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

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.

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 |