07-1352/Class Notes for February 6 - Revision history

Drorbn at 20:33, 27 February 2007

2007-02-27T20:33:46Z

← Older revision		Revision as of 16:33, 27 February 2007
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	{{07-1352/Navigation}}		{{07-1352/Navigation}}

	~~[[Image:~~07-1352 Kontsevich Integral~~.png\|center\|480px]]~~		{{07-1352/Schematics of the Kontsevich Integral}}

	{{Equation*\|<math>Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}</math>}}


	==Today's (and Next Weeks') Agenda==		==Today's (and Next Weeks') Agenda==

Drorbn at 21:34, 6 February 2007

2007-02-06T21:34:45Z

← Older revision		Revision as of 17:34, 6 February 2007
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	{{07-1352/Navigation}}		{{07-1352/Navigation}}
	{{In Preparation}}

	[[Image:07-1352 Kontsevich Integral.png\|center\|480px]]		[[Image:07-1352 Kontsevich Integral.png\|center\|480px]]

Drorbn: /* Today's (and Next Weeks') Agenda */

2007-02-06T19:08:37Z

Today's (and Next Weeks') Agenda

← Older revision		Revision as of 15:08, 6 February 2007
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	* The Kontsevich integral for unframed knots.		* The Kontsevich integral for unframed knots.
	** Convergence.		** Convergence.
			** Factorization.
	** Invariance under horizontal deformations:		** Invariance under horizontal deformations:
	*** Using connections and curvature.		*** Using connections and curvature.

Drorbn at 19:07, 6 February 2007

2007-02-06T19:07:47Z

← Older revision		Revision as of 15:07, 6 February 2007
Line 1:		Line 1:
	{{07-1352/Navigation}}		{{07-1352/Navigation}}
	{{In Preparation}}		{{In Preparation}}

			[[Image:07-1352 Kontsevich Integral.png\|center\|480px]]

		⚫	{{Equation*\|<math>Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}</math>}}


	==Today's (and Next Weeks') Agenda==		==Today's (and Next Weeks') Agenda==
	* A bit about genus, finite type invariants and the Alexander polynomial.		* A bit about genus, finite type invariants and the Alexander polynomial.

			* The Kontsevich integral for unframed knots.
⚫	* The Kontsevich integral for unframed knots: {{Equation*\|<math>Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}</math>}}
	** Convergence.		** Convergence.
	** Invariance under horizontal deformations:		** Invariance under horizontal deformations:

Drorbn: /* Today's (and Next Weeks') Agenda */

2007-02-06T15:46:14Z

Today's (and Next Weeks') Agenda

← Older revision		Revision as of 11:46, 6 February 2007
Line 6:		Line 6:

	* The Kontsevich integral for unframed knots: {{Equation*\|<math>Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}</math>}}		* The Kontsevich integral for unframed knots: {{Equation*\|<math>Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}</math>}}
			** Convergence.
	** Invariance under horizontal deformations:		** Invariance under horizontal deformations:
	*** Using connections and curvature.		*** Using connections and curvature.

Drorbn: /* Today's (and Next Weeks') Agenda */

2007-02-06T15:45:51Z

Today's (and Next Weeks') Agenda

← Older revision		Revision as of 11:45, 6 February 2007
Line 5:		Line 5:
	* A bit about genus, finite type invariants and the Alexander polynomial.		* A bit about genus, finite type invariants and the Alexander polynomial.

		⚫	* The Kontsevich integral for unframed knots: {{Equation*\|<math>Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}</math>}}
	* The Kontsevich integral for unframed knots:
⚫	{{Equation*\|<math>Z_0(K)=\ \ \ \ \ \ \ \ \ \ \int\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{m;\ t_1<\ldots<t_m;\ P=\{(z_i,z'_i)\}} \frac{(-1)^{\#P_{\downarrow}}}{(2\pi i)^m} D_P \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}</math>}}
	** Invariance under horizontal deformations:		** Invariance under horizontal deformations:
	*** Using connections and curvature.		*** Using connections and curvature.

Drorbn at 15:44, 6 February 2007

2007-02-06T15:44:39Z

@@ Line 2: / Line 2: @@
 {{In Preparation}}
-==Today's Agenda==
+==Today's (and Next Weeks') Agenda==
 * A bit about genus, finite type invariants and the Alexander polynomial.
 ==Genus and the Alexander Polynomial==
@@ Line 35: / Line 56: @@
 +4z^2, 1+6z^2, 1+7z^2, 1-2z^2, 1-4z^2, 1-6z^2, 1+5z^2, 1+8z^2, 1+10z^2, 1+9z^2,
 -2z^2, 1-5z^2}</nowiki>}}

Drorbn at 15:05, 6 February 2007

2007-02-06T15:05:23Z

@@ Line 12: / Line 12: @@
 Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.}}
-{{In|n=2|Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]}}
+{{In|n=2|in=Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]}}
 {{Message|KnotTheory::credits: The 3-genus data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.}}
@@ Line 23: / Line 23: @@
  Knot[11, Alternating, 343], Knot[11, Alternating, 362], Knot[11, Alternating, 363],
  Knot[11, NonAlternating, 139], Knot[11, NonAlternating, 141]}</nowiki>}}

Drorbn at 14:56, 6 February 2007

2007-02-06T14:56:48Z

New page

{{07-1352/Navigation}}
{{In Preparation}}

==Today's Agenda==
* A bit about genus, finite type invariants and the Alexander polynomial.

==Genus and the Alexander Polynomial==

{{In|n=1|in=<< KnotTheory`}}

{{Print|Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.}}

{{In|n=2|Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]}}

{{Message|KnotTheory::credits: The 3-genus data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.}}

{{Message|KnotTheory::loading: Loading precomputed data in IndianaData`.}}

{{Out|n=2|out=<nowiki>{Knot[3, 1], Knot[4, 1], Knot[5, 2], Knot[6, 1], Knot[7, 2],
Knot[7, 4], Knot[8, 1], Knot[8, 3], Knot[9, 2], Knot[9, 5], Knot[9, 35],
Knot[9, 46],Knot[10, 1], Knot[10, 3], Knot[11, Alternating, 247],
Knot[11, Alternating, 343], Knot[11, Alternating, 362], Knot[11, Alternating, 363],
Knot[11, NonAlternating, 139], Knot[11, NonAlternating, 141]}</nowiki>}}