Quadrisecant

In three-dimensional Euclidean space, every nontrivial tame knot or link has a quadrisecant. Originally established in the case of knotted polygons and smooth knots by Erika Pannwitz,[5] this result was extended to knots in suitably general position and links with nonzero linking number,[6] and later to all nontrivial tame knots and links.[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. The existence of at least one quadrisecant follows from the fact that any such disk must have at least one singularity.[5][10] 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.[6][10] Counterexamples to this conjecture have since been discovered.[10]

Two-component links have quadrisecants in which the points on the quadrisecant appear in alternating order between the two components,[6] 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.[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.[11] Quadrisecants have also been used to find lower bounds on the ropelength of knots.[12]

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 intersect ${\displaystyle K}$, in the same order as they appear on ${\displaystyle K}$. 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.[13]

It has been conjectured that every wild knot has an infinite number of quadrisecants.[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.[7] For a curve of degree ${\displaystyle d}$ and genus ${\displaystyle g}$, the number of quadrisecants is[14]

$${\displaystyle {\frac {(d-2)(d-3)^{2}(d-4)}{12}}-{\frac {g(d^{2}-7d+13-g)}{2}}.}$$

This formula assumes that the given curve is non-singular; adjustments may be necessary if it has singular points.[15][16]

The Schläfli double six

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. Any three of the four lines determine a hyperboloid, 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. 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.[17] In spaces with complex number coordinates rather than real coordinates, four skew lines always have exactly two quadrisecants.[8]

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 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.[18][19]

An arrangement of ${\displaystyle n}$ complex 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 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.[8]