Invariants of plane algebraic curves via representations of the braid groups

Abstract

Recently new remarkable polynomial invariants of knots in 3-sphere were introduced using some representations of the braid groups which were shown to be effective in distinguishing knots ([-J], [-FYHLMO]). The purpose of this note is to introduce, using representation of the braid groups, an invariant of continuous equisingular families of plane algebraic curves. A pleasant feature is that this invariant is defined for any representation of the braid group (no coherence of representations for different number of strings is required), though for many representations this invariant is trivial. In the case of the reduced Burau representation, this invariant essentially coincides with the Alexander polynomial of plane curves introduced in [-L 1]. This fact is the counterpart of the well-known relation between the Alexander polynomial of closed braids and the reduced Burau representation. The definition of this invariant depends on braid monodromy associated with the curve ([M]). In particular we obtain a direct (not involving calculations of the fundamental group, though computationally cumbersome) method of computation of the Alexander polynomial via braid monodromy. In the next section we shall recall the background on continuous equisingular families of plane algebraic curves and their braid monodromies. In the Sect. 3 we introduce the invariant and in the last section we consider the case of reduced Burau representations.