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Studying links via closed braids. I: A finiteness theorem. (English) Zbl 0724.57001

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

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
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