Invariants of plane algebraic curves via representations of the braid groups
type: Item
instance of: scholarly article
title: Invariants of plane algebraic curves via representations of the braid groups
zbMATH Open document ID: 0674.14015
DOI: 10.1007/BF01394143
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143644
review text: If two plane curves of the same degree have one to one corresponding singularities which are topologically equivalent, they are said to be equisingular. An equisingular family need not be connected. It is interesting to find invariants which may distinguish the connected components of a family. In a previous paper by the author [Duke Math. J. 49, 833-851 (1982; Zbl 0524.14026)], the Alexander polynomial was shown to have this property. In the note under review the author introduces another invariant to do the job. This invariant, which is defined for any representation of the braid group, is a Laurant polynomial determined up to a unit in the ring \({\mathbb{Q}}[t,t^{-1}]\). In the case of the reduced Burau representation, this invariant essentially coincides with the Alexander polynomial.
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 4103232
zbMATH Keywords: equisingular families
zbMATH Keywords: representation of the braid group
zbMATH Keywords: Laurant polynomial
zbMATH Keywords: Alexander polynomial
author: Anatoly Libgober
MaRDI profile type: Publication
cites work: A polynomial invariant for knots via von Neumann algebras
cites work: A new polynomial invariant of knots and links
cites work: On the Severi problem
cites work: Alexander polynomial of plane algebraic curves and cyclic multiple planes
cites work: Q3481837
cites work: On the homotopy of the complement to plane algebraic curves.
cites work: Q3933100
OpenAlex ID: W2107434495
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description: scientific article; zbMATH DE number 4103232