(78) Osculating Tubes and Self-Linking Numbers of Curves on the Three-Sphere

Accepted for publication in the refereed proceedings of the Alfred Gray Memorial Conference, Bilbao, Spain 2000, Contemporary Math., Amer. Math .Soc., Vol. 288 (2001) 10-19.

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ABSTRACT

This paper defines the Normal Euler Number and the Normal Euler Class for polyhedral surfaces in 4-space by means of singularities of projections into hyperplanes. There exist polyhedral analogues of nearly all of Whitney's theorems on Normal Euler Classes of surfaces smoothly immersed in 4-space. However, although the Normal Euler Number of a smooth embedding of the real projective plane in 4-space must be plus or minus 2, the Normal Euler Number of a (locally knotted) polyhedral embedding of the real projective plane can be any integer congruent to 2 modulo 4.