AN ELEMENTARY PROOF THAT ALL SPANNING SURFACES OF A LINK ARE TUBE-EQUIVALENT

Abstract

. The standard proof that the potential function models the Alexander-Conway polynomial depends on the fact that all spanning surfaces of a link are tube-equivalent [K1]. Proofs that all spanning surfaces of a link are tube-equivalent use machinery such as the Thom-Pontrjagin construction [L]. Here we present an elementary geometric argument in the case of links in S 3 which allows one to visualize the additions and removals of tubes. Contents 1. Introduction 1 2. Some background and preliminary work 2 3. Proof of Theorem 1 4 References 7 1. Introduction Let L be an oriented 1-dimensional link in S 3 : A spanning surface for L is a compact, oriented two-dimensional manifold S in S 3 such that S has oriented boundary L (for technical convenience, we've dropped the connectivity assumption. This only makes our results below more general). Two spanning surfaces are called tube-equivalent or S-equivalent if they are related up to ambient isotopy by the addition or removal of tu...