Celebratio Mathematica

J. S. Birman and W. W. Menasco: “Studying links via closed braids, IV: Composite links and split links,” Invent. Math. 102 : 1 (December 1990), pp. 115–139. An erratum for this article was published in Invent. Math. 160:2 (2005); Parts I, III and VI were published in Pac. J. Math. 154:1 (1992), 161:1 (1993) and 156:2 (1992); Part II was published in Topology Appl. 40:1 (1991); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1069243 Zbl 0711.57006 article

The main result concerns changing an arbitrary closed braid representative of a split or composite link to one which is obviously recognizable as being split or composite. Exchange moves are introduced; they change the conjugacy class of a closed braid without changing its link type or its braid index. A closed braid representative of a composite (respectively split) link is composite (split) if there is a 2-sphere which realizes the connected sum decomposition (splitting) and meets the braid axis in 2 points. It is proved that exchange moves are the only obstruction to representing composite or split links by composite or split closed braids. A special version of these theorems holds for 3 and 4 braids, answering a question of H. Morton. As an immediate Corollary, it follows that braid index is additive (resp. additive minus 1) under disjoint union (resp. connected sum).

@article {key1069243m,
	AUTHOR = {Birman, Joan S. and Menasco, William
		 W.},
	TITLE = {Studying links via closed braids, {IV}:
		 {C}omposite links and split links},
	JOURNAL = {Invent. Math.},
	FJOURNAL = {Inventiones Mathematicae},
	VOLUME = {102},
	NUMBER = {1},
	MONTH = {December},
	YEAR = {1990},
	PAGES = {115--139},
	DOI = {10.1007/BF01233423},
	NOTE = {An erratum for this article was published
		 in \textit{Invent. Math.} \textbf{160}:2
		 (2005); Parts I, III and VI were published
		 in \textit{Pac. J. Math.} \textbf{154}:1
		 (1992), \textbf{161}:1 (1993) and \textbf{156}:2
		 (1992); Part II was published in \textit{Topology
		 Appl.} \textbf{40}:1 (1991); Part V
		 was published in \textit{Trans. Am.
		 Math. Soc.} \textbf{329}:2 (1992). MR:1069243.
		 Zbl:0711.57006.},
	ISSN = {0020-9910},
	CODEN = {INVMBH},
}

J. S. Birman and W. W. Menasco: “Studying links via closed braids, II: On a theorem of Bennequin,” Topology Appl. 40 : 1 (June 1991), pp. 71–82. Parts I, III, and VI were published in Pac. J. Math. 154:1 (1992), 161:1 (1993) and 156:2 (1992); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1114092 Zbl 0722.57001 article

J. S. Birman and W. W. Menasco: “Studying links via closed braids, V: The unlink,” Trans. Am. Math. Soc. 329 : 2 (February 1992), pp. 585–606. Parts I, III and VI were published in Pac. J. Math. 154:1 (1992), 161:1 (1993) and 156:2 (1992); Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990). MR 1030509 Zbl 0758.57005 article

The main result is a version of Markov’s Theorem which does not involve stabilization, in the special case of the \( r \)-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most 1. To state the result, we need the concept of an “exchange move”, which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theorem 1 shows that exchange moves are the only obstruction to reducing a closed \( n \)-braid representative of the \( r \)-component unlink to the standard closed \( r \)-braid representative, through a sequence of braids of nonincreasing braid index.

J. S. Birman and W. W. Menasco: “Studying links via closed braids, I: A finiteness theorem,” Pacific J. Math. 154 : 1 (May 1992), pp. 17–36. Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1154731 Zbl 0724.57001 article

This paper is the first in a series which study the closed braid representatives of an oriented link type \( \mathscr{L} \) in oriented 3-space. A combinatorial symbol is introduced which determines an oriented spanning surface \( F \) for a representative \( L \) of \( \mathscr{L} \). The surface \( F \) is in a special position in 3-space relative to the braid axis \( A \) and the fibers in a fibration of the complement of \( A \). The symbol simultaneously describes \( F \) as an embedded surface and \( L \) as a closed braid. Therefore it is both geometrically and algebraically meaningful. Using it, a complexity function is introduced. It is proved that \( \mathscr{L} \) is described by at most finitely many combinatorial symols, and thus by finitely many conjugacy classes in each braid group \( B_n \) when the complexity is minimal.

J. S. Birman and W. W. Menasco: “A calculus on links in the 3-sphere,” pp. 625–631 in Knots 90 (Osaka, 15–19 August 1990). Edited by A. Kawauchi. de Gruyter (Berlin), 1992. MR 1177450 Zbl 0764.57005 incollection

Akio Kawauchi	Related
William Wyatt Menasco	Related

J. S. Birman and W. W. Menasco: “Studying links via closed braids, VI: A nonfiniteness theorem,” Pacific J. Math. 156 : 2 (December 1992), pp. 265–285. Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1186805 Zbl 0739.57002 article

Exchange moves were introduced in an earlier paper by the same authors. They take one closed \( n \)-braid representative of a link to another, and can lead to examples where there are infinitely many conjugacy classes of \( n \)-braids respresenting a single links type.

