From Rotman's algebraic topology: In the picture below in example $7.16$, how is the underlined statement in red true given the definition above it? The elementary move of $(0,1)(1,2) \rightarrow (0,2)$ should still apply but the book is saying they can't be homotopic?
Additional definitions: Edge path $\alpha$ is a sequence of finite edges $\alpha = e_1e_2\dots e_n$.
Edge $e=(p,q)$ in a simplicial complex is an ordered pair of vertexes lying in a simplex of $K$.
Hint: The definition says "where $\{p,q,r\}$ lie in a simplex of $K$" - which simplex of the skeleton do $\{p_0,p_1,p_2\}$ lie in?