Math Bulletin
Recent News |  Archives |  Tags |  Newsletter |  Message Board/Forum |  About |  Links |  Subscribe to AgingBulletin.com RSS Feed Subscribe


More Articles
Improvements in fuel cell designImprovements in fuel cell design

Rediscovering Venus to find faraway earthsRediscovering Venus to find faraway earths

Archaeologists discover bronze remains of Iron Age chariotArchaeologists discover bronze remains of Iron Age chariot

Researchers resolve the Karakoram glacier anomaly, a cold case of climate scienceResearchers resolve the Karakoram glacier anomaly, a cold case of climate science

Fish tale: New study evaluates antibiotic content in farm-raised fishFish tale: New study evaluates antibiotic content in farm-raised fish

New 3-D display technology promises greater energy efficiencyNew 3-D display technology promises greater energy efficiency

Researchers break nano barrier to engineer the first protein microfiberResearchers break nano barrier to engineer the first protein microfiber

Magnetic mirrors enable new technologies by reflecting light in uncanny waysMagnetic mirrors enable new technologies by reflecting light in uncanny ways

Structure of an iron-transport protein revealedStructure of an iron-transport protein revealed

First step: From human cells to tissue-engineered esophagusFirst step: From human cells to tissue-engineered esophagus

Lift weights, improve your memoryLift weights, improve your memory

Spiders: Survival of the fittest groupSpiders: Survival of the fittest group

Autophagy helps fast track stem cell activationAutophagy helps fast track stem cell activation

Myelin vital for learning new practical skillsMyelin vital for learning new practical skills

More physical activity improved school performanceMore physical activity improved school performance

Around the world in 400,000 years: The journey of the red foxAround the world in 400,000 years: The journey of the red fox

Engineering new vehicle powertrainsEngineering new vehicle powertrains

Active aging is much more than exerciseActive aging is much more than exercise

Study: New device can slow, reverse heart failureStudy: New device can slow, reverse heart failure

Are the world's religions ready for ET?Are the world's religions ready for ET?

Gut bacteria, artificial sweeteners and glucose intoleranceGut bacteria, artificial sweeteners and glucose intolerance

Recreating the stripe patterns found in animals by engineering synthetic gene networksRecreating the stripe patterns found in animals by engineering synthetic gene networks

Hold on, tiger momHold on, tiger mom

Nature's designs inspire research into new light-based technologiesNature's designs inspire research into new light-based technologies

Missing piece found to help solve concussion puzzleMissing piece found to help solve concussion puzzle

Biologists delay the aging process by 'remote control'Biologists delay the aging process by 'remote control'

Geography matters: Model predicts how local 'shocks' influence U.S. economyGeography matters: Model predicts how local 'shocks' influence U.S. economy

Identified for the first time what kind of explosive has been used after the detonationIdentified for the first time what kind of explosive has been used after the detonation

Copied from nature: Detecting software errors via genetic algorithmsCopied from nature: Detecting software errors via genetic algorithms

A revolution in knot theory (11/11/2011)

Tags:
knots
This knot has Gauss code O1U2O3U1O2U3. -  Graphic by Sam Nelson.
This knot has Gauss code O1U2O3U1O2U3. - Graphic by Sam Nelson.

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.

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."

Note: This story has been adapted from a news release issued by the American Mathematical Society

Post Comments:

Search
New Articles
Risk analysis for a complex world

Recommendation theory

Network physicists create model to predict traffic patterns

Back to basicsBack to basics

Raising cryptography's standards

Spy master prays for the return of Alan TuringSpy master prays for the return of Alan Turing

Saving lots of computing capacity with a new algorithm

Study will teach algebra via student-authored stories that draw on their own interests

'Data smashing' could unshackle automated discovery

The mathematics behind the Ebola epidemic

Laying the groundwork for data-driven scienceLaying the groundwork for data-driven science

New frontier in error-correcting codes

Untangling how cables coil

Adding uncertainty to improve mathematical models

How things coil



Archives
November 2014
October 2014
September 2014
August 2014
July 2014
June 2014
May 2014
April 2014
March 2014
February 2014
January 2014
December 2013
November 2013
October 2013
September 2013
August 2013
July 2013
June 2013
May 2013
April 2013
March 2013
February 2013
January 2013
December 2012
November 2012
October 2012
September 2012
August 2012
July 2012
June 2012
May 2012
April 2012
March 2012
February 2012
January 2012
December 2011
November 2011
October 2011
September 2011
August 2011
July 2011
June 2011
May 2011
April 2011
March 2011


Science Friends
Agriculture News
Astronomy News
Sports Tech
Biology News
Biomimicry Science
Cognitive Research
Chemistry News
Tissue Engineering
Cancer Research
Cybernetics Research
Electonics Research
Fossil News
Forensics Report
Genetic Archaeology
Genetics News
Geology News
Microbiology Research
Nanotech News
Physics News
Parenting News


  Archives |  Submit News |  Advertise With Us |  Contact Us |  Links
Use of this site constitutes acceptance of our Terms of Service and Privacy Policy. All contents © 2000 - 2015 Web Doodle, LLC. All rights reserved.