  The cyclic sieving phenomenon (CSP) was introduced by Reiner, Stanton, and
White to study combinatorial structures with actions of cyclic groups. The
crucial step is to find a polynomial, for example a q-analog, that satisfies
the CSP conditions for an action. This polynomial will give us a lot of
information about the symmetry and structure of the set under the action. In
this paper, we study the cyclic sieving phenomenon of the cyclic group $C_{2n}$
acting on $P_{n,k}$, which is the set of matchings of $2n$ points on a circle
with $k$ crossings. The noncrossing matchings ($k=0$) was recently studied as a
Catalan object. In this paper, we study more general cases, the matchings with
more number of crossings. We prove that there exists $q$-analog polynomials
$f_{n,k}(q)$ such that $(P_{n,k},f_{n,k},C_{2n})$ exhibits the cyclic sieving
phenomenon for $k=1,2,3$. In the proof, we also introduce an efficient
representation of the elements in $P_{n,k}$, which helps us to understand the
symmetrical structure of the set.
