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==The Fundamental Theorem of Finite Type Invariants==

{{06-1350/The Fundamental Theorem}}

===The Mysterious <math>\nu</math>===

It remains to define <math>\nu\in{\mathcal A}(\uparrow)</math>. Well, it is the element often called "the invariant of the unknot", for indeed, by a long chain of reasoning, it is the invariant of the unknot. It is also given by the following explicit formula of {{ref|BGRT}} and {{ref|BLT}}:

<center><math>\nu=\chi\left(\exp_\cup\left(\sum_{n=1}^\infty b_{2n}\omega_{2n}\right)\right).</math></center>

In the above formula \chi denotes the PBW "symmetrization" map, <math>\exp_\cup</math> means "exponentiation in the disjoint union sense", <math>\omega_{2n}</math> is the "wheel with <math>2n</math> legs" (so <math>\omega_2=\twowheel,</math> <math>\omega_4=\fourwheel,</math> etc.) and the <math>b_{2n}</math>'s are the "modified Bernoulli numbers" defined by the power series expansion

<center><math>\sum_{n=0}^\infty b_{2n}x^{2n} = \frac12\log\frac{\sinh x/2}{x/2}</math></center>

(so <math>b_2=1/48</math>, <math>b_4=-1/5760</math>, <math>b_6=1/362880</math>, etc.).

===Some values of the invariant===

While these are not necessary for the statement of the theorem, it is worthwhile to note that the invariants of the unknot, the unknotted <math>\theta</math>-graph and the unknotted dumbbell are as follows:

[[Image:UnknotThetaDumbbell.png|thumb|center|640px|Some noteworthy invariants]]

===References===

{{note|BGRT}} D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, ''Wheels, wheeling, and the Kontsevich integral of the unknot'', Israel Journal of Mathematics '''119''' (2000) 217-237, {{arXiv|q-alg/9703025}}.

{{note|BLT}} D. Bar-Natan, T. Q. T. Le and D. P. Thurston, ''Two applications of elementary knot theory to Lie algebras and Vassiliev invariants'', [http://www.msp.warwick.ac.uk/gt/2003/07/p001.xhtml Geometry and Topology '''7-1''' (2003) 1-31], {{arXiv|math.QA/0204311}}.

{{note|MO}} J. Murakami and T. Ohtsuki, ''Topological Quantum Field Theory for the Universal Quantum Invariant'', Communications in Mathematical Physics '''188''' (1997) 501-520.