In the paper "The Volume Conjecture and Topological Strings" (0903.2084v2) 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?


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