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 M...

❙✉♠♠❛r2 ❚❤❡ ❋♦✉r✐❡r Ps❡✉❞♦s♣❡❝tr❛❧ ❚✐♠❡ ❉♦♠❛✐♥ ✭❋♦✉r✐❡r✲P❙❚❉✮ ♠❡t❤♦❞ ✇❛s s❤♦✇♥ t♦ ❜❡ ❛♥ ❡✛❡❝t✐✈❡ ✇❛2 ♦❢ ♠♦❞❡❧❧✐♥❣ ✇❛✈❡ ♣r♦♣❛❣❛t✐♦♥✳ ❋♦✉r✐❡r✲P❙❚❉ ✐s ❜❛s❡❞ ♦♥ ❋♦✉r✐❡r ❛♥❛❧2s✐s ❛♥❞ s2♥t❤❡s✐s t♦ ❝♦♠♣✉t❡ t❤❡ s♣❛t✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❣♦✈❡r♥✐♥❣ ✇❛✈❡ ❡q✉❛t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♠❡t❤♦❞ s✉✛❡rs ❢r♦♠ t❤❡ ✇❡❧❧✲❦♥♦✇♥ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥ ✇❤❡♥ ❝♦♠♣✉t✐♥❣ ❛ ♥♦♥✲s♠♦♦t❤ ♦r ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✳ ❚❤✐s ❧✐♠✐ts ✐ts ♣♦ss✐...