Celebratio Mathematica — Birman

My work with Joan Birman

Works connected to Jerome Paul Levine

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J. S. Birman and W. W. Menasco: “On Markov’s theorem,” pp. 295–310 in Knots 2000 Korea (Volume 1) (Yongpyong, Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 3. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Singapore), 2002. MR 1905686 Zbl 1059.57002 incollection

Let \( \chi \) be an oriented link type in the oriented 3-sphere \( S^3 \) or \[ \mathbb{R}^3 = S^3 - \{\infty\} .\] A representative \( X \in \chi \) is said to be a closed braid if there is an unknotted curve \[ \mathbf{A} \subset S^3 - X \] (the axis) and a choice of fibration \( \mathscr{H} \) of the open solid torus \( S^3 - \mathbf{A} \) by meridian discs \[ \{H_{\theta}: \theta\in [0, 2\pi]\} ,\] such that whenever \( X \) meets a fiber \( H_{\theta} \) the intersection is transverse.

Closed braid representations of \( \chi \) 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.

Ki Hyoung Ko	Related
Yukio Matsumoto	Related
William Wyatt Menasco	Related
Cameron McAllan Gordon	Related	Volume
Akio Kawauchi	Related
Louis H. Kauffman	Related
Jerome Paul Levine	Related
Gyo Taek Jin	Related