07-1352/Class Notes for February 6: Difference between revisions
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{{07-1352/Navigation}} |
{{07-1352/Navigation}} |
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{{In Preparation}} |
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[[Image:07-1352 Kontsevich Integral.png|center|480px]] |
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{{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>}} |
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{{07-1352/Schematics of the Kontsevich Integral}} |
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==Today's (and Next Weeks') Agenda== |
==Today's (and Next Weeks') Agenda== |
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* The Kontsevich integral for unframed knots. |
* The Kontsevich integral for unframed knots. |
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** Convergence. |
** Convergence. |
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** Factorization. |
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** Invariance under horizontal deformations: |
** Invariance under horizontal deformations: |
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*** Using connections and curvature. |
*** Using connections and curvature. |
Latest revision as of 15:33, 27 February 2007
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Today's (and Next Weeks') Agenda
- A bit about genus, finite type invariants and the Alexander polynomial.
- The Kontsevich integral for unframed knots.
- Convergence.
- Factorization.
- Invariance under horizontal deformations:
- Using connections and curvature.
- Using Stokes' Theorem.
- Sliding critical points.
- canceling critical points and the correction factor .
- The Kontsevich integral of the unknot.
- Re-introducing framing:
- Using counter-terms in the original Kontsevich integral.
- Using further algebra on :
- The Milnor-Moore Theorem.
- Using and .
- Unzipping a circle, the error terms and and their cancellation following [Le_Murakami_97].
- The extension to knotted trivalent graphs following [Murakami_Ohtsuki_97].
- The delete, unzip and connected sum operations.
Genus and the Alexander Polynomial
In[1]:=
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<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[2]:=
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Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]
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KnotTheory::credits: The 3-genus data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[2]=
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{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]}
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In[3]:=
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Conway[#][z] & /@ Ks
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[3]=
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{1+z^2, 1-z^2, 1+2z^2, 1-2z^2, 1+3z^2, 1+4z^2, 1-3z^2, 1-4z^2,
1+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,
1-2z^2, 1-5z^2}
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References
[Le_Murakami_97] ^ T. Q. T. Le and J. Murakami, Parallel Version of the Universal Vassiliev-Kontsevich Invariant, Journal of Pure and Applied Algebra 121 (1997) 271-291.
[Murakami_Ohtsuki_97] ^ J. Murakami and T. Ohtsuki, Topological Quantum Field Theory for the Universal Quantum Invariant, Communications in Mathematical Physics 188-3 (1997) 501-520.