According to the Baez-Dolan cobordism hypothesis, an extended TQFT is determined by its value on a single point. This value a fully dualizable object of a symmetric monoidal $n$ category (a fully dualizable object is a higher categorical analogue of a finite dimensional vector space). The Alexander polynomial is a quantum invariant, and comes from a TQFT.

How can an "Alexander polynomial" TQFT be put into an extended TQFT, and what is its value at a single point?

The question I just asked is closely related to this question. I also asked the question on the ldtopology blog here, and Theo Johnson-Freyd suggested that MO might be the place to ask it.

Briefly, I will summarize what an extended TQFT is. A TQFT as a symmetric monoidal functor Z:

**Cob**(n)->

**Vect**(k) from the tensor category of $n-1$ dimensional manifolds and cobordisms between them to the tensor category of vector spaces over a field k. An extended TQFT is a symmetric monoidal functor Z:

**Cob**

_{k}(n)->

**C**from the n-category of cobordisms to a symmetric monoidal n-category

**C**. I vote for the introduction to Lurie's expository account of his work proving the Baez-Dolan cobordism hypothesis as the best place to read about why extended TQFT's are natural objects, to understand their motivation, and to understand why people are so excited about them. An extended TQFT assigns a fully dualizable object to a point, and a higher “trace” on this object to a closed n-dimensional manifolds.

reallywant to know whether two notions of 2-category are the same, you have to compare the 3-categories that they form --- and there your headache begins... $\endgroup$ – Tom Leinster Feb 4 '10 at 3:44