geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a quadrisecant or quadrisecant line of a

space 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 ...

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 the quadrisecants form a discrete set of lines. Quadrisecants have been studied for curves of several types: * Knots and links in

knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

, when

nontrivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group (mathematics), group, topological space). The n ...

, 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 mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...

can be computed by a formula derived by

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...

. *Quadrisecants of arrangements of

skew lines In three-dimensional geometry, skew lines are two Line (geometry), lines that do not Line-line intersection, intersect and are not Parallel (geometry), parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges ...

touch subsets of four lines from the arrangement. They are associated with

ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...

s and the Schläfli double six configuration.

Definition and motivation

A quadrisecant is a line that intersects a curve, surface, or other set in four distinct points. It is analogous to a

secant line In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant inter ...

, 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. Compared to secants and trisecants, quadrisecants are especially relevant for

s, 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 A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is '' simple harmonic motion''; as rotation, it correspon ...

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. For generic space curves, the quadrisecants form a discrete set of lines. In contrast, when trisecants occur, they form continuous families of lines. One explanation for this phenomenon is visual: looking at a space curve from far away, the space of such points of view can be described as a two-dimensional sphere, one point corresponding to each direction. 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 In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...

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. 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, algebraic curves, or arrangements of lines.

For special classes of curves

Knots and links

In three-dimensional

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 ...

, every

tame

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

or link has a quadrisecant. Originally established in the case of knotted

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s 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, 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. The existence of at least one quadrisecant follows from the fact that any such disk must have at least one singularity. conjectured that the number of distinct quadrisecants of a given knot is always at least

n(n-1)/2

, where

n

is the crossing number of the knot. 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

abcd

on the knot, appear in order

acbd

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

K

has finitely many quadrisecants,

K

can be approximated with an equivalent polygonal knot with its vertices at the points where the quadrisecants intersect

K

, in the same order as they appear on

K

. 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. It has been conjectured that every

wild knot In the mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus S^1\times D^2 into the 3-sphere. A knot is tame if and only if it can be represented as a fin ...

has an infinite number of quadrisecants.

Algebraic curves

derived a formula for the number of quadrisecants of an

in three-dimensional

, as a function of its degree and

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

. For a curve of degree

d

and genus

g

, the number of quadrisecants is

\frac-\frac.

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

Skew lines

In three-dimensional

, every set of four

in general position has either two quadrisecants (also in this context called transversals) or none. Any three of the four lines determine a

hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...

, 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, it has two points of intersection, because the hyperboloid is defined by a

quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...

. 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. In spaces with

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

coordinates rather than real coordinates, four skew lines always have exactly two quadrisecants. The quadrisecants of sets of lines play an important role in the construction of the Schläfli double six, a

configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice board ...

of twelve lines intersecting each other in 30 crossings. If five lines are given in three-dimensional space, such that all five are intersected by a common line

b_6

but are otherwise in general position, then each of the five quadruples of the lines

a_i

has a second quadrisecant

b_i

, and the five lines

b_i

formed in this way are all intersected by a common line

a_6

. These twelve lines and the 30 intersection points

a_ib_j

form the double six. An arrangement of

n

complex lines with a given number of pairwise intersections and otherwise skew may be interpreted as an algebraic curve with degree

n

and with genus determined from its number of intersections, and Cayley's aforementioned formula used to count its quadrisecants. The same result as this formula can also be obtained by classifying the quadruples of lines by their intersections, counting the number of quadrisecants for each type of quadruple, and summing over all quadruples of lines in the given set.

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