[Ku1] W. Kühnel, Tight and 0-tight polyhedral embeddings of surfaces, Invent. Math.. 58 (1980) 161-177.This paper shows that the Banchoff bound on the essential co-dimension of a tight embedding is sharp for all surfaces but the Klein bottle, and gives explicit dimension ranges for the existence of tight embeddings for all surfaces, with explicit construction techniques. The paper also treats surfaces with boundary.
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