Geometry & Topology 10 (2006) 413–540 413 Stabilization in the braid groups I: MTWS JOAN SBIRMAN WILLIAM WMENASCO Choose any oriented link type X and closed braid representatives X ; X of X , C where X has minimal braid index among all closed braid representatives of 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 C them with closed braids X 0 ; X 0 ) there is a sequence of closed braid representatives X X 1 X 2 CX r X such that each passage X i X i 1 is strictly 0 D ! ! ! D 0 ! C complexityC reducing and non-increasing on braid index. The templates which define the passages X i X i 1 include 3 familiar ones, the destabilization, exchange move ! C and flype templates, and in addition, for each braid index m 4 a finite set T .m/ of new ones. The number of templates in T .m/ is a non-decreasing function of m. We give examples of members of T .m/; m 4, but not a complete listing. There are applications to contact geometry, which will be given in a separate paper[6]. 57M25, 57M50 1 Introduction 1.1 The problem Let X be an oriented link type in the oriented 3–sphere S 3 or R3 S 3 : D n f1g A representative X X is said to be a closed braid if there is an unknotted curve 2 A .S 3 X / (the axis) and a choice of fibration H of the open solid torus S 3 A n n by meridian discs H Œ0; 2 , such that whenever X meets a fiber H the f I 2 g intersection is transverse. We call the pair (A; H) a braid structure The fact that X is a closed braid with respect to H implies that the number of points in X H is \ independent of . We call this number the braid index of X , and denote it by the symbol b.X /. The braid index of X , denoted b.X /, is the minimum value of b.X / over all closed braid representatives X X . 2 Closed braid representations of X are not unique, and Markov’s well-known theorem (see the book by Birman[3], and the papers by Birman and Menasco[12], Lam- bropoulou and Rourke[22], Markov[23], Morton[28] and Traczyk[31]) asserts that Published: 27 April 2006 DOI: 10.2140/gt.2006.10.359 414 Joan S Birman and William W Menasco any two are related by a finite sequence of elementary moves. One of the moves is braid isotopy by which we mean an isotopy of the pair (X; R3 A) which preserves the n condition that X is transverse to the fibers of H. The other two moves are mutually inverse, and are illustrated in Figure 1. Both take closed braids to closed braids. We call them destabilization and stabilization where the former decreases braid index by one and the latter increases it by one. The ‘weight’ w denotes w parallel strands, relative to the given projection. The braid inside the box which is labeled P is an arbitrary .w 1/–braid. Later, it will be necessary to distinguish between positive and negative C destabilizations, so we illustrate both now. destabilize destabilize w w w w 1 1 1 1 P P P P A A A A C stabilize stabilize Figure 1: The two destabilization moves Theorem 1 (Markov’s Theorem (MT)[23]) Let X ; X be closed braid representa- tives of the same oriented link type X in oriented 3–space,C with the same braid axis A. Then there exists a sequence (1–1) X X1 X2 Xr X C D ! ! ! D of closed braid representatives of X such that, up to braid isotopy, each Xi 1 is C obtained from Xi by a single stabilization or destabilization. It is easy to find examples of subsequences Xj Xj k of (1–1) in Theorem 1 ! ! C such that b.Xj / b.Xj k /, but Xj and Xj k are not braid isotopic. Call such D C C a sequence a Markov tower. The stabilization and destabilization moves are very simple, but sequences of stabilizations, braid isotopies and destabilizations can have unexpected consequences. In the braid groups these moves are ‘site dependent’, unlike the stabilization–destabilization move in the Reidemeister–Singer Theorem. (For an example the reader should refer ahead to the specified site of the stabilization in the sequence in Figure 5.) Until now these moves have been predominately used to develop link invariants, but the Markov towers themselves have been ‘black boxes’. One of Geometry & Topology, Volume 10 (2006) Stabilization in the braid groups I: MTWS 415 the main motivating ideas of this work is to open up the black box and codify Markov towers. Markov’s Theorem is typical of an entire class of theorems in topology where some form of stabilization and destabilization