Publication: An infinite family of non-separable represented knots. (Spanish)
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1973
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Real Sociedad Matemática Española;Consejo Superior de Investigaciones Científicas. Instituto "Jorge Juan" de Matemáticas
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Abstract
Let Fg denote a closed orientable surface of genus g≥1. The author proves first that Fg×S1 is not a 2-fold cyclic covering of S3 branched over a link. (The special case g=1 was established by R. H. Fox [same Rev. (4) 32 (1972), 158–166; MR0331360 (48 #9694)]. Since the appearance of the paper the author has obtained more general results pertaining to Seifert fibre spaces [Bol. Soc. Mat. Mexicana (2) 18 (1973), 1–32; MR0341467 (49 #6218)] and to p-fold cyclic coverings [Proc. Amer. Math. Soc. 47 (1975), 495–500;].) Then a represented link (Lg,ω) is exhibited for each g≥1 such that the associated (4-fold) covering of S3 branched over Lg is Fg×S1. These two facts involve the fact that the represented links (Lg,ω) are not separable, whereas the author had previously conjectured that any represented link is separable; for the definition of separability see, e.g., R. H. Fox [op. cit.].