  It is proved that a multiset of permissible arcs over a tiling is uniquely
determined by its intersection vector under a mild condition. This generalizes
a classical result over marked surfaces with triangulations. We apply this
result to study $\tau$-tilting theory of gentle algebras and denominator
conjecture in cluster algebras. In the case of gentle algebras, it is proved
that different $\tau$-rigid $A$-modules over a gentle algebra $A$ have
different dimension vectors if and only if $A$ has no even oriented cycle with
full relations. For cluster algebras, the denominator conjecture has been
established for cluster algebras of type $\mathbb{A}\mathbb{B}\mathbb{C}$.
