Binding Numbers of Tight Contact Structures on $L(n,1)$

We study binding numbers of tight contact structures on the lens spaces $L(n,1)$.

Using the $d_3-$invariant, together with restrictions on planar monodromy factorizations and the Durst-Kegel algorithm for computing $d_3$ from open books, we obtain lower bounds for the number of binding components of planar open books supporting these contact structures.

As a consequence, we compute the binding number of the universally tight contact structures on $L(n,1)$, showing that it is equal to $n$.