What is knot theory ?

[KLscilevothma]

Hi, I'll attend a lecture around 14 hours later addressed by a Nobel laureate and here's the brief description of the lecture.

"Starting from a Parlor Game, I shall show how a deep mathematical problem can be formulated in an elementary way. The steps are understandable to high school students. Applications to knot theory, word problems and to statistical mechanics are indicated."

Part of the students attending the lecture will be high school students like me. I want to do a little bit preparation before attending the lecture, so I would like to know what knot theory is. Also what is a Parlor Game? Can anyone please provide me with some elementary information about these or just give me some links with simple descriptions? Thanks.

[HallsofIvy]

Well first, it doesn't have anything to do with general physics! This might have been better in the "mathematics" section.

"Knot theory" is, in its simplest sense, just what it says: imagine tying a rope up in some complicated knot: Is it possible to tell whether that knot is or is not the same as another knot which was tied in a different way? In particular, there are a variety of "knots" which look very complicated but such that, if you pull in a certain way, they collapse into nothing (magicians love those!). All such knots are essentially the same as no knot at all. Is there any way to tell that two knots are the same?

In a more general sense, like any mathematical study, it can be applied to many things. "Complexity" whether of a road network, communications among various persons in a company or a data system can all be studied using knot theory.

I'm not certain which "parlor game" is intended (the phrase itself just means a simple game that can be played in a small room) but I suspect something like the commercial game "Twistor" in which people have to place hands and feet at certain spots (while trying NOT to pass through one another!).

[GRQC]

Originally posted by HallsofIvy
[B]
"Knot theory" is, in its simplest sense, just what it says: imagine tying a rope up in some complicated knot: Is it possible to tell whether that knot is or is not the same as another knot which was tied in a different way? In particular, there are a variety of "knots" which look very complicated but such that, if you pull in a certain way, they collapse into nothing (magicians love those!). All such knots are essentially the same as no knot at all. Is there any way to tell that two knots are the same?


To be a bit more specific: knots are *closed* curves, i.e. they are different objects from braids (which is what you would get if you knoted up a string. The "un-knot", or the identity knot, is a simple closed loop (circle). There are two basic transformations (Reidemeister moves) which you can make on knots to see whether or not they belong to a specific class.

Knot theory is ripe with topological and algebraic invariants, the most important of which are the J- and A-polynomials (computed from the number of cross-overs in the knot, etc...).

For more information, see e.g.

mathworld.wolfram.com (http://mathworld.wolfram.com/KnotTheory.html)

Also, do a search on "Hoste", who is a knot theory expert.

[KLscilevothma]

Argh, I should have noticed that knot theory belongs to the mathematics forum. [t)]

In a more general sense, like any mathematical study, it can be applied to many things. "Complexity" whether of a road network, communications among various persons in a company or a data system can all be studied using knot theory.
I suspect knot theory may have some connections with graph theory, right? Their applications are quite similar.

Thanks for the replies.

[KLscilevothma]

The lecture was pretty good. There were 6 subtopics in the lecture, they are Braid Algebra, Dirac's Game, topology, physics of a neutron, knot theory and statistical mechanics (many body problems). However he didn't have time to talk about statistical mechanics.

I think the J-polynomials and K-polynomials GRQC refered to were Jones and Kanffman polynomial, which are tools to help us classifying knots if I'm not mistaken.

Under the topic physics of a neutron, the prof. told us an experiment which verifys a QM's prediction that when a neutron rotates 360 degrees, it does not come back to itself but acquires a phase of -1. However when a neutron roates 720 degrees, it can come back to itself.

I think the most difficult part was when he talked about toplogy and introduced the geometry of SO3, I couldn't really understand what he said. What I could understand was the SO3 is some kind of operation related to rotation (a 3 X 3 matrice?).

[Hurkyl]

Elements of SO3 are rotations about the origin.

If you write these transformations in matrix notation, they are all 3x3 orthogonal matrices with determinant one.

[Janitor]

Did he make any mention of Conway polynomials? One of my heroes since my own high-school days is the mathematician John Horton Conway. In addition to knot theory, Conway also has made important contributions to group theory. Some day, if you study group theory, you may hear about the Enormous Theorem, classifying all the finite simple groups. One such group is named after Conway, its discoverer. I first read about Conway in connection with a cellular automata game he invented, called Life. I really got into trying out my own Life patterns using graph paper and pencil, in those pre- personal computer days. I tried to interest some of my buddies at school in it, but they never caught the fire the way I did.

[Janitor]

Rotation in more than two dimensions is a non-commutative operation. You probably know that multiplying square matrices is a non-commutative operation. It turns out that certain matrices can model rotation, as Hurkyl has pointed out.

A topic worth looking into some day, if you care to, would be spinors.

[GRQC]

Originally posted by KLscilevothma
\
I think the J-polynomials and K-polynomials GRQC refered to were Jones and Kanffman polynomial, which are tools to help us classifying knots if I'm not mistaken.


A-polynomial, not K. It's the Alexander polynomial. The J- is the Jones, yes.

[meteor]

Yeah, the Jones polynomial was great in the epoch of its discovery, but now there exists another still more powerful, the HOMFLY polynomial