Invariants of degree one of almost generic plane immersed curves
type: Item
instance of: scholarly article
title: Invariants of degree one of almost generic plane immersed curves
zbMATH Open document ID: 0974.57017
DOI: 10.1142/S0218216500000621
author: Sara Mohanty
publication date: 2001-12-16T00:00:00Z
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R45
zbMATH DE Number: 1582264
zbMATH Keywords: plane curve
zbMATH Keywords: generic immersion
zbMATH Keywords: singularity
zbMATH Keywords: bifurcation diagram
zbMATH Keywords: discriminant
MaRDI profile type: Publication
published in: Journal of Knot Theory and its Ramifications
review text: Let \(\Omega\) be the infinite dimensional space of all immersions of a circle into the plane, \(\Sigma\) its discriminant (stratified) hypersurface, consisting of non-generic immersions, and \(\Sigma^1\) the codimension 1 strata of \(\Sigma\), consisting of immersions with only one singularity of degree 1. In [Adv. Sov. Math. 21, 33-91 (1994; Zbl 0864.57027)] \textit{V. I. Arnold} makes use of the properties of \(\Sigma^1\) to introduce the three basic (integer) invariants \(St,J^+,J^-\) of plane generic immersed curves (having only ordinary double points of transversal intersections). In analogy, the present paper introduces seven new invariants of degree 1 of curves in \(\Sigma^1\), by making use of the properties of the codimension 2 strata \(\Sigma^2\) (consisting of immersions with only one singularity of degree 2). Moreover, the author provides an axiomatic description of these invariants, which are dependent on the orientation of the immersed circle and take value either in \(\mathbb{Z}\) (five of them) or in \(\mathbb{Z}_3\) (two of them).
reviewed by: Maria Rita Casali
description: scientific article