On computational complexity of plane curve invariants

The theory of generic smooth closed plane curves initiated by Vladimir Arnold is a beautiful fusion of topology, combinatorics, and analysis. The theory remains fairly undeveloped. We review existing methods to describe generic smooth closed plane curves combinatorially, introduce a new one, and giv...

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Bibliographic Details
Main Authors: Tao, Biaoshuai, Duzhin, Fedor
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2015
Subjects:
Online Access:https://hdl.handle.net/10356/107188
http://hdl.handle.net/10220/25396
http://analytic-combinatorics.org/index.php/ojac/article/view/P2
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Institution: Nanyang Technological University
Language: English
Description
Summary:The theory of generic smooth closed plane curves initiated by Vladimir Arnold is a beautiful fusion of topology, combinatorics, and analysis. The theory remains fairly undeveloped. We review existing methods to describe generic smooth closed plane curves combinatorially, introduce a new one, and give an algorithm for efficient computation of Arnold's invariants. Our results provide a good source of future research projects that involve computer experiments with plane curves. The reader is not required to have background in topology and even undergraduate students with basic knowledge of differential geometry and graph theory will easily understand our paper.