On approximability by embeddings of cycles in the plane

We obtain a criterion for approximability of PL maps S1→R2 by embeddings, analogous to the one proved by Minc for PL maps I→R2. m. Let ϕ :S1→R2 be a PL map, which is simplicial for some triangulation of S1 with k vertices. The map ϕ is approximable by embeddings if and only if for each i=0,…,k the ith derivative ϕ(i) (defined by Minc) neither contains transversal self-intersections nor is the standard winding of degree, ∉{−1,0,1}. uce from the Minc result the completeness of the van Kampen obstruction to approximability by embeddings of PL maps I→R2 (Corollary 1.4). We also generalize these criteria to simplicial maps T→S1⊂R2, where T is a graph without vertices of degree >3 (Theorem 1.5).