Crosscaps and knotes
Bradd Evans Clark · International Journal of Mathematics and Mathematical Sciences · 1978
IntroductionSeifert demonstrated in 1934 that every knot can be spanned by an orientable surface.These Seifert surfaces lead to numerous knot invariants.The purpose of this paper is to demonstrate the existence of a parallel theory concerning connected nonorientable surfaces.These surfaces give rise to additional knot invariants.If X is a point set we let cl(X) stand for the closure of X, int(X) stand for the interior of X, and 8X stand for the boundary of X.If S is a surface, let x(S) stand for the Euler characteristic of S.If K is a knot in Euclidean 3-dimensional space, E3, let g(k) stand for the genus of k as defined by Seifert in [4].This paper deals with piecewise linear topology.As such, all manifolds and maps will be considered to be piecewise linear. Z. The Crosscap NumberLet k be a knot in the orthogonal projection.