## Geometry & Topology

### Stabilization in the braid groups I: MTWS

#### Abstract

Choose any oriented link type $X$ and closed braid representatives $X+,X−$ of $X$, 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 them with closed braids $X+′,X−′$) there is a sequence of closed braid representatives $X+′=X1→X2→⋯→Xr=X−′$ such that each passage $Xi→Xi+1$ is strictly complexity reducing and non-increasing on braid index. The templates which define the passages $Xi→Xi+1$ include 3 familiar ones, the destabilization, exchange move 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.

#### Article information

Source
Geom. Topol., Volume 10, Number 1 (2006), 413-540.

Dates
Accepted: 25 January 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799713

Digital Object Identifier
doi:10.2140/gt.2006.10.413

Mathematical Reviews number (MathSciNet)
MR2224463

Zentralblatt MATH identifier
1128.57003

#### Citation

Birman, Joan S; Menasco, William W. Stabilization in the braid groups I: MTWS. Geom. Topol. 10 (2006), no. 1, 413--540. doi:10.2140/gt.2006.10.413. https://projecteuclid.org/euclid.gt/1513799713

#### References

• J W Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923) 93–95
• D Bennequin, Entrelacements et équations de Pfaff, from: “Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)”, Astérisque 107, Soc. Math. France, Paris (1983) 87–161
• J S Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J. (1974)
• J S Birman, E Finkelstein, Studying surfaces via closed braids, J. Knot Theory Ramifications 7 (1998) 267–334
• J S Birman, M D Hirsch, A new algorithm for recognizing the unknot, Geom. Topol. 2 (1998) 175–220
• J S Birman, W W Menasco, Stabilization in the braid groups II: Applications to transverse knots
• J S Birman, W W Menasco, Studying links via closed braids. IV. Composite links and split links, Invent. Math. 102 (1990) 115–139
• J S Birman, W W Menasco, Studying links via closed braids. I. A finiteness theorem, Pacific J. Math. 154 (1992) 17–36
• J S Birman, W W Menasco, 1030509
• J S Birman, W W Menasco, Studying links via closed braids. VI. A nonfiniteness theorem, Pacific J. Math. 156 (1992) 265–285
• J S Birman, W W Menasco, Studying links via closed braids. III. Classifying links which are closed $3$-braids, Pacific J. Math. 161 (1993) 25–113
• J S Birman, W W Menasco, On Markov's theorem, J. Knot Theory Ramifications 11 (2002) 295–310
• J S Birman, M Rampichini, P Boldi, S Vigna, Towards an implementation of the B-H algorithm for recognizing the unknot, J. Knot Theory Ramifications 11 (2002) 601–645
• J S Birman, N C Wrinkle, On transversally simple knots, J. Differential Geom. 55 (2000) 325–354
• G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co., Berlin (1985)
• P R Cromwell, Embedding knots and links in an open book. I. Basic properties, Topology Appl. 64 (1995) 37–58
• I Dynnikov, Arc presentations of links: monotonic simplification
• T Fiedler, A small state sum for knots, Topology 32 (1993) 281–294
• J Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J. (1976)
• K Kawamuro, Failure of the Morton–Franks–Williams inequality
• R Kirby, A calculus for framed links in $S\sp{3}$, Invent. Math. 45 (1978) 35–56
• S Lambropoulou, C P Rourke, Markov's theorem in $3$-manifolds, Topology Appl. 78 (1997) 95–122
• A A Markov, Uber die freie Aquivalenz geschlossener Zopfe, Recueil Mathematique Moscou 1 (1935) 73–78
• W W Menasco, Closed braids and Heegaard splittings, from: “Knots, braids, and mapping class groups–-papers dedicated to Joan S. Birman (New York, 1998)”, AMS/IP Stud. Adv. Math. 24, Amer. Math. Soc., Providence, RI (2001) 131–141
• W W Menasco, On iterated torus knots and transversal knots, Geom. Topol. 5 (2001) 651–682
• H R Morton, Infinitely many fibred knots having the same Alexander polynomial, Topology 17 (1978) 101–104
• H R Morton, An irreducible $4$-string braid with unknotted closure, Math. Proc. Cambridge Philos. Soc. 93 (1983) 259–261
• H R Morton, Threading knot diagrams, Math. Proc. Cambridge Philos. Soc. 99 (1986) 247–260
• M Scharlemann, A Thompson, Thin position and Heegaard splittings of the $3$-sphere, J. Differential Geom. 39 (1994) 343–357
• J Singer, }, Trans. Amer. Math. Soc. 35 (1933) 88–111
• P Traczyk, A new proof of Markov's braid theorem, from: “Knot theory (Warsaw, 1995)”, Banach Center Publ. 42, Polish Acad. Sci., Warsaw (1998) 409–419
• F Waldhausen, Heegaard-Zerlegungen der $3$-Sphäre, Topology 7 (1968) 195–203
• J Zablow, Loops, waves and an algebra for Heegaard splittings, PhD thesis, City University of New York (1999)