Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

A revolution in knot theory

10.11.2011
In the 19th century, Lord Kelvin made the inspired guess that elements are knots in the "ether". Hydrogen would be one kind of knot, oxygen a different kind of knot---and so forth throughout the periodic table of elements.

This idea led Peter Guthrie Tait to prepare meticulous and quite beautiful tables of knots, in an effort to elucidate when two knots are truly different. From the point of view of physics, Kelvin and Tait were on the wrong track: the atomic viewpoint soon made the theory of ether obsolete. But from the mathematical viewpoint, a gold mine had been discovered: The branch of mathematics now known as "knot theory" has been burgeoning ever since.


This knot has Gauss code O1U2O3U1O2U3.
Credit: Graphic by Sam Nelson

In his article "The Combinatorial Revolution in Knot Theory", to appear in the December 2011 issue of the Notices of the AMS,, Sam Nelson describes a novel approach to knot theory that has gained currency in the past several years and the mysterious new knot-like objects discovered in the process.

As sailors have long known, many different kinds of knots are possible; in fact, the variety is infinite. A *mathematical* knot can be imagined as a knotted circle: Think of a pretzel, which is a knotted circle of dough, or a rubber band, which is the "un-knot" because it is not knotted. Mathematicians study the patterns, symmetries, and asymmetries in knots and develop methods for distinguishing when two knots are truly different.

Mathematically, one thinks of the string out of which a knot is formed as being a one-dimensional object, and the knot itself lives in three-dimensional space. Drawings of knots, like the ones done by Tait, are projections of the knot onto a two-dimensional plane. In such drawings, it is customary to draw over-and-under crossings of the string as broken and unbroken lines. If three or more strands of the knot are on top of each other at single point, we can move the strands slightly without changing the knot so that every point on the plane sits below at most two strands of the knot. A planar knot diagram is a picture of a knot, drawn in a two-dimensional plane, in which every point of the diagram represents at most two points in the knot. Planar knot diagrams have long been used in mathematics as a way to represent and study knots.

As Nelson reports in his article, mathematicians have devised various ways to represent the information contained in knot diagrams. One example is the Gauss code, which is a sequence of letters and numbers wherein each crossing in the knot is assigned a number and the letter O or U, depending on whether the crossing goes over or under. The Gauss code for a simple knot might look like this: O1U2O3U1O2U3.

In the mid-1990s, mathematicians discovered something strange. There are Gauss codes for which it is impossible to draw planar knot diagrams but which nevertheless behave like knots in certain ways. In particular, those codes, which Nelson calls *nonplanar Gauss codes*, work perfectly well in certain formulas that are used to investigate properties of knots. Nelson writes: "A planar Gauss code always describes a [knot] in three-space; what kind of thing could a nonplanar Gauss code be describing?" As it turns out, there are "virtual knots" that have legitimate Gauss codes but do not correspond to knots in three-dimensional space. These virtual knots can be investigated by applying combinatorial techniques to knot diagrams.

Just as new horizons opened when people dared to consider what would happen if -1 had a square root---and thereby discovered complex numbers, which have since been thoroughly explored by mathematicians and have become ubiquitous in physics and engineering---mathematicians are finding that the equations they used to investigate regular knots give rise to a whole universe of "generalized knots" that have their own peculiar qualities. Although they seem esoteric at first, these generalized knots turn out to have interpretations as familiar objects in mathematics. "Moreover," Nelson writes, "classical knot theory emerges as a special case of the new generalized knot theory."

Related to this subject are an upcoming issue of the Journal of Knot Theory and its Ramifications, devoted to virtual knot theory, and the upcoming Knots in Washington conference at George Washington University, December 2-4, 2011, which will focus on on "Categorification of Knots, Algebras, and Quandles; Quantum Computing".

Specific questions are best directed to the author, whose email address is given in the article. General questions can be directed to:

Mike Breen and Annette Emerson
AMS Public Awareness Office
Email: paoffice@ams.org
Telephone: 401-455-4000
Founded in 1888 to further mathematical research and scholarship, today the more than 30,000 member American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.
American Mathematical Society
201 Charles Street
Providence, RI 02904
401-455-4000

Mike Breen | EurekAlert!
Further information:
http://www.ams.org

More articles from Science Education:

nachricht How Humans and Machines Navigate Complex Situations
19.11.2018 | Max-Planck-Institut für Bildungsforschung

nachricht A gene activated in infant and young brains determines learning capacity in adulthood
13.11.2018 | Universitätsklinikum Hamburg-Eppendorf

All articles from Science Education >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: Self-repairing batteries

UTokyo engineers develop a way to create high-capacity long-life batteries

Engineers at the University of Tokyo continually pioneer new ways to improve battery technology. Professor Atsuo Yamada and his team recently developed a...

Im Focus: Quantum Cloud Computing with Self-Check

With a quantum coprocessor in the cloud, physicists from Innsbruck, Austria, open the door to the simulation of previously unsolvable problems in chemistry, materials research or high-energy physics. The research groups led by Rainer Blatt and Peter Zoller report in the journal Nature how they simulated particle physics phenomena on 20 quantum bits and how the quantum simulator self-verified the result for the first time.

Many scientists are currently working on investigating how quantum advantage can be exploited on hardware already available today. Three years ago, physicists...

Im Focus: Accelerating quantum technologies with materials processing at the atomic scale

'Quantum technologies' utilise the unique phenomena of quantum superposition and entanglement to encode and process information, with potentially profound benefits to a wide range of information technologies from communications to sensing and computing.

However a major challenge in developing these technologies is that the quantum phenomena are very fragile, and only a handful of physical systems have been...

Im Focus: A step towards probabilistic computing

Working group led by physicist Professor Ulrich Nowak at the University of Konstanz, in collaboration with a team of physicists from Johannes Gutenberg University Mainz, demonstrates how skyrmions can be used for the computer concepts of the future

When it comes to performing a calculation destined to arrive at an exact result, humans are hopelessly inferior to the computer. In other areas, humans are...

Im Focus: Recording embryonic development

Scientists develop a molecular recording tool that enables in vivo lineage tracing of embryonic cells

The beginning of new life starts with a fascinating process: A single cell gives rise to progenitor cells that eventually differentiate into the three germ...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

VideoLinks
Industry & Economy
Event News

SEMANTiCS 2019 brings together industry leaders and data scientists in Karlsruhe

29.04.2019 | Event News

Revered mathematicians and computer scientists converge with 200 young researchers in Heidelberg!

17.04.2019 | Event News

First dust conference in the Central Asian part of the earth’s dust belt

15.04.2019 | Event News

 
Latest News

Discovering unusual structures from exception using big data and machine learning techniques

17.05.2019 | Materials Sciences

ALMA discovers aluminum around young star

17.05.2019 | Physics and Astronomy

A new iron-based superconductor stabilized by inter-block charger transfer

17.05.2019 | Materials Sciences

VideoLinks
Science & Research
Overview of more VideoLinks >>>