Open Access

ARTICLE

The Geometric Interpretation of Linking Number, Writhe and Twist for a Ribbon

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1 School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore. Email: mckau@ntu.edu.sg

Computer Modeling in Engineering & Sciences 2008, 29(3), 151-162. https://doi.org/10.3970/cmes.2008.029.151

Abstract

Ribbons may be used for the modeling of DNAs and proteins. The topology of a ribbon can be described by the linking number, while its geometry is represented by the writhe and the twist. These quantities are integrals and are related by the Cǎlugǎreanu's theorem from knot theory. This theorem also describes the relationship between the various conformations. The heart of the Cǎlugǎreanu's theorem rests in the Gauss Integral. Due to the large number of molecules, the topology and the geometry of a ribbon model can be very complicated. As a result, these integrals are commonly evaluated by numerical methods. The writhe of a ribbon is usually computed by mapping its self-crossing onto a unit sphere. The twist of a ribbon is mainly due to the geometric relationship between its central spine and boundary. This article offers a geometric interpretation of the twist of a ribbon in terms of crossing between the central spine and boundary of the ribbon. This approach facilitates the calculation of the twist of a ribbon. Some basics in visibility will enhance the intuition.

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