geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 a ...

is a

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

that passes through four points of the curve. This is the largest possible number of intersections that a

generic Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other ...

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 A knot is a fastening in rope or interwoven lines. Knot may also refer to: Places * Knot, Nancowry, a village in India Archaeology * Knot of Isis (tyet), symbol of welfare/life. * Minoan snake goddess figurines#Sacral knot Arts, entertainme ...

and links in

knot theory In the mathematical field of 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 ...

, 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 an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...

, always have quadrisecants, and the existence and number of quadrisecants has been studied in connection with

knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

s including the minimum

total curvature In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length: :\int_a^b k(s)\,ds. The total curvature of a closed curve i ...

and the

ropelength In physical knot theory, each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are ...

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 c ...

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 prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, C ...

. *Quadrisecants of arrangements of

skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the sa ...

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

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...

s and the

Schläfli double six In geometry, the Schläfli double six is a configuration of 30 points and 12 lines, introduced by . The lines of the configuration can be partitioned into two subsets of six lines: each line is disjoint from ( skew with) the lines in its own subse ...

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 Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ...

, 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 just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...

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. 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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

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 Charles Terence Clegg "Terry" Wall (born 14 December 1936) is a British mathematician, educated at Marlborough College, Marlborough and Trinity College, Cambridge. He is an :wikt:emeritus, emeritus professor of the University of Liverpool, where ...

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. Additionally, for generic space curves, the quadrisecants form a discrete set of lines that in contrast to the trisecants which, when they occur, form continuous families of lines. 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, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

, every

tame Tame may refer to: *Taming, the act of training wild animals *River Tame, Greater Manchester *River Tame, West Midlands and the Tame Valley *Tame, Arauca, a Colombian town and municipality * "Tame" (song), a song by the Pixies from their 1989 alb ...

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

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

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

s and

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

knots by

Erika Pannwitz Erika Pannwitz (May 26, 1904 in Lychen, Hohenlychen, Germany – November 25, 1975 in Berlin) was a German mathematician who worked in the area of geometric topology. During World War II, Pannwitz worked as a Cryptanalysis, cryptanalyst in the ...

, this result was extended to knots in suitably

general position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that ar ...

and links with nonzero

linking number In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...

, and later to all nontrivial tame knots and links. Pannwitz proved more strongly that, for a

locally flat In topology, a branch of mathematics, local flatness is smoothness condition that can be imposed on topological submanifold, submanifolds. In the Category (mathematics), category of topological manifolds, locally flat submanifolds play a role sim ...

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 In the mathematical theory of knots, the Fáry–Milnor theorem, named after István Fáry and John Milnor, states that three-dimensional smooth curves with small total curvature must be unknotted. The theorem was proved independently by Fáry i ...

, a

lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

on the

of a nontrivial knot. Quadrisecants have also been used to find lower bounds on the

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 Wild, wild, wilds or wild may refer to: Common meanings * Wild animal * Wilderness, a wild natural environment * Wildness, the quality of being wild or untamed Art, media and entertainment Film and television * ''Wild'' (2014 film), a 2014 Am ...

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 Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

and

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

. 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 In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...

; adjustments may be necessary if it has singular points.

Skew lines

In three-dimensional

, every set of four

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 defo ...

, a

doubly ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, th ...

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

. 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 form ...

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

, 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.

References

{{reflist, refs= {{citation , last1 = Bai , first1 = Sheng , last2 = Wang , first2 = Chao , last3 = Wang , first3 = Jiajun , doi = 10.1142/S0218216518500220 , issue = 2 , journal = Journal of Knot Theory and Its Ramifications , mr = 3770471 , at = 1850022 , title = Counterexamples to the quadrisecant approximation conjecture , volume = 27 , year = 2018, arxiv = 1605.00538 , s2cid = 119601013 {{citation , last = Cayley , first = Arthur , author-link = Arthur Cayley , jstor = 108806 , pages = 453–483 , title =

Philosophical Transactions of the Royal Society of London ''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journa ...

, volume = 153 , year = 1863, publisher = The Royal Society {{citation , last = Coxeter , first = H. S. M. , authorlink = Harold Scott MacDonald Coxeter , contribution = An absolute property of four mutually tangent circles , doi = 10.1007/0-387-29555-0_5 , location = New York , mr = 2191243 , pages = 109–114 , publisher = Springer , series = Math. Appl. (N. Y.) , title = Non-Euclidean geometries , volume = 581 , year = 2006; Coxeter repeats Schläfli's construction, and provides several references to simplified proofs of its correctness {{citation , last = Denne , first = Elizabeth Jane , arxiv = math/0510561 , publisher =

University of Illinois at Urbana-Champaign The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Univ ...

, series = Ph.D. thesis , title = Alternating quadrisecants of knots , year = 2004, bibcode = 2005math.....10561D {{citation , last = Denne , first = Elizabeth , editor1-last = Blatt , editor1-first = Simon , editor2-last = Reiter , editor2-first = Philipp , editor3-last = Schikorra , editor3-first = Armin , contribution = Quadrisecants and essential secants of knots , doi = 10.1515/9783110571493-006 , mr = 3915943 , pages = 138–158 , publisher = De Gruyter, Berlin , series = Partial Differential Equations and Measure Theory , title = New directions in geometric and applied knot theory , year = 2018, s2cid = 128222971 , doi-access = free {{citation , last1 = Denne , first1 = Elizabeth , last2 = Diao , first2 = Yuanan , last3 = Sullivan , first3 = John M. , author3-link = John M. Sullivan (mathematician) , doi = 10.2140/gt.2006.10.1 , journal =

Geometry & Topology ''Geometry & Topology'' is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sc ...

