Trivial pretzel links Is there a simple rule to check whether a pretzel link P(n_1,...,n_k) is a trivial link?
I am interested in the 2-component case but every information would be helpful. 
 A: A pretzel link has as many components as the pretzel link with coefficients reduced $(\mod 2)$. So it will have two components if and only if there are precisely two even coefficients. 
A conceptual classification is given by considering the 2-fold branched cover, or the $\pi$-orbifold, obtained by taking the orbifold quotient of the 2-fold branched cover. For pretzel links, the 2-fold branched cover is a connect sum of Seifert fibered spaces (this is more generally true for Montesinos links). Then a 2-component link is trivial if and only if the two-fold branched cover is $S^2\times S^1$. A theorem of Bonahon and Siebenmann classifies Seifert fibered orbifolds, so in principal you can classify pretzel links from their classification. 
A: In principle, normal surface theory answers your question - it can detect if a link is split and can decide if a knot is unknotted.  In practice, just draw your link in SnapPea.  But neither if these could be called a "simple rule."  You might look at Purcell's thesis work/early papers (eg "Volumes of highly twisted knots and links"), using hyperbolic geometry, to give a simple rule implying that a link is non-trivial.
