Arithmetic of weighted Catalan numbers

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.