecant, a line that intersects a curve or surface in three points.^[1]

However, quadrisecants are especially relevant for space curves, because they have the largest possible number of intersection points of a line with a generic curve. 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. In contrast, if an arbitrary space curve is perturbed by a small distance to make it generic, there will be no lines through five or more points of the perturbed curve. Nevertheless, any quadrisecants of the original space curve will remain present nearby in its perturbation. C. T. C. Wall writes that the fact that generic space curves are only crossed a bounded number of times by lines is "one of the simplest theorems of the kind", a model case for analogous theorems on higher-dimensional transversals.^[2]

Additionally, for generic space curves, the quadrisecants form a discrete set of lines that, in many cases, is finite, in contrast to the trisecants which, when they occur, form continuous families of lines.^[3] This makes it of interest to determine conditions for the existence of quadrisecants or bounds on their number in various special cases, such as knotted curves,^[4][5] algebraic curves,^[6] or arrangements of lines.^[7]

For special classes of curves[]

Knots and links[]

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,^[4] this result was extended to knots in suitably general position and links with nonzero linking number,^[5] and later to all nontrivial tame knots and links.^[8]

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.^[4][9] Morton & Mond (1982) conjectured that the number of distinct quadrisecants of a given knot is always at least ${\displaystyle n(n-1)/2}$, where ${\displaystyle n}$ is the crossing number of the knot.^[5][9] However, counterexamples to this conjecture have since been discovered.^[9]

Two-component links have quadrisecants in which the points on the quadrisecant appear in alternating order between the two components,^[5] and nontrivial knots have quadrisecants in which the four points, ordered cyclically as ${\displaystyle abcd}$ on the knot, appear in order ${\displaystyle acbd}$ along the quadrisecant.^[10] 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.^[10] Quadrisecants have also been used to find lower bounds on the ropelength of knots.^[11]

G. T. Jin and H. S. Kim conjectured that, when a knotted curve ${\displaystyle K}$ has finitely many quadrisecants, ${\displaystyle K}$ can be approximated with an equivalent polygonal knot with its vertices at the points where the quadrisecants meet ${\displaystyle K}$, in the same order as they appear on ${\displaystyle K}$. 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.^[12]

Algebraic curves[]

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.^[6] For a curve of degree ${\displaystyle d}$ and genus ${\displaystyle g}$, the number of quadrisecants is^[13]

This formula assumes that the given curve is non-singular; adjustments may be necessary if it has singular points.^[14][15]

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.^[16] In spaces with complex number coordinates rather than real coordinates, four skew lines always have exactly two quadrisecants.^[7]

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 ${\displaystyle a_{i}}$ (for ${\displaystyle i=1,2,3,4,5}$) are given in a three-dimensional space, such that all five are intersected by a common line ${\displaystyle b_{6}}$ but are otherwise in general position, then each of the five quadruples of the lines ${\displaystyle a_{i}}$ has a second quadrisecant ${\displaystyle b_{i}}$, and the five lines ${\displaystyle b_{i}}$ formed in this way are all intersected by a common line ${\displaystyle a_{6}}$. These twelve lines and the 30 intersection points ${\displaystyle a_{i}b_{j}}$ form the double six.^[17][18]

An arrangement of ${\displaystyle n}$ lines with a given number of pairwise intersections and otherwise skew may be interpreted as an algebraic curve with degree ${\displaystyle n}$ and with genus determined from its number of intersections, and Cayley's formula used to count its quadrisecants. One obtains the same result by counting two quadrisecants per quadruple of skew lines and fewer for quadruples with intersections, and summing over all types of quadruples of lines in the given set.^[7]

References[]