Geometry of C?lug?reanu's theorem

Geometry of C?lug?reanu's theorem

A central result in the space geometry of closed twisted ribbons is C?lug?reanu's theorem (also known as White's formula, or the C?lug?reanu–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.

ribbon, linking number, twist, writhe, Gauss map, curve framing

3245-3254

Dennis, M.R.

ff55cf66-eb8b-4eb9-83eb-230c2f223d61

Hannay, J.H.

90d8c251-4201-42a3-80df-2eaa42eaedd5

2005

Dennis, M.R.

ff55cf66-eb8b-4eb9-83eb-230c2f223d61

Hannay, J.H.

90d8c251-4201-42a3-80df-2eaa42eaedd5

Dennis, M.R. and Hannay, J.H.
(2005)
Geometry of C?lug?reanu's theorem.
*Proceedings of the Royal Society A*, 461 (2062), .
(doi:10.1098/rspa.2005.1527).

## Abstract

A central result in the space geometry of closed twisted ribbons is C?lug?reanu's theorem (also known as White's formula, or the C?lug?reanu–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.

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## More information

Published date: 2005

Keywords:
ribbon, linking number, twist, writhe, Gauss map, curve framing

## Identifiers

Local EPrints ID: 29395

URI: http://eprints.soton.ac.uk/id/eprint/29395

ISSN: 1364-5021

PURE UUID: cbf38891-a199-4b85-814f-958136a83c40

## Catalogue record

Date deposited: 11 May 2006

Last modified: 16 Dec 2019 19:21

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## Contributors

Author:
M.R. Dennis

Author:
J.H. Hannay

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