play a central role. Other examples are: (1) The Reidemeister–Singer Theorem[30] relates any two Heegaard diagrams of the same 3–manifold, by a finite sequence of very simple elementary changes on Heegaard diagrams. The stabilization–destabilization move adds or deletes a pair of simple closed curves a; b in the defining Heegaard diagram, where a b \ = 1 point and neither a nor b intersects any other curve ai; bj in the Heegaard diagram. (2) The Kirby Calculus[21] gives a finite number of moves which, when applied repeatedly, suffice to change any surgery presentation of a given 3–manifold into any other, at the same time keeping control of the topological type of a 4– manifold which the given 3–manifold bounds. The stabilization–destabilization move is the addition-deletion of an unknotted component with framing 1 to ˙ the defining framed link. (3) Reidemeister’s Theorem (see Burde–Zieschang[15]) relates any two diagrams of the same knot or link, by a finite sequence of elementary moves which are known as RI, RII, RIII. The stabilization–destabilization move is RI. It is easy to see that Markov’s Theorem implies Reidemeister’s Theorem. These theorems are all like Markov’s Theorem in the sense that while the stabilization and destabilization moves are very simple, a sequence of these moves, combined with the appropriate isotopy, can have very non-trivial consequences. Here are other examples in which the stabilization move is not used, at the expense of restricting attention to a special example: (4) W Haken proved that any Heegaard diagram for a non-prime 3–manifold is equivalent to a Heegaard diagram which is the union of two separate Heegaard diagrams, one for each summand, supported on disjoint subsets of the given Heegaard surface. See Scharlemann–Thompson[29] for a very pleasant proof. (5) Waldhausen[32] proved that any two Heegaard diagrams of arbitrary but fixed genus g for the 3–sphere S 3 are equivalent. In the course of an effort which we began in 1990 to discover the theorem which will be the main result of this paper (see Theorem 2 below) the authors made several related contributions to the theory of closed braid representatives of knots and links: Geometry & Topology, Volume 10 (2006) 416 Joan S Birman and William W Menasco (40 ) A split (resp. composite) closed n–braid is an n–braid which factorizes as a product XY where the sub-braid X involves only strands 1;:::; k and the sub- braid Y involves only strands k 1;:::; n (resp. k;:::; n). In the manuscript C [7] the authors proved that if X is a closed n–braid representative of a split or composite link, then up to (braid-index preserving) isotopy and exchange moves, as in Figure 2, X may be assumed to be a split or composite closed braid. w w 1 1 Q P Q P Figure 2: The exchange move (50 ) In the manuscript[9] the authors proved that if X is a closed braid representative of the –component unlink X , then a finite sequence of braid isotopies, exchange moves and destabilization can be found which change X to the closure of the identity braid in the braid group B . (6) In the manuscript[11] the authors discovered that there is another move, the 3–braid flype (see Figure 3) with the property that if X is a closed 3–braid representative of a knot or link type X which cannot be represented by a 1–braid or 2–braid, then either X has a unique conjugacy class or X has exactly two conjugacy classes, and these two classes are related by a 3–braid flype. They also showed that the exchange move can be replaced by braid isotopy for prime links of braid index 3. P P P P R R R R A C C A A A Q Q Q Q positive negative flype flype Figure 3: The two flype moves The authors also established two fundamental facts which gave strong evidence that a more general result might be true: Geometry & Topology, Volume 10 (2006) Stabilization in the braid groups I: MTWS 417 (7) In[8] the authors introduced a complexity function on closed braid representatives of X and proved that, up to exchange moves, there are at most finitely many conjugacy classes of representatives of minimum complexity. (8) In[10] the authors proved that if a link type X has infinitely many conjugacy classes of closed braid representatives of the same braid index, then, up to exchange moves, they fall into finitely many equivalence classes.