# 07-1352/Class Notes for February 6

${\displaystyle Z_{0}(K)=\ \ \ \ \ \ \ \ \ \ \int \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum _{m;\ t_{1}<\ldots

## 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 ${\displaystyle \nu }$.
• The Kontsevich integral of the unknot.
• Re-introducing framing:
• Using counter-terms in the original Kontsevich integral.
• Using further algebra on ${\displaystyle {\mathcal {A}}}$:
• The Milnor-Moore Theorem.
• Using ${\displaystyle {\hat {\theta }}}$ and ${\displaystyle {\frac {d}{d\theta }}}$.
• Unzipping a circle, the error terms ${\displaystyle a}$ and ${\displaystyle b}$ and their cancellation following .
• The extension to knotted trivalent graphs following .
• 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.