ON MARKOV'S THEOREM

Abstract

Let χ be an oriented link type in the oriented 3-sphere S3 or ℝ3 = S3 - {∞}. A representative X ∈ χ is said to be a closed braid if there is an unknotted curve A ⊂ S3 - X (the axis) and a choice of fibration ℋ of the open solid torus S3 - A by meridian discs {Hθ : θ ∈ [0, 2 π]}, such that whenever X meets a fiber Hθ the intersection is transverse. Closed braid representations of χ are not unique, and Markov's well-known theorem asserts that any two are related by a finite sequence of elementary moves. The main result in this paper is to give a new proof of Markov's theorem. We hope that our new proof will be of interest because it gives new insight into the geometry.