Studying links via closed braids. I: A finiteness theorem

Joan Birman and William W. Menasco

Vol. 154 (1992), No. 1, 17–36

DOI: 10.2140/pjm.1992.154.17

Abstract

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

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 5 March 1990

Revised: 29 May 1991

Published: 1 May 1992