Genera of the alternating links
One of the fundamental results in link theory is \textit{H. Seifert}'s [Math. Ann. 110, 571-592 (1934; Zbl 0010.13303)] constructive proof of the fact that every tame link in the three-sphere is the oriented boundary of an orientable surface. It is a theorem of \textit{R. H. Crowell} [Ann. Math., II. Ser. 69, 258-275 (1959; Zbl 0111.358)] and \textit{K. Murasugi} [J. Math. Soc. Japan 10, 94-105 (1958; Zbl 0084.193); ibid. 235-248 (1958; Zbl 0106.167)] that Seifert's construction, applied to an alternating link diagram, always results in a spanning surface that is of minimal genus for that link type. Their proofs depend on the relationship between the degree of the Alexander polynomial and the genus of a spanning surface. In this paper the author presents a very different proof of this theorem, using elementary homology theory and geometric arguments. Regrettably, the closing passages of the paper leave this reader uncertain about several important aspects of the proof.
scientific article; zbMATH DE number 4026349