Abstract
IN [7] Seifert presents a method for constructing a knot in the three-sphere S3, by imbedding a surface (with a single boundary component) in S” with prescribed self-linking characteristics. This technique was used to characterize the Alexander polynomial of a knot, in an algebraic manner. But the Alexander polynomial is only the first of a sequence of polynomials which arise as knot invariants, and it is natural to consider the problem of characterizing them algebraically. Seifert’s construction does not seem delicate enough to carry out this task.f It is the aim of this note to present a new technique for constructing knots, also based on a prescription of linking information, which can resolve the problem. The techniques to be presented here generalize to give similar characterizations of suitably defined invariants of higher-dimensional knots. This entire subject will be dealt with in a future paper; also see [4, Part II].
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