07-1352/Class Notes for February 6

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$Z_{0}(K)=\ \ \ \ \ \ \ \ \ \ \int \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum _{m;\ t_{1}<\ldots <t_{m};\ P=\{(z_{i},z'_{i})\}}{\frac {(-1)^{\#P_{\downarrow }}}{(2\pi i)^{m}}}D_{P}\bigwedge _{i=1}^{m}{\frac {dz_{i}-dz'_{i}}{z_{i}-z'_{i}}}$

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 extension to knotted trivalent graphs following [Murakami_Ohtsuki_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}

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.