The braid index of an algebraic link. (English) Zbl 0673.57003

Williams, R. F.

Braids, AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 78, 697-703 (1988).

[For the entire collection see Zbl 0651.00010.]
Let L be an algebraic link, i.e. the intersection of a small sphere about the origin in \({\mathbb{C}}^ 2\) with a complex algebraic surface \(f(z,w)=0\) in \({\mathbb{C}}^ 2\) which has an isolated singular point at the origin. D. Eisenbud and W. Neumann [Three-dimensional link theory and invariance of plane curve singularities, Ann. Math. Stud. 110 (Princeton, New Jersey) (1985; Zbl 0628.57002)], devised a procedure, based on the polynomial f, for representing L as an “iterated torus link” and in particular as a positive closed braid (“positive” means that all crossings are of the same sign). In the present article the author proves that this braid is actually “very positive”, i.e. contains the full twist as a factor, and is therefore a minimal closed braid representative of L, by a theorem of J. Franks and the author [Trans. Am. Math. Soc. 303, 97-108 (1987; Zbl 0647.57002)] and H. R. Morton [Closed braid representatives for a link and its Jones-Conway polynomial, Preprint (1985)]. This means that one can calculate the braid index of L directly from f.

Reviewer: J.Vrabec

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
14B05	Singularities in algebraic geometry
32S05	Local complex singularities

Keywords:

isolated singularity of a complex curve; algebraic link; plane curve singularities; iterated torus link; positive closed braid; minimal closed braid representative

Citations:

Zbl 0651.00010; Zbl 0628.57002; Zbl 0647.57002