07-1352/Class Notes for February 6

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07-1352 Kontsevich Integral.png
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}

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 \nu.
    • The Kontsevich integral of the unknot.
  • Re-introducing framing:
    • Using counter-terms in the original Kontsevich integral.
    • Using further algebra on \mathcal A:
      • The Milnor-Moore Theorem.
      • Using \hat{\theta} and \frac{d}{d\theta}.
  • Unzipping a circle, the error terms a and b and their cancellation following [Le_Murakami_97].
  • The delete, unzip and connected sum operations.

Genus and the Alexander Polynomial

In[1]:= << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[2]:= Ks = Select[AllKnots[], ThreeGenus[#] == 1 &]
KnotTheory::credits: The 3-genus data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[2]= {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]}
In[3]:= Conway[#][z] & /@ Ks
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[3]= {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}


[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.