''Not Knot'' is a 16-minute film on the mathematics of

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 ...

and

low-dimensional topology In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot th ...

, centered on and titled after the concept of a

knot complement In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...

. It was produced in 1991 by mathematicians at the Geometry Center at the

University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...

, directed by Charlie Gunn and Delle Maxwell, and distributed on

videotape Videotape is magnetic tape used for storing video and usually sound in addition. Information stored can be in the form of either an analog or digital signal. Videotape is used in both video tape recorders (VTRs) and, more commonly, videocassett ...

with a 48-page paperback booklet of supplementary material by

A K Peters A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science. They published the journals ''Experimental Mathematics'' and the '' Journal ...

Topics

The video is structured into three parts. It begins by introducing knots, links, and their classification, using the

trefoil knot In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kno ...

figure-eight knot The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...

, and

Borromean rings In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...

as examples. It then describes the construction of two-dimensional surfaces such as cones and cylinders by gluing together the edges of flat sheets of paper, the internal geometry of the resulting

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s or

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

s, and the behavior of light rays within them. Finally, it uses a three-dimensional version of the same construction method to focus in more depth on the

link complement In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...

of the Borromean rings and on the

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

of this complementary space, which has a high degree of symmetry and is closely related to classical

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

. The view of this space, constructed as the limit of a process of pushing the rings out "to infinity", is immersive, rendered and lit accurately, "like flying through hyperbolic space". The supplementary material includes a complete script of the video, with black-and-white reproductions of many of its frames, accompanied by explanations at two levels, one set aimed at high school students and another at more advanced mathematics students at the late undergraduate or early graduate level.

Audience and reception

Reviewer James M. Kister writes that making these topics understandable to non-mathematicians in this format, as this video attempts, is "virtually impossible", and in this case "only partially successful". Kister writes of pre-high-school students entranced by the visual images in the video but with no understanding of their meaning, and of academics in non-mathematical disciplines who were equally bewildered. He suggests that the true audience for this video is the mathematics students for whom the more detailed supplementary material was intended. On the other hand, while agreeing that the material is fully understandable only with significant mathematical background, L. P. Neuwirth writes that "value may surely be found for elementary school students". Knot theorist Mark Kidwell suggests that, even if the details are not understood, the video could be helpful in dispelling the popular misconception that knot theory is not mathematics. And in a review published over ten years after the initial release of this video, Charles Ashbacher writes that the visual effects in this video "are still capable of stunning you", that the mathematics they depict can be clearly followed, and that it should be viewed by "all mathematics students".

References

{{reflist, refs= {{citation , last = Abbott , first = Steve , date = July 1997 , doi = 10.2307/3619248 , issue = 491 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, jstor = 3619248 , pages = 340–342 , title = Review of ''Not Knot'' and ''Supplement to Not Knot'' , volume = 81 {{citation , last = Ashbacher , first = Charles , date = Spring 2003 , issue = 2 , journal = Mathematics and Computer Education , pages = 263–264 , title = Review of ''Not Knot'' , volume = 37 {{citation , last = Emmer , first = Michele , date = June–August 1992 , issue = 3–4 , journal =

Leonardo Leonardo is a masculine given name, the Italian, Spanish, and Portuguese equivalent of the English, German, and Dutch name, Leonard Leonard or ''Leo'' is a common English masculine given name and a surname. The given name and surname originate ...

, pages = 390–391 , title = ''Not Knot'' by Charlie Gunn, et al. (review) , url = https://muse.jhu.edu/article/607116/summary , volume = 25, doi = 10.2307/1575876 , jstor = 1575876 {{citation , last = Kidwell , first = Mark , date = March 1993 , department = Media Highlights , doi = 10.1080/07468342.1993.11973528 , issue = 2 , journal = The College Mathematics Journal , pages = 191–198 , title = Review of ''Not Knot'' and ''Supplement to Not Knot'' , volume = 24 {{citation , last = Kister , first = James M. , mr = 1176795 , title = Review of ''Not Knot'' and ''Supplement to Not Knot'' , work = MathSciNet , year = 1994 {{citation , last = Neuwirth , first = L. P. , title = Review of ''Not Knot'' and ''Supplement to Not Knot'' , work = zbMATH , zbl = 0769.57001 {{citation , last = Stewart , first = Ian , author-link = Ian Stewart (mathematician) , date = January 1994 , department = Mathematical Recreations , issue = 1 , magazine =

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...

, jstor = 24942566 , pages = 152–154 , title = Knots, links and videotape , volume = 270 Films about mathematics 1991 short films 1991 films