Some Useful Links: Knots in Physics and Number Theory

In modern times, “knots” have been important objects of study in mathematics. These “knots” are akin to the ones we encounter in ordinary life, except that they don’t have loose ends. For a better idea of what I mean, consider the following picture of what is known as a “trefoil knot“:

250px-TrefoilKnot_01.svg

More technically, a knot is defined as the embedding of a circle in 3-dimensional space. For more details on the theory of knots, the reader is referred to the following Wikipedia pages:

Knot on Wikipedia

Knot Theory on Wikipedia

One of the reasons why knots have become such a major part of modern mathematical research is because of the work of mathematical physicists such as Edward Witten, who has related them to the Feynman path integral in quantum mechanics (see Lagrangians and Hamiltonians).

Witten, who is very famous for his work on string theory (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) and for being the first, and so far only, physicist to win the prestigious Fields medal, himself explains the relationship between knot theory and quantum mechanics in the following article:

Knots and Quantum Theory by Edward Witten

But knots have also appeared in other branches of mathematics. For example, in number theory, the result in etale cohomology known as Artin-Verdier duality states that the integers are similar to a 3-dimensional object in some sense. In particular, because it has a trivial etale fundamental group (which is kind of an algebraic analogue of the fundamental group discussed in Homotopy Theory and Covering Spaces), it is similar to a 3-sphere (recall the common but somewhat confusing notation that the ordinary sphere we encounter in everyday life is called the 2-sphere, while a circle is also called the 1-sphere).

Note: The fact that a closed 3-dimensional space with a trivial fundamental group is a 3-sphere is the content of a very famous conjecture known as the Poincare conjecture, proved by Grigori Perelman in the early 2000’s.  Perelman refused the million-dollar prize that was supposed to be his reward, as well as the Fields medal.

The prime numbers, because their associated finite fields have one cover for every integer, are like circles, and recalling the definition of knots mentioned above, are therefore like knots on this 3-sphere. This analogy, originally developed by David Mumford and Barry Mazur, is better explained in the following post by Lieven le Bruyn on his blog neverendingbooks:

What is the Knot Associated to a Prime on neverendingbooks

Finally, given what we have discussed, could it be that knot theory can “tie together” (pun intended) physics and number theory? This is the motivation behind the new subject called “arithmetic Chern-Simons theory” which is introduced in the following paper by Minhyong Kim:

Arithmetic Chern-Simons Theory I by Minhyong Kim

Of course, it must also be clarified that this is not the only way by which physics and number theory are related. It is merely another way, a new and not yet thoroughly explored one, by which the unity of mathematics manifests itself via its many different branches helping one another.

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One thought on “Some Useful Links: Knots in Physics and Number Theory

  1. Pingback: A share – cfldgeometry

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