For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with finite volume). However, not every link complement can be endowed with such a structure. Does anybody know if there is a criterion for a link to have a corresponding hyperbolic structure? Since this problem is fundamental to the definition of hyperbolic invariants of knots and links, I guess people have already studied it thoroughly. However, I couldn't find such results. Could you please help me? Thank you very much!



Is there a criterion for a link complement to have a hyperbolic structure with finite volume
There is a topological criterion due to Thurston. Using the JSJ machine (and work of many others) this criterion can also be phrased algebraically. I'll essay these below. Please note that the situation is much simpler for knots. To answer your question most directly, here is the desired reference to Wikipedia. 


<http://en.wikipedia.org/wiki/Hyperbolic_link>


This page refers to the books of Colin Adams and William Thurston. Both are excellent. 


Now, here is Thurston's criterion. (EDIT: exposition improved after reading Bruno Martelli's answer.)


Suppose that $L$ is the link and $X$ is the link complement. Suppose $\pi = \pi\_1(X)$. We assume the following properties (and each property assumes the proceeding ones). $\newcommand{\ZZ}{\mathbb{Z}}$


* $L$ is not a split link. Equivalently, $X$ is contains no essential two-sphere. Equivalently, $\pi$ is not a free product.
* $L$ is not the unknot. Equivalently, $X$ contains no essential disk. Equivalently, $\pi$ is not $\ZZ$.
* $L$ has no component that is an "undisturbed satellite knot". Equivalently, $X$ contains no essential torus.
* $L$ is not a torus knot. Equivalently, $X$ contains no essential annulus. These last two topological properties are equivalent to $\pi$ not containing a copy of $\ZZ^2$.


Then $X$ admits a hyperbolic structure.


