4
$\begingroup$

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer labelings of the edges of $T$ certain $q$-polynomials.

For a closed $3$-manifolds $q$ needs to be a root of unity for the invariant to be defined, but I think that for a general 3-manifold with boundary TV(M) could be constructed as a $q$-difference module by convolving 6j-symbols.

Question: is function $TV_n(M)$ known (expected) to be $q$-holonomic?

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ The TV construction, with $q$ not a root of unity, only works if $M$ is a handlebody (i.e. can be built out of 0- and 1-handles). If $M$ requires 2-handles, the sums defining the TV invariant become infinite and are not obviously convergent. $\endgroup$
    – Kevin Walker
    Commented Jun 2, 2019 at 1:22
  • $\begingroup$ Thanks for pointing that out! I will be grateful for any reference. $\endgroup$
    – Daniil Rudenko
    Commented Jun 2, 2019 at 2:27
  • $\begingroup$ But I think that the question still makes sense: one should be able to construct TV(M) as a q-difference module by subsequently convoluting 6-j symbols. Probably, the holonomicity would follow from holonomicity of 6j-symbol. $\endgroup$
    – Daniil Rudenko
    Commented Jun 2, 2019 at 2:30

0

Reset to default

You must log in to answer this question.