Geometry of Călugăreanu's theorem
Geometry of Călugăreanu's theorem
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.
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 Călugăreanu'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 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.
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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
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Date deposited: 11 May 2006
Last modified: 15 Mar 2024 07:31
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Author:
M.R. Dennis
Author:
J.H. Hannay
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