Ribbon (mathematics)

Geometrical smooth strip

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 ) {\displaystyle (X,U)} includes a curve  X {\displaystyle X} given by a three-dimensional vector X ( s ) {\displaystyle X(s)} , depending continuously on the curve arc-length s {\displaystyle s} ( a ≤ s ≤ b {\displaystyle a\leq s\leq b} ), and a unit vector  U ( s ) {\displaystyle U(s)} perpendicular to  ∂ X ∂ s ( s ) {\displaystyle {\partial X \over \partial s}(s)} at each point.[1] Ribbons have seen particular application as regards DNA.[2]

## Properties and implications

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

The ribbon concept plays an important role in the Călugăreanu formula,[3] [4] that states that

L k = W r + T w , {\displaystyle Lk=Wr+Tw,}

where  L k {\displaystyle Lk} is the asymptotic (Gauss) linking number, the integer number of turns of the ribbon around its axis;  W r {\displaystyle Wr} denotes the total writhing number (or simply writhe), a measure of non-planarity of the ribbon's axis curve; and  T w {\displaystyle 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.

## See also

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* Bollobás–Riordan polynomial
* Knots and graphs
* Knot theory
* DNA supercoil
* Möbius strip

## References

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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 5279522.815-819&rft.date=1971&rft_id=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC389050#id-name=PMC&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0278197#id-name=MR&rft_id=info:bibcode/1971PNAS...68..815B&rft_id=info:pmid/5279522&rft_id=info:doi/10.1073/pnas.68.4.815&rft.aulast=Fuller&rft.aufirst=F.+Brock&rft_id=http://www.pnas.org/content/68/4/815.full.pdf&rfr_id=info:sid/en.wikipedia.org:Ribbon+(mathematics)" class="Z3988">
4. ^ Dennis, M. R.; Hannay, J. H. (26 August 2005). "Geometry of Călugăreanu's theorem". Proceedings of the Royal Society A. 461 (2062): 3245–3254.3245-3254&rft.date=2005-08-26&rft.aulast=Dennis&rft.aufirst=M.+R.&rft.au=Hannay,+J.+H.&rft_id=https://royalsocietypublishing.org/doi/10.1098/rspa.2005.1527&rfr_id=info:sid/en.wikipedia.org:Ribbon+(mathematics)" class="Z3988">

## Bibliography

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* 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 01318465-20&rft.date=1959&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0131846#id-name=MR&rft.aulast=Călugăreanu&rft.aufirst=Gheorghe&rfr_id=info:sid/en.wikipedia.org:Ribbon+(mathematics)" class="Z3988">
* 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 0149378588-625&rft.date=1961&rft_id=info:doi/10.21136/CMJ.1961.100486&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0149378#id-name=MR&rft.aulast=Călugăreanu&rft.aufirst=Gheorghe&rfr_id=info:sid/en.wikipedia.org:Ribbon+(mathematics)" class="Z3988">
* 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 0253264693-728&rft.date=1969&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0253264#id-name=MR&rft_id=https://www.jstor.org/stable/2373348#id-name=JSTOR&rft_id=info:doi/10.2307/2373348&rft.aulast=White&rft.aufirst=James+H.&rfr_id=info:sid/en.wikipedia.org:Ribbon+(mathematics)" class="Z3988">