A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric 'energies' given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.

Ricca, R., Oberti, C. (2016). On torus knots and unknots. JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, 25(6) [10.1142/S021821651650036X].

On torus knots and unknots

RICCA, RENZO
;
2016

Abstract

A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric 'energies' given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.
Articolo in rivista - Articolo scientifico
curvature; linking number; parametric equations; torsion; Torus knots; writhe;
curvature; linking number; parametric equations; torsion; Torus knots; writhe
English
Ricca, R., Oberti, C. (2016). On torus knots and unknots. JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, 25(6) [10.1142/S021821651650036X].
Ricca, R; Oberti, C
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/135485
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