On Approximability by Embeddings of Cycles in the Plane

We obtain a criterion for approximability of piecewise linear maps S1 → R2 by embeddings, analogous to the one proved by Minc for piecewise linear maps I → R2. Theorem. Let φ : S1 → R2 be a piecewise linear 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 i-th derivative φ(i) (defined by Minc) neither contains transversal self-intersections nor is the standard winding of degree 6∈ {−1, 0, 1}. We deduce from the Minc result the completeness of the van Kampen obstruction to approximability by embeddings of piecewise linear maps I → R2. We also generalize these criteria to simplicial maps T → S1 ⊂ R2, where T is a graph without vertices of degree > 3.