  A $k$-positive matrix is a matrix where all minors of order $k$ or less are
positive. Computing all such minors to test for $k$-positivity is inefficient,
as there are $\sum_{\ell=1}^k \binom{n}{\ell}^2$ of them in an $n\times n$
matrix. However, there are minimal $k$-positivity tests which only require
testing $n^2$ minors. These minimal tests can be related by series of
exchanges, and form a family of sub-cluster algebras of the cluster algebra of
total positivity tests. We give a description of the sub-cluster algebras that
give $k$-positivity tests, ways to move between them, and an alternative
combinatorial description of many of the tests.
