 the following two types of elementary operations over $c$. If $e$ is an oriented edge of $c$, $e^{-0}$ is the edge with an opposite to $e$ orientation, and $ee^{-0}$ is a subpath of $c$ so that $c = c_0 ee^{-0} c_0$, where $c_0, c_0$ are subpaths of $c$, then the operation $c \to c_0 c_0$ over $c$ is called an [*[elementary homotopy]{} of type 0*]{}. Suppose that $c =c_0 u c_0$, where $c_0, u, c_0$ are subpaths of $c$, and a boundary path ${\partial}s$ of a unit square $s$ of ${\mathcal{T}}$ is ${\partial}s = uv$, where $u, v$ are subpaths of ${\partial}s$ and either of $u, v$ could be of zero length, i.e., either of $u, v$ could be a single vertex of ${\partial}s$. Then the operation $c \to c_0 v^{-0} c_0$ over $c$ is called an [*[elementary homotopy]{} of type 0*]{}.  