Calabi-Yau manifolds and knot theory

Question

In the paper "The Volume Conjecture and Topological Strings" it is said that the mirror Calabi-Yau threefold is given by

$X := \{ (x,y,u,v) \in \mathbb{C^* \times\mathbb{C^*} \times \mathbb{C} \times \mathbb{C}} $ : uv = A(x,y) }, where A(x,y) is the A-polynomial for a given knot.

Let us take the A-polynomial for the trefoil knot: $A(x,y)=(y-1)(y+x^6)$ which gives us Calabi-Yau manifold defined by the equation:

$uv = (y-1)(y+x^6)$

My questions are: 1) Why is this a Calabi-Yau manifold ? There are some very specific conditions to be satisfied, how do I check them or make sure that they are satisfied for manifolds defined this way? If I'm correct, these conditions would require for me to know for example a curvature form of the tangent bundle. I also don't know the metric so it would be hard to even compute something.

2) Is there any universal condition that can tell me if this is a Calabi-Yau manifold? Maybe something from knot theory?

Answer 1

$X$ is certainly Calabi-Yau in the algebraic sense: the canonical line bundle is trivialized by the holomorphic volume form

$\Omega=\frac{dx}{x} \wedge \frac{dy}{y} \wedge \frac{du}{u}$

(this comes from the fact that $X$ is a conic bundle over $(\mathbb{C}^{*})^2$, degenerate over the curve $A=0$. The form $\Omega$ is constructed from the standard holomorphic volume form on $(\mathbb{C}^{*})^2$ and from the standard holomorphic volume on the $\mathbb{C}^{*}$ fiber).

In the compact Kähler case, this would be enough to insure the metric Calabi-Yau property, i.e. the existence of a Ricci flat metric, by Yau's theorem. But here, $X$ is not compact, so one has to be careful about question of completeness. My guess would be that in this particular example, there really exists a complete Ricci-flat metric but I don't know if/where it has been proved (if true).

In any case, what I have written is for any Laurent polynomial $A(x,y)$ and I don't think that the fact that $A$ comes from a knot should play a role (for this particular question).

EDIT (taking into account Vivek Shende's comment to the question): in my answer, I am assuming $A$ generic so that the curve $A=0$ is smooth and then $X$ is also smooth. But if $A=0$ is singular, as it is typical for the case coming from a knot (for a knot, there is always a factor $(y-1)$ if I remember correctly so the curve $A=0$ is always reducible), then $X=0$ will also be singular and so one has to be careful about what one means by Calabi-Yau).