The Basics of Knot Theory
A Mathematical Method of Deciphering Loops, Tangles and Braids
Mar 17, 2008
A basic knowledge of knots is invaluable for sailors. It is crucial for fisherman. It can save the life of a mountain climber.
But what good are they to a mathematician?
According to famed German Mathematician Carl Friederich Gauss (perhaps most famous for his contributions to the geometrical coordinates used in Einstein’s general theory of relativity) and all of those who would follow him in his studies, they are very important.
The History of Knot Theory
There are interesting coincidences one might notice when investigating the history of the knot theory. The most striking of these is the fact that a surprising number of the mathematicians who worked in this field have also proven themselves crucial to the progress of theoretical physics over the years.
One might not be as surprised after learning a bit about knot theory, however, as it possesses many of the same characteristic problems that lie at the bottom of physics, and uses many similar methods of problem solving. In addition, there have been several physical theories over the years which viewed certain aspects of reality in terms of knots (specifically one which viewed atoms as tiny knots in a swirling vortex).
Such figures who fit into this category are Gauss (who has already been mentioned, and whose work was continued by his student, Johann Listing), Sir William “Lord Kelvin” Thompson (whose legacy survives in the temperature scale that bears his name), James Clerk Maxwell (the “King” of electromagnetism in the nineteenth century) and Henri Poincare (famed French Mathematician known for his work on the “three body problem,” and his additions to the theory of relativity).
Knots as Topology
In mathematics, topology is the study of geometrical space; how it is utilized and how it can be described using systems of numbers, equations and functions. One might say that it is the mathematical equivalent of the work of M.C. Escher.
In knot theory, mathematicians seek to better understand complicated closed loops (with no dangling ends). Most often these knots are studied on paper, but most of them can also be physically made, simply by tying a knot into a rope or string and then tying the ends together.
Knot theorists attempt to determine the nature of the knot by describing other identical knots (an identical knot is one which the first knot can be exactly transformed into without cutting or breaking it – it must have the exact same pattern of cross-overs and –unders), by discovering the knot’s simplest form (many knots appear more complex than they actually are, and can be simplified by shifting the knot around).
In the end, though, knot theory attempts to classify as many forms of knots as possible, and in so doing acquire precise mathematical understandings for them all.
Applications of Knot Theory
Knot theory has many applications, not many of which are obvious on the outset.
To mathematicians, knot theory provides a unique blending of geometry, algebra, topology, set theory, number theory and other branches as well. It has proven to be an effective stimuli in prodding certain areas of mathematics forward toward solving brand new forms of problems.
In physics and chemistry (besides the ill-fated attempt to define atoms as knots in swirling vortices during the nineteenth century), knot theory has been used to understand complex molecules such as DNA and other long-chain polymers which through microscopic analysis are shown to tie themselves up into knots, which can be defined and explained using knot theory.
In certain realms of modern physics as well, such as in the theory of “quantum gravity,” knots play an important role, as well as in some variations of string theory (for reasons which should be rather obvious).
Knot theory is still being worked on at this point, and is in no way a “closed” science. Surely there will be many more improvements upon the theory in the coming years, and many more applications that science will find for these remarkable advancements.
References:
“Knot Theory.” Mega Math.
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