On the number of closed braids obtained as a result of single stabilizations and destabilizations of a closed braid

Malyutin, A.

doi:10.1090/S1061-0022-07-00980-6

On the number of closed braids obtained as a result of single stabilizations and destabilizations of a closed braid
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by A. V. Malyutin
Translated by: the author

St. Petersburg Math. J. 18 (2007), 1011-1020

DOI: https://doi.org/10.1090/S1061-0022-07-00980-6

Published electronically: October 2, 2007

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Abstract:

Sufficient conditions for a closed $n$-braid $\widehat {\beta }$ to have infinite sets ${\mathfrak {D}}(\widehat {\beta })$ and ${\mathfrak {S}}(\widehat {\beta })$ are given, where ${\mathfrak {D}}(\widehat {\beta })$ denotes the set of all closed $(n-1)$-braids that are obtained from $\widehat {\beta }$ via Markov destabilization, while ${\mathfrak {S}}(\widehat {\beta })$ denotes the set of all closed $(n+1)$-braids that are obtained from $\widehat {\beta }$ via Markov stabilization. New integer-valued conjugacy invariants for the braid group are introduced.

References

J. W. Alexander, A lemma on system of knotted curves, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 93–95.
E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg 4 (1925), 47–72.
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
A. A. Markov, Über die freie Äquivalenz der geschlossenen Zöpfe, Mat. Sb. (N.S.) 1 (43) (1936), no. 1, 73–78. (German)