Knot Theory 3: Simplify, Simplify!

The Story So Far: we've defined knots and links, and identified what it means for two knots to be equivalent. To study them more carefully, we've begun looking at plane representations of knots, and seen what that notion of equivalence translates to for plane representations: manipulation of the arcs between crossing points ("plane isotopy") and certain manipulations of the crossing points themselves ("Reidemeister moves").

Here's the problem. if two knots (or links) are equivalent, it's possible to manipulate any given plane representation of one, as just mentioned, so that it looks just like any given plane representation of the other. This is great, if you can come up with such a manipulation, but what if you can't? The best you can say, in that case, is that you don't know whether the knots are equivalent. We need a different, complementary approach, that of invariants. An invariant is a way of assigning, to any given knot or link, something - usually an algebraic object - in such a way that if two knots are equivalent the invariant assigns the same object to both. If you have an invariant, you can apply it to each of two knots; if the results are different, that tells you that the knots are not equivalent. (See the difference? The manipulation test is a way of showing that two knots are equivalent; invariants provide a way of showing that they aren't. The best of all situations is to have a complete set of invariants - one or more invariants with the property that, if each of them gives the same results to Knot1 and Knot2, then Knot1 and Knot2 are equivalent. Unfortunately, we don't have a complete set of invariants for knots as yet.)

To verify that a proposed invariant actually is an invariant, all we have to do is show that it isn't changed by plane isotopy or by Reidemeister moves. Since plane isotopy only affects the arcs in between crossing points, we can save time by looking at possible invariants which only look at the crossing points; then we only have to worry about the Reidemeister moves.

There are a number of known invariants for knots, but I'm only going to talk about one of them, the ( Jones polynomial. )

There are several other known invariants, all computed in similar ways; I hope the description under the cut gives some idea of the flavor of this sort of investigation.

Ain't Technology Grand?

I subscribe to Dish Network. The satellite box is controlled, of course, by an IR remote.

As of about ten minutes ago, the remote is not working. There are several possible explanations; the most benign has to do with dead batteries. The remote requires four AAA batteries, and for all I know the failure of one of them suffices to render the remote inert. I have two spares; that is, I have six AAA batteries, of which an unknown number are dead. There are fifteen different ways of choosing four batteries from a set of six; I could, I suppose, test all fifteen to see if I can find a subset that works. Success would, of course, be satisfying, but failure would tell me nothing, since I don't know how many (if any) of the batteries are dead. (Yes, I'm aware of the existence of battery testers. I don't have any handy.) All that, of course, assumes that it is the battery and not the IR source or some other component.

What this means is that, for the moment, I can't change channels. I'm stuck on the local WB affiliate. BO-ring. Fortunately, all of the series that I've been watching have finished their seasons. But there are other shows...

Fortunately again, I received a nice haul of books today. From the Library of America (a very nice book club, I must say), I received a volume of Louisa May Alcott: Little Women, Little Men, and Jo's Boys; I've never read any of her work, and had been idly toying with looking into them, so this is serendipitous. Also, part of my Amazon order arrived: Jasper Fforde's Lost in a Good Book, Susanna Clarke's Jonathan Strange and Mr. Norrell, Dorothy Dunnett's Scales of Gold (the fourth House of Niccolo book), and A Pirate of Exquisite Mind by Diana and Michael Preston. (The last is a biography of William Dampier.)

I finished the Chanur series, and am now reading Richard Fortey's Earth: An Intimate History and rereading M. Z. Bradley's Sharra's Exile. (I've never been much of a Bradley fan. The Heritage of Hastur is excellent, but none of her other works that I've read - Thendara House, The Shattered Chain, and Sharra's Exile - have impressed me. However, I do happen to have a copy of the last, and decided to give it another shot. Just a whim.)

[Addendum: the remote is working again, for no discernible reason. Damn gremlins.]
[Further Addendum: the reason the remote appeared not to be working was that I had done something stupid with another remote. Damn multiple wands.]