Studying links via closed braids. V. The unlink

Authors:
Joan S. Birman and William W. Menasco

Journal:
Trans. Amer. Math. Soc. **329** (1992), 585-606

MSC:
Primary 57M25; Secondary 20F36

DOI:
https://doi.org/10.1090/S0002-9947-1992-1030509-1

MathSciNet review:
1030509

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main result is a version of Markov's Theorem which does not involve stabilization, in the special case of the -component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most . 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 shows that exchange moves are the only obstruction to reducing a closed -braid representative of the -component unlink to the standard closed -braid representative, through a sequence of braids of nonincreasing braid index.

**[A]**J. W. Alexander,*A lemma on systems of knotted curves*, Proc. Nat. Acad. Sci. U.S.A.**9**(1923), 93-95.**[Be]**Daniel Bennequin,*Entrelacements et equations de Pfaff*, Asterisque**107-108**(1983), 87-161. MR**753131 (86e:58070)****[Bi]**Joan S. Birman,*Braids, links and mapping class groups*, Ann. of Math. Stud., no. 82, Princeton Univ. Press, Princeton, N.J., 1974. MR**0375281 (51:11477)****[B-M,I]**Joan S. Birman and William W. Menasco,*Studying links via closed braids.*I:*A finiteness theorem*, Pacific J. Math. (to appear). MR**1154731 (93f:57009)****[B-M,II]**-,*Studying links via closed braids.*II:*On a theorem of Bennequin*, Topology Appl. (to appear). MR**1114092 (92g:57009)****[B-M,III]**-,*Studying links via closed braids.*III:*Classifying links which are closed*-*braids*, preprint, 1989.**[B-M,IV]**-,*Studying links via closed braids.*IV:*Split and composite links*, Invent. Math.**102**Fase. 1 (1990), 115-139. MR**1069243 (92g:57010a)****[B-M.VI]**-,*Studying links via closed braids.*VI:*A non-finiteness theorem*, Pacific J. Math. (to appear).**[J]**V. Jones,*Hecke algebra representations of braid groups and link polynomials*, Ann. of Math.**126**(1987), 335-388. MR**908150 (89c:46092)****[G]**F. Garside,*The braid groups and other groups*, Quart. J. Math. Oxford**20**235-254. MR**0248801 (40:2051)****[Ma]**A. A. Markov,*Uber die freie Aquivalenz der geschlossenen Zopfe*, Rec. Soc. Math. Moscou**43**(1936), 73-78.**[Mo,1]**Hugh R. Morton,*An irreducible*-*string braid with unknotted closure*, Math. Proc. Cambridge Philos. Soc.**93**(1983), 259-261. MR**691995 (84m:57006)****[Mo,2]**-,*Threading knot diagrams*, Math. Proc. Cambridge Philos. Soc.**99**(1986), 247-260. MR**817666 (87c:57007)****[R]**L. Rudolph,*Braided surfaces and Seifert ribbons for closed braids*, Comment. Math. Helv.**58**(1983), 1-37. MR**699004 (84j:57006)****[W]**N. Weinberg,*Sur l'equivalence libre des tresses fermees*, Dokl. Akad. Sci. USSR**23**(1939), no. 3.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57M25,
20F36

Retrieve articles in all journals with MSC: 57M25, 20F36

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1030509-1

Keywords:
Knot,
link,
closed braid,
Markov equivalence,
stabilization

Article copyright:
© Copyright 1992
American Mathematical Society