Drorbn at 10:46, 24 June 2008

2008-06-24T10:46:17Z

← Older revision		Revision as of 06:46, 24 June 2008
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	eliminate all crossings using the rule:		eliminate all crossings using the rule:
	<br><br>		<br><br>
	<center>		<center>[[Image:Kauffman.png\|200px]]
	http://katlas.math.toronto.edu/drorbn/images/6/64/Kauffman.png
	<br>		<br>
	</center>		</center>

69.197.142.199: /* Jones Polynomial */

2006-10-22T02:30:21Z

Jones Polynomial

← Older revision		Revision as of 22:30, 21 October 2006
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	In the right hand side, the first bracket is called the <i>0-smoothing</i> and the second		In the right hand side, the first bracket is called the <i>0-smoothing</i> and the second
	is called the <i>1-smoothing</i>. To calculate the Kauffman bracket we must sum over all		is called the <i>1-smoothing</i>. To calculate the Kauffman bracket we must sum over all
	possible smoothings. For instance, for the ~~trifoil~~, we 2<sup>3</sup> = 8 summands, one of which will		possible smoothings. For instance, for the trefoil, we 2<sup>3</sup> = 8 summands, one of which will
	be:		be:
	<br><br>		<br><br>

Drorbn at 00:07, 21 September 2006

2006-09-21T00:07:29Z

← Older revision		Revision as of 20:07, 20 September 2006
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	The Mathematica notebook we wrote is [http://katlas.math.toronto.edu/svn/06-1350/ComputingTheJonesPolynomial.nb here].		The Mathematica notebook we wrote is [http://katlas.math.toronto.edu/svn/06-1350/ComputingTheJonesPolynomial.nb here].


			===Jones Polynomial===

			The simplest way to define the <i>Jones polynomial</i> is via the <i>Kauffman bracket</i>. The idea is to
			eliminate all crossings using the rule:
			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/6/64/Kauffman.png
			<br>
			</center>
			<br>
			In the right hand side, the first bracket is called the <i>0-smoothing</i> and the second
			is called the <i>1-smoothing</i>. To calculate the Kauffman bracket we must sum over all
			possible smoothings. For instance, for the trifoil, we 2<sup>3</sup> = 8 summands, one of which will
			be:
			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/8/8b/Trifoil-smoothing.png
			<br>
			</center>
			<br>
			Each summand will have no crossing and thus will be a union of (possibly nested) unknots.
			We define the bracket polynomial of k unknots to be d<sup>k-1</sup> for some indeterminate d. Our hopes
			that our polynomial in ℤ[d,A,B] will be an invariant under the Reidmeister moves. We first
			verify R2.
			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/c/cd/Jones-r2.png
			<br>
			</center>
			<br>

			Collecting like terms and comparing we find AB = 1 and A<sup>2</sup> + B<sup>2</sup> + dAB = 0. Thus, we must
			have B = A<sup>-1</sup> and d = -(A<sup>2</sup> + A<sup>2</sup>). Things are looking bad, we still have two moves to verify
			and we already lost two of our variables.
			<br><br>
			We now verify R3. For this we remark that
			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/f/fb/Jones-r3.png
			<br>
			</center>
			<br>

			The two diagrams with coefficients B coincide; and the diagrams with coefficients A differ by
			two R2 moves. Now we verify R1:
			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/9/99/Jones-r1.png
			<br>
			</center>
			<br>
			The right hand side evaluates to A + A<sup>-1</sup>(-A<sup>2</sup> -A<sup>-2</sup>) = -A<sup>3</sup> of the desired. This is
			unfortunate. One could salvage something by taking A to be one of the cube roots of -1;
			however, the right way out of this is to define another invariant which fails in the exactly the
			same way, and multiply it with the bracket polynomial.

			===Writhe===

			The invariant we are looking for is called the <i>writhe</i>. If D is a diagram of an oriented knot, we
			define
			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/8/88/Mathtext1.png
			</center>
			<br>
			We choose +1 if the crossing is positive (the overpass goes over the underpass from left to right) and -1 if the crossing is negative (otherwise):
			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/3/38/Crossings.png
			<br>
			</center>
			<br>

			Lets have an example. Notice that the orientation of the knot is actually irrelevant.

			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/c/c5/Writhe-example.png
			<br>
			</center>
			<br>
			We now show that the writhe is also invariant under R2 and R3, under R1 we gain ±1
			depending which way we apply it. In R2, the two crossings have opposite signs; and in R3, the
			diagonal crossing doesn’t change sign and the other two are reversed.

			<br><br>
			<center>
			http://katlas.math.toronto.edu/drorbn/images/f/f9/Writhe-reid.png
			<br>
			</center>
			<br>

			It follows that 〈D〉⋅(-A<sup>-3</sup>)<sup>w(d)</sup> is a knot invariant. This is a polynomial in A, we now
			substitute q<sup>
			http://katlas.math.toronto.edu/drorbn/images/2/23/Mathtext2.png</sup> for A and call this the <i>Jones polynomial</i> (which strictly speaking, is not really a
			polynomial).

Drorbn at 20:05, 14 September 2006

2006-09-14T20:05:07Z

New page

{{06-1350/Navigation}}

The Mathematica notebook we wrote is [http://katlas.math.toronto.edu/svn/06-1350/ComputingTheJonesPolynomial.nb here].

06-1350/Class Notes for Thursday September 14 - Revision history

Drorbn at 10:46, 24 June 2008

69.197.142.199: /* Jones Polynomial */

Drorbn at 00:07, 21 September 2006

Drorbn at 20:05, 14 September 2006