If a link type has infinitely many conjugacy classes of closed \( n \)-braid representatives, then \( n\geq 4 \) and the infinitely many classes divide into finitely many equivalence classes under the equivalence relation generated by exchange moves.

This theorem is the last of the preliminary steps in the authors’ program for the development of a calculus on links in \( S^3 \).

Choose integers \( n \), \( g\geq 1 \). Then there are at most finitely many link types with braid index \( n \) and genus \( g \).

J. S. Birman and W. W. Menasco: “Studying links via closed braids, III: Classifying links which are closed 3-braids,” Pacific J. Math. 161 : 1 (November 1993), pp. 25–113. Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1237139 Zbl 0813.57010 article

A complete solution is given to the classification problem for oriented links which are closed three-braids. The Classification Theorem asserts that, up to a finite list of exceptional cases, links which can be represented by closed 3-braids are represented by a unique conjugacy class in the group of 3-braids. The exceptional cases are the expected ones (links of braid index 1 and 2) and an unexpected infinite family of invertible links, each member of which has two 3-braid axes. The two axes correspond to diagrams which are related by “braid-preserving flypes”.

An algorithm is given which begins with an arbitrary closed 3-braid (or alternatively any link diagram with 3 Seifert circles), and converts it into a normal form which characterizes its oriented link type in oriented 3-space. One can decide from the normal form whether the link is prime or composite, split or irreducible, amphicheiral and or invertible. One can decide if the braid index is 3, 2 or 1. Using related results of P. J. Xu, one may determine the genus and construct a surface of maximum Euler characteristic with boundary the given link.

It is proved that the stabilization index of a link which is represented by a closed 3-braid is \( \leq 1 \), i.e. any two 3-braid representatives of the same link type become conjugate after a single stabilization to \( B_4 \).

J. S. Birman and W. W. Menasco: “Special positions for essential tori in link complements,” Topology 33 : 3 (July 1994), pp. 525–556. An erratum for this article was published in Topology 37:1 (1998). MR 1286930 Zbl 0833.57004 article

The decomposition of links into non-split components, by cutting along essential 2-spheres, is a fundamental step in any attempt to understand the link problem. In an earlier paper [1990] the authors studied that problem from the point of view of braid theory. A somewhat more subtle decomposition of a link complement involves splitting along essential tori. By a fundamental theorem which is due to Alexander (see p. 107 of [Rolfsen 1976]) every embedded torus \( T \) in \( S^3 \) is the boundary of a solid torus \( V \) on at least one side. The solid torus \( V \) may, however, be knotted, and this makes the study of embedded tori much more difficult than embedded 2-spheres, since the latter cannot be knotted. The first major attempt to understand embedded tori in link complements was a groundbreaking paper by Schubert [1953]. The seminal role which is played in the topology and geometry of link complements by embedded tori was later underscored in the important work of Jaco and Shalen [1979], Johansson [1975] and Thurston [1982], who showed that if \( M^3 \) is a 3-manifold, then there is a finite collection \( \Omega \) of essential, non-peripheral tori \( T_1,\dots \),\( T_q \), in \( M^3 \) such that each component of \( M^3 \) split open along the tori in \( \Omega \) is either Seifert-fibered or hyperbolic. Our goal in this paper is to apply the techniques of [Birman and Menasco 1990] to the study of essential tori in link complements.

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 and W. W. Menasco: “Stabilization in the braid groups, I: MTWS,” Geom. Topol. 10 : 1 (2006), pp. 413–540. MR 2224463 Zbl 1128.57003 article

Choose any oriented link type \( \mathscr{X} \) and closed braid representatives \( X_+ \), \( X_- \) of \( \mathscr{X} \), where \( X_- \) has minimal braid index among all closed braid representatives of \( \mathscr{X} \). The main result of this paper is a ‘Markov theorem without stabilization’. It asserts that there is a complexity function and a finite set of ‘templates’ such that (possibly after initial complexity-reducing modifications in the choice of \( X_+ \) and \( X_- \) which replace them with closed braids \( X^{\prime}_+ \), \( X^{\prime}_- \)) there is a sequence of closed braid representatives \[ X^{\prime}_+ = X^1 \to X^2 \to \cdots \to X^r = X^{\prime}_- \] such that each passage \( X^i\to X^{i+1} \) is strictly complexity reducing and non-increasing on braid index. The templates which define the passages \( X^i \to X^{i+1} \) include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index \( m\geq 4 \) a finite set \( \mathscr{T}(m) \) of new ones. The number of templates in \( \mathscr{T}(m) \) is a non-decreasing function of \( m \). We give examples of members of \( \mathscr{T}(m) \), \( m\geq 4 \), but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.

J. S. Birman and W. W. Menasco: “Stabilization in the braid groups, II: Transversal simplicity of knots,” Geom. Topol. 10 : 3 (2006), pp. 1425–1452. MR 2255503 Zbl 1130.57005 ArXiv math.GT/0310280 article

J. S. Birman and W. W. Menasco: “A note on closed 3-braids,” Commun. Contemp. Math. 10 : supplement 1 (November 2008), pp. 1033–1047. MR 2468377 Zbl 1158.57006 article

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