Celebratio Mathematica

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

J. S. Birman, M. Rampichini, P. Boldi, and S. Vigna: “Towards an implementation of the B–H algorithm for recognizing the unknot,” pp. 601–645 in Knots 2000 Korea (Volume 2) (Yongpyong, Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 4. 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 (Hackensack, NJ), 2002. MR 1915499 Zbl 1007.57004 incollection

In the manuscript [Birman and Hirsch 1998] the first author and Michael Hirsch presented a then-new algorithm for recognizing the unknot. The first part of the algorithm required the systematic enumeration of all discs which support a ‘braid foliation’ and are embeddable in 3-space. The boundaries of these ‘foliated embeddable discs’ (FEDs) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FEDs. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list.

In this paper we describe implementations of the first and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FEDs have unexplored and interesting structure. Open question are interspersed throughout the manuscript.

The third part of the algorithm was studied in [Birman, Ko and Lee 1998] and [Birman, Ko and Lee 2001], and implemented by S. J. Lee [preprint]. At this writing his algorithm is polynomial for \( n\leq 4 \) and exponential for \( n\geq 5 \).

Ki Hyoung Ko	Related
Paolo Boldi	Related
Sebastiano Vigna	Related
Yukio Matsumoto	Related
Cameron McAllan Gordon	Related	Volume
Akio Kawauchi	Related
Louis H. Kauffman	Related
Marta Rampichini	Related
Jerome Paul Levine	Related
Gyo Taek Jin	Related

Knots 2000 Korea (Volume 1) (Yongpyong, South 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. Zbl 0995.00510 book

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

Knots 2000 Korea (Volume 3) (Yongpyong, South Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 6. 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), September 2002. Zbl 1018.00508 book

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

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