A cluster algebra is the commutative algebra generated by all cluster variables obtained from
an initial seed by repeated seed mutations. In the
coefficient-free case, a seed consists of a cluster 
, meaning an 
-tuple of algebraically
independent cluster variables, and a skew-symmetrizable
matrix 
with integer entries. A seed mutation replaces one cluster variable and simultaneously
transforms 
.
The Laurent phenomenon states that every cluster variable is a Laurent polynomial in the variables of any cluster. For cluster algebras from triangulated surfaces, cluster variables can be expressed using perfect matchings of associated snake graphs. De Loera Chávez (2026) gave a version of this expansion as a determinant. Here, an arc is a non-self-intersecting curve in the surface between marked points, considered up to isotopy. A plain arc has plain rather than notched tags at its endpoints, and the crossed arcs are the arcs of the initial triangulation that it crosses. For cluster variables associated with plain arcs, the matching sum is the determinant of a weighted biadjacency matrix of the snake graph-the matrix whose rows and columns are indexed by the two vertex classes of the bipartite graph and whose entries are the corresponding edge weights-divided by a monomial in the initial cluster variables corresponding to the crossed arcs.
