Unknots with highly knotted control polygons

An example is presented of a cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bézier curve. These examples complement known upper bo...

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Bibliographic Details
Published inComputer aided geometric design Vol. 28; no. 3; pp. 212 - 214
Main Authors Bisceglio, J., Peters, T.J., Roulier, J.A., Séquin, C.H.
Format Journal Article
LanguageEnglish
Published Kidlington Elsevier B.V 01.03.2011
Elsevier
Subjects
Bezier
Bezier curves
Complement
Computer aided design
Construction
Exact sciences and technology
Isotopy
Knots
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Polygons
Sciences and techniques of general use
Subdivisions
Theorems
Upper bounds
Bezier curves
Knots
Isotopy
Upper bound
Bézier curve
Cubics
Subdivision
Polygon
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Summary:An example is presented of a cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bézier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its Bézier curve. ► We create an unknottted Bezier curve which has a knotted control polygon. ► We extend the construction to a highly unknotted control polygon. ► Our examples complement subdivision theorems on eventual equivalence of knot types.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0167-8396
1879-2332
DOI:10.1016/j.cagd.2011.01.001