Finite-type invariants for curves on surfaces
Noboru Ito
TL;DR
This paper introduces finite-type invariants for curves on surfaces, extending Arnold's invariants for plane curves, using Turaev's word theory to classify stable homeomorphism classes.
Contribution
It develops a higher-order generalization of Arnold's invariants for curves on surfaces via Turaev's word theory, broadening the understanding of curve classification.
Findings
Constructed finite-type invariants for curves on surfaces.
Extended Arnold's invariants to higher orders.
Applied word theory to classify stable homeomorphism classes.
Abstract
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-order invariants for plane immersed curves. The invariants in this theory are developed using the word theory proposed by Turaev.
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Taxonomy
TopicsGeometric and Algebraic Topology · Algebraic Geometry and Number Theory · Geometric Analysis and Curvature Flows