Geometry of Călugăreanu's theorem

Author:

Dennis M.R1,Hannay J.H2

Affiliation:

1. School of Mathematics, University of SouthamptonHighfield, Southampton SO17 1BJ, UK

2. H.H. Wills Physics Laboratory, University of BristolTyndall Avenue, Bristol BS8 1TL, UK

Abstract

A central result in the space geometry of closed twisted ribbons is Călugăreanu's theorem (also known as White's formula, or the Călugăreanu–White–Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately)—the ‘local’ crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve—the ‘writhe framing’ ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero.

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

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