Avril Smith

]]>Thanks, Daniel! I should add that the approach I’m describing above is actually now carried out for adequate knots. See arXiv:1002.0256.

]]>Another super question!

I have no idea. The Alexander polynomial is a Reidemeister torsion in a natural way (and so is the hyperbolic volume), but no K-theory interpretation of the coloured Jones polynomial is known, and I assume that you would need something of this sort to “lift” Poincare duality.

I wonder what the physicists say…

This is a very nice question, which I should write a proper post to answer… let’s distinguish between three things:

**Quantum invariants**: Associated to an R-matrix, coming from a representation of a quantum group (or a ribbon Hopf algebra). The (coloured) Jones polynomial and the Alexander polynomial are both examples of quantum invariants, associated to different quantum groups.

**Finite type invariants**: Associated to Jacobi diagrams (formal finite sums of graphs) plus a “weight system”, which in this context is a map from Jacobi diagrams to satisfying certain relations. Coefficients of the coloured Jones polynomial is an example, or coefficients of the Alexander polynomial.

**The Kontsevich invariant**: A formal sum of Jacobi diagrams over . It is universal with respect to both finite type invariants and quantum invariants. A weight system gives rules to contract Jacobi diagrams and get numbers, and you can get any finite type invariant this way. An R-matrix gives (closely related) rules to contract Jacobi diagrams and get a polynomial (for example), and you can get any quantum invariant this way.

This is what I think John meant- the Alexander polynomial is a quantum invariant, but the Blanchfield pairing is not. However, the Blanchfield pairing **is** a (very important) part of the Kontsevich invariant. This is why MMR is true conceptually, and once you see this, MMR becomes easy.

The Blanchfield pairing corresponds to the coefficients of Jacobi diagrams with precisely one loop (the “wheels part” of the Kontsevich invariant). If you choose an R-matrix, these will contract and you will be left with the Alexander polynomial (or some reduction of it)- the presentation matrix will collapse to its (Kadison-Fuglede) determinant. That’s how you prove MMR.

The signature function plays an important conjectural part in the theory of the coloured Jones polynomial. The key paper is http://arxiv.org/abs/math/0310203. Additionally, the Casson invariant of the infinite cyclic cover can be recovered from the expansion of the coloured Jones of a knot in powers of (q-1) as the “second line” (the bottom line is the Alexander polynomial). The Casson invariant of the infinite cyclic cover contains the “total signature”.

Stoimenow seems to have written a lot about Tristram-Levine signatures and finite type invariants, which I do not understand well conceptually. I assume an answer to the conjecture hinted at in the previous paragraph would make that all clear.

These are some of the questions I am most interested in in quantum topology, and a good mathematical understanding of the role of signatures, Minkowski units, Nakanishi index, and such would really clarify the coloured Jones polynomial as it relates to classical topology (I think).

The result (apparently due to Bar Natan) that the Alexander polynomial is determined by the coloured Jones polynomial, can it been souped-up to give a proof of the symmetry of the Alexander polynomial? ie: can you “lift” Poincare duality to a theorem at the level of the coloured Jones polynomial?

]]>I should say more… I would argue that the Alexander polynomial is not only a quantum invariant, but the prototypical quantum invariant. This is the main reason I care about the Alexander polynomial (!)… I don’t completely agree with what John said- assuming vanishing anomalies, path integrals, and other physicist magic, saying something is the universal finite-type invariant for some Lie superalgebra is the same as saying it’s a partition function, which encodes the statistical mechanics of the space. So the Alexander polynomial would give you pressure, entropy, free energy, and other important statistical mechanical quantities… of a manifold equipped with G-gauge fields where G is the simplest Lie superalgebra you can think of ( is my vote).

]]>Thanks!! Clearly I must read CCGLS urgently (for many reasons)…

]]>Thanks Dave… this is beautiful and simple.

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