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. 



Trivial pretzel links
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](http://www.ams.org/mathscinet-getitem?mr=827297) classifies Seifert fibered orbifolds, so in principal you can classify pretzel links from their classification. 


