A progress report on the study of links via closed braids. (English) Zbl 0890.57010

Summary: The \(n\)-string braid groups \(\{B_n\), \(n=1,2,3,\dots\}\) were introduced into the mathematical literature in 1925 in a foundational paper by Emil Artin [Abh. Math. Semin. Univ. Hamb. 4, 47-72 (1925; JFM 51.0450.01)]. Earlier, it had been proved by J. W. Alexander [Proc. Natl. Acad. Sci. USA 9, 93-95 (1923; JFM 49.0408.03)] that every knot or link could be represented (in many ways) as a closed braid. Artin proposed that the braid groups might be useful as part of a program to study and ultimately classify knots and links. The goal of this report is to discuss progress on Artin’s program. The discovery in 1984 of new knot and link invariants (i.e. the Jones polynomial and its various generalizations) via the theory of braids is discussed. The author’s recent contributions in joint work with W. W. Menasco [Pac. J. Math. 154, No. 1, 17-36 (1992; Zbl 0724.57001); Topology Appl. 40, No. 1, 71-82 (1991; Zbl 0722.57001); Pac. J. Math. 161, No. 1, 25-113 (1993; Zbl 0813.57010); Invent. Math. 102, No. 1, 115-139 (1990; Zbl 0711.57006); Trans. Am. Math. Soc. 329, No. 2, 585-606 (1992; Zbl 0758.57005); Pac. J. Math. 156, No. 2, 265-286 (1992; Zbl 0739.57002)] to the study of links via braids is discussed, too.


57M25 Knots and links in the \(3\)-sphere (MSC2010)