User:Kwantus/fubar

i'll write articles here until they're semifit to submit. Then i'm not (less likely) tripping over other editors and getting "use complete sentences" remarks =p and to *@$^ with history/biography, i'll stick to maths.

...well, I shoulda seen THAT coming. Stole the page before even Alexander was ready...

A knot polynomial is a particular knot invariant. It can also be viewed as a hash function. The coefficient are the important part; The polynomial is not meant to be evaluated, but merely a way of indexing a set of numbers.

Justification

Why bother? For one thing, a polynomial is much easier to communicate than a knot, or even a drawing of a knot.

For another, it's far easier to compare two polynomials for equivalence than two knots. If the knot-to-polynomial mapping can be calculated from elements of the knot and is sufficiently discriminating, two complicated knots can be checked for identity algorithmically.

The latter condition is the harder to satisfy.

It's also possible that elementary polynomial operations could turn out to have analogues in knot manipulations. (For instance, the Alexander polynomial of the sum of two knots is the product of the Alexander polynomials of the knots.)

Of course polynomials are not the only things available; another hash on a knot is the least number of crossings needed in a diagram of it. But that does not discriminate knots at all well. Another hash is the Fukuhara/O'Hara energy, which discriminate fairly well—an energy E corresponds to at most 0.264×1.658^E knots—but is hard to compute.[1] actually it looks like E increases rather rapidly, wrt to crossings, so "rather well" may be optimistic Yet another hash is the genus of the knot, the least genus of all its Siefert surfaces.

Alexander-Conway polynomial

1960s looks like the same thing with a less abstruse construction[2]

Jones Polynomial

1984 Vaughn F. R. Jones Can distinguish a knot from its reflection

Example 3 (by skein relations)

Another shortcut to the Alexander polynomials comes via skein relations (SRs), thanks to Conway. As a recursion, it's not quite so direct as the other way, but the other knot polynomials are defined through SRs; it is well to learn SRs here where a distinct method is available as a check.

Using the notations of the skein relations page: Let function P from diagrams to Laurent polynomials (in x) be such that $P({\rm {unknot}})=1$ and a triple of skein-relation diagrams $(D_{-},D_{0},D_{+})$ satisfies the equation

P(D_{-})=(x^{-1/2}-x^{1/2})P(D_{0})+P(D_{+})

Then P maps a knot to one of its Alexander polynomials.

We'll now work the "figure-8" knot. For convenience let $A=x^{-1/2}-x^{1/2}$ . Patching the eastern crossing gives

P(File:Skein-relation-figure8-minus-sm.png) = A×P(File:Skein-relation-figure8-zero-sm.png) + P(File:Skein-relation-figure8-plus-sm.png)

homfly or homflypt polynomial

1985 Jones' discovery inspired a hunt for a structure above his polynomial and Alexander's. Five collaborations found one essentially simultaneously; four published jointly rather than fight over priority. "Homfly" is derived from their initials: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter. Some authors write "homflypt" to include the pair of Poles, Prztycki and Traczyk, who got left out due to slow mail service.

WTF Siefert matrices?

Ref

Ivars Peterson the Mathematical Tourist (1988) p70–80

[3]