Low Dimensional Topology

February 19, 2013

It came from K2

Filed under: Uncategorized — dmoskovich @ 11:34 am

At the “Mathematics of Knots 5” conference at Waseda University, I attended a most interesting talk by Takefumi Nosaka. Nosaka’s work always gives me the impression of being robust and sophisticated, and this talk was no exception. This time he was in the process constructing new topological invariants of links as images of longitudes in K_2 of a ring.

Given a link L\subset S^3 and a ring F with unit, consider the set of representations \pi_1\left(S^3\setminus L\right)\overset{f}{\longrightarrow} SL_2(F) of the knot group onto the special linear group of 2\times 2 matrices over F with determinant one. In analogy with the case F=\mathbb{R}, you can think of these as being representations of the link group as orientation and volume preserving (whatever than means) linear transformations of F^2.

Which invariants of L can we identify in the K-groups of SL_2(F)? Quite a few, actually.

  • K_1 is a stand-in for the determinant (honest determinants don’t exist unless F is commutative), so you get twisted Reidemeister torsions, which are closely related to twisted Alexander polynomials. Alexander polynomials are good solid classical topological invariants. The Alexander polynomial is also the archetypal quantum invariant. so we’re off to a good start!
  • An element of K_3 for the case F=\mathbb{C} gives rise to the hyperbolic volume of a hyperbolic link, via work of Quillen and Goncharov (what does the parallel construction do for other rings?). The hyperbolic volume is another very good invariant, and it is supposed to arise as a limit of coloured Jones polynomials, so again we have obtained an important and useful topological invariant.

Given that 2 is between 1 and 3, one naturally wonders which knot invariants correspond to values of special elements (such as longitudes) of link groups in K_2(F). One would almost be surprised if the images of longitudes in K_2 were not useful topological invariants, given the impeccable pedigree of invariants coming from K_1 and from K_3.

I like this idea a lot (I’ve long had a secret, or non-so-secret, longing to understand stuff in quantum topology via K-theory). People have studied similar ideas in relation to the Volume Conjecture (see for example this paper by Li and Wang), and it is a natural idea to be thinking about, but Nosaka presented it in a way I really liked. With a nice quandle argument he showed that, of you’re lifting \pi_1\left(S^3\setminus L\right)\overset{f}{\longrightarrow} SL_2(F) to a map \tilde{f} to K_2(F), then the natural object to consider is in fact the image of the longitude not in K_2(F), but rather in K_2(F)\times U, where U denotes the group of elliptic matrices \left(\begin{matrix}1 & a\\ 0 & 1\end{matrix}\right) where a is in F. One suggestion (which made a lot of sense to me) was that the invariants he was constructing might also arise as quandle cocycles.

It would be nice to calculate some examples for topologically relevant F, which is what he is doing right now, I believe. This looks to me like a real approach to bridge between the geometric and the quantum, or at least to discover useful and natural link invariants in the territory between the two.

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