, mr = 2207788 , pages = 1–26 , title = Quadrisecants give new lower bounds for the ropelength of a knot , url = http://msp.warwick.ac.uk/gt/2006/10/p001.xhtml , volume = 10 , year = 2006, arxiv = math/0408026, s2cid = 5770206 {{citation , last1 = Eisenbud , first1 = David , author1-link = David Eisenbud , last2 = Harris , first2 = Joe , author2-link = Joe Harris (mathematician) , doi = 10.1017/CBO9781139062046 , isbn = 978-1-107-60272-4 , mr = 3617981 , page = 377 , publisher = Cambridge University Press , location = Cambridge, UK , title = 3264 and All That: A second course in algebraic geometry , url = https://books.google.com/books?id=au67CwAAQBAJ&pg=PA377 , year = 2016 {{citation , last1 = Griffiths , first1 = Phillip , last2 = Harris , first2 = Joseph , isbn = 9781118030776 , page = 296 , publisher = John Wiley & Sons , series = Wiley Classics Library , title = Principles of Algebraic Geometry , url = https://books.google.com/books?id=Sny48qKdW40C&pg=PA296 , volume = 52 , year = 2011 {{citation , last1 = Hilbert , first1 = David , author1-link = David Hilbert , last2 = Cohn-Vossen , first2 = Stephan , author2-link = Stephan Cohn-Vossen , edition = 2nd , isbn = 978-0-8284-1087-8 , location = New York , page = 164 , publisher = Chelsea , title = Geometry and the Imagination , year = 1952 {{citation , last = Jin , first = Gyo Taek , contribution = Quadrisecants of knots with small crossing number , doi = 10.1142/9789812703460_0025 , mr = 2197955 , pages = 507–523 , publisher = World Scientific Publishing , location = Singapore , series = Ser. Knots Everything , title = Physical and numerical models in knot theory , url = https://mathsci.kaist.ac.kr/~jin/papers/quad-small-xing/quad-small-xing.pdf , volume = 36 , year = 2005 {{citation , last = Jin , first = Gyo Taek , editor1-last = Reiter , editor1-first = Philipp , editor2-last = Blatt , editor2-first = Simon , editor3-last = Schikorra , editor3-first = Armin , contribution = Polygonal approximation of unknots by quadrisecants , date = December 2017 , doi = 10.1515/9783110571493-008 , pages = 159–175 , publisher = De Gruyter Open , title = New Directions in Geometric and Applied Knot Theory, doi-access = free {{citation , last = Kuperberg , first = Greg , author-link = Greg Kuperberg , arxiv = math/9712205 , doi = 10.1142/S021821659400006X , journal =

Journal of Knot Theory and Its Ramifications The ''Journal of Knot Theory and Its Ramifications'' was established in 1992 by Louis Kauffman and was the first journal purely devoted to knot theory. It is an interdisciplinary journal covering developments in knot theory, with emphasis on creat ...

, mr = 1265452 , pages = 41–50 , title = Quadrisecants of knots and links , volume = 3 , year = 1994, s2cid = 6103528 {{citation , last = Maxwell , first = Edwin A. , date = July 1935 , doi = 10.1017/s0305004100013086 , issue = 3 , journal = Mathematical Proceedings of the Cambridge Philosophical Society , pages = 324–326 , title = Note on the formula for the number of quadrisecants of a curve in space of three dimensions , volume = 31, s2cid = 122279811 {{citation , last1 = Morton , first1 = Hugh R. , last2 = Mond , first2 = David M. Q. , doi = 10.1016/0040-9383(82)90007-6 , journal =

Topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, mr = 0649756 , pages = 235–243 , title = Closed curves with no quadrisecants , volume = 21 , year = 1982, issue = 3 , doi-access = free {{citation , last = Pannwitz , first = Erika , author-link = Erika Pannwitz , doi = 10.1007/BF01452857 , issue = 1 , journal =

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

, pages = 629–672 , title = Eine elementargeometrische Eigenschaft von Verschlingungen und Knoten , volume = 108 , year = 1933, s2cid = 123026724 {{citation , last = Schläfli , first = Ludwig , authorlink = Ludwig Schläfli , editor-last = Cayley , editor-first = Arthur , editor-link = Arthur Cayley , journal = Quarterly Journal of Pure and Applied Mathematics , pages = 55–65, 110–120 , title = An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface , url = http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0002 , volume = 2 , year = 1858 {{citation , last = Wall , first = C. T. C. , author-link = C. T. C. Wall , editor1-last = Palis , editor1-first = Jacob , editor2-last = do Carmo , editor2-first = Manfredo , contribution = Geometric properties of generic differentiable manifolds , doi = 10.1007/BFb0085382 , mr = 0494233 , pages = 707–774 , series = Lecture Notes in Mathematics , title = Geometry and Topology: Proceedings of the Latin American School of Mathematics (ELAM III) held at the Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, July 1976 , volume = 597 , year = 1977 {{citation , last = Welchman , first = W. G. , date = April 1932 , doi = 10.1017/s0305004100010872 , issue = 2 , journal = Mathematical Proceedings of the Cambridge Philosophical Society , pages = 206–208 , title = Note on the trisecants and quadrisecants of a space curve , volume = 28, s2cid = 120725025 {{citation , last = Wong , first = B. C. , doi = 10.1090/S0002-9904-1934-05854-3 , issue = 4 , journal = Bulletin of the American Mathematical Society , mr = 1562839 , pages = 291–296 , title = Enumerative properties of

r

-space curves , volume = 40 , year = 1934, doi-access = free Knot theory Algebraic geometry