  In this paper, we study arithmetic properties of weighted Catalan numbers.
Previously, Postnikov and Sagan found conditions under which the $2$-adic
valuations of the weighted Catalan numbers are equal to the $2$-adic
valutations of the Catalan numbers. We obtain the same result under weaker
conditions by considering a map from a class of functions to $2$-adic integers.
These methods are also extended to $q$-weighted Catalan numbers, strengthening
a previous result by Konvalinka. Finally, we prove some results on the
periodicity of weighted Catalan numbers modulo an integer and apply them to the
specific case of the number of combinatorial types of Morse links. Many open
questions are mentioned.
