Ribbon (mathematics)

In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by $(X,U)$ includes a curve $X$ given by a three-dimensional vector $X(s)$ , depending continuously on the curve arc-length $s$ ( $a\leq s\leq b$ ), and a unit vector $U(s)$ perpendicular to $X$ at each point.^[1] Ribbons have seen particular application as regards DNA.^[2]

Properties and implications[edit]

The ribbon $(X,U)$ is called simple if $X$ is a simple curve (i.e. without self-intersections) and closed and if $U$ and all its derivatives agree at $a$ and $b$ . For any simple closed ribbon the curves $X+\varepsilon U$ given parametrically by $X(s)+\varepsilon U(s)$ are, for all sufficiently small positive $\varepsilon$ , simple closed curves disjoint from $X$ .

The ribbon concept plays an important role in the Călugăreanu-White-Fuller formula,^[3] that states that

Lk=Wr+Tw,

where $Lk$ is the asymptotic (Gauss) linking number, the integer number of turns of the ribbon around its axis; $Wr$ denotes the total writhing number (or simply writhe), a measure of non-planarity of the ribbon's axis curve; and $Tw$ is the total twist number (or simply twist), the rate of rotation of the ribbon around its axis.

Ribbon theory investigates geometric and topological aspects of a mathematical reference ribbon associated with physical and biological properties, such as those arising in topological fluid dynamics, DNA modeling and in material science.

References[edit]

^ Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
^ Vologodskiǐ, Aleksandr Vadimovich (1992). Topology and Physics of Circular DNA (First ed.). Boca Raton, FL. p. 49. ISBN 978-1138105058. OCLC 1014356603.{{cite book}}: CS1 maint: location missing publisher (link)
^ Fuller, F. Brock (1971). "The writhing number of a space curve" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 68 (4): 815–819. Bibcode:1971PNAS...68..815B. doi:10.1073/pnas.68.4.815. MR 0278197. PMC 389050. PMID 5279522.

Bibliography[edit]

Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 0-8218-3678-1, MR 2079925
Călugăreanu, Gheorghe (1959), "L'intégrale de Gauss et l'analyse des nœuds tridimensionnels", Revue de Mathématiques Pure et Appliquées, 4: 5–20, MR 0131846
Călugăreanu, Gheorghe (1961), "Sur les classes d'isotopie des noeuds tridimensionels et leurs invariants", Czechoslovak Mathematical Journal, 11: 588–625, doi:10.21136/CMJ.1961.100486, MR 0149378
White, James H. (1969), "Self-linking and the Gauss integral in higher dimensions", American Journal of Mathematics, 91 (3): 693–728, doi:10.2307/2373348, JSTOR 2373348, MR 0253264

[1] Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495

[2] Vologodskiǐ, Aleksandr Vadimovich (1992). Topology and Physics of Circular DNA (First ed.). Boca Raton, FL. p. 49. ISBN 978-1138105058. OCLC 1014356603.{{cite book}}: CS1 maint: location missing publisher (link)

[3] Fuller, F. Brock (1971). "The writhing number of a space curve" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 68 (4): 815–819. Bibcode:1971PNAS...68..815B. doi:10.1073/pnas.68.4.815. MR 0278197. PMC 389050. PMID 5279522.

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