I dont see immediately how I can talk about knot theory(adsbygoogle = window.adsbygoogle || []).push({});

here at PF because I cant 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

Code (Text):

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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 dont have to be concentric, simply unlinked) is a useful fact.

We can use it to find out what Jones of a trefoil knot is.

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# Knots and Jones polynomial

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