How Does the Jones Polynomial Relate to Different Knot Configurations?

[marcus]

I don't see immediately how I can talk about knot theory
here at PF because I can't easily draw pictures. this is a test.
I'll try to discuss the (Vaughn) Jones polynomial by first
drawing 3 simple PARTS of knots and naming them and
then trying to get some results. Like about the trefoil.
this is following the AMS knot website as best I can with limited graphics.

Here are downover and upover and nocross

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/   \

The skein relation talks about three knots that look exactly the same except at one location, where one has downover, one has upover, and one has nocross.

the Jones of the three knots are related as follows


t^-1 Jones(downover) - t Jones(upover) = (t^1/2 - t^-1/2) Jones(nocross)

If you do a change of variable and substitute 1/t for t you will see that the roles of downover and upover can be exchanged.

THE JONES AXIOM ----- Jones(unknot) = 1

An unknot is just a circle or some equivalent simple loop. Twist an unknot and lay one lobe inside the other. Then it looks like two concentric circles except for a crossing. Call that unknot'

Twist it the other way and again lay one lobe inside the other. Again two concentric circles except for (the other kind of) crossing. Call that unknot''. By the mighty Jones axiom we have that:

J(unknot') = J(unknot'') = 1

Notice that if in either of these two you were to replace the solitary crossing by a "nocross" then you would get two concentric circles or anyway two concentric unknots.

BUT BY THE VERITABLE SKEIN RELATION we have that:

t^-1 J(unknot') - t J(unknot'') = (t^1/2 - t^-1/2) J(two concentric unknots)

Now we apply the AXIOM and on the LHS have simply t^-1 - t

We are going to SOLVE for J(two concentric unknots)

t^-1 - t = (t^1/2 - t^-1/2)J(two concentric unknots)

But (t^-1 - t )/(t^-1/2 - t^1/2) = (t^-1/2 + t^1/2)

So, not losing track of the minus sign, we have

J(two concentric unknots) = - t^1/2 - t^-1/2

This fact about the Jonesj polynomial evaluated on two unknots (they don't have to be concentric, simply unlinked) is a useful fact.
We can use it to find out what Jones of a trefoil knot is.

 

[marcus]

So we just got this result that the Jones polynomial of any two unlinked loops----its a topological thing, they don't have to be concentric----is

J(two separate unknots) = - t^1/2 - t^-1/2

How about two LINKED loops? The skein relation works here too:

By drawing a picture you can see that changing one crossing from upover to downover or vice versa will separate the two---unlink them. Also replacing that crossing by a nocross will just produce one long unknot.

t^-1 Jones(two linked unknots) - t Jones(two unlinked unknots) = (t^1/2 - t^-1/2) Jones(unknot)

Substitute 1 for J(unknot) and substitute
- t^1/2 - t^-1/2
for J(two unlinked loops) and we have:

t^-1 Jones(two linked unknots) - t (- t^1/2 - t^-1/2) = (t^1/2 - t^-1/2)

t^-1 Jones(two linked unknots) + t^3/2 + t^1/2 = t^1/2 - t^-1/2

Jones(two linked unknots) + t^5/2 + t^3/2 = t^3/2 - t^1/2

Jones(two linked unknots) = - t^5/2 - t^1/2

A chain with just two links has that polynomial.

The Jones polynomial is a topological invariant and key to a lot of knot theory. Witten was awarded the Fields medal for discovering a relation between the Jones polynomial and Quantum Field Theory. My understanding is he won it for his work in QFT, of which the connection to knots is maybe the most far-reaching. Relating QFT to knots happened in 1989 or so, I think.

We should be able to get the invariant of the trefoil next.

 

[marcus]

J(two separate unknots) = - t^1/2 - t^-1/2

J(any unknot) = 1

J(two linked unknots) = - t^5/2 - t^1/2


The way a trefoil knot is usually drawn it has 3 lobes and 3 crossings.

If you pick one crossing and replace it by the other kind of crossing you get an unknot (draw a picture to see)

And if you replace the crossing by a nocross you get two linked loops----linked unknots!

So one application of the skein relation should do it:


t^-1 Jones(trefoil) - t Jones(one long unknot) = (t^1/2 - t^-1/2) Jones(two linked unknots)

Jones(trefoil) - t² Jones(one long unknot) = (t^3/2 - t^1/2) Jones(two linked unknots)

Then applying the "axiom" to the one long unknot and what we found out already about two links:

Jones(trefoil) - t² = (t^3/2 - t^1/2) (- t^5/2 - t^1/2)

Jones(trefoil) - t² = - t⁴ + t³ + t - t²

Jones(trefoil) = - t⁴ + t³ + t

I should go to the AMS site and check this. Am wondering if there are two inequivalent kinds of trefoil and maybe this is just the polynomial for one of them.

Yeah, I looked. What I did was the RIGHTHAND trefoil, and it is correct. Jones studied oriented knots where there is an arrow along the rope. If you grasp the rope with your right hand, thumb in the direction of the arrow, then in the righthand trefoil all the crossings are with your curl of your fingers.

Jones(righthand trefoil) = - t⁴ + t³ + t