Skip to main content

Topological Quantities: Calculating Winding, Writhing, Linking, and Higher Order Invariants

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1973)

Abstract

Many topological calculations can be done most easily using the basic idea of winding number. This chapter demonstrates the use of winding number techniques in calculating writhe, linking number, twist, and higher order braid invariants. The writhe calculation works for both closed and open curves. These measures have applications in molecular biology, materials science, fluid mechanics and astrophysics.

Keywords

  • Tangent Vector
  • Boundary Plane
  • Axis Curve
  • Relative Position Vector
  • Topological Quantity

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AKT95] Aldinger J, Klapper I, & Tabor M: Formulae for the calculation and estimation of writhe. J. Knot Theory Ram., 4, 343–372 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. [B91] Berger M A: Third order braid invariants. J. Physics A: Mathematical and General, 24, 4027–4036 (1991)

    CrossRef  ADS  MATH  Google Scholar 

  3. [B01] Berger M A: Topological invariants in braid theory. Letters in Math. Physics, 55, 181–192 (2001)

    CrossRef  MATH  Google Scholar 

  4. [BP06] Berger M A & Prior P: The writhe of open and closed curves. J. Physics A: Mathematical and General, 39, 8321–8348 (2006)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  5. [Ba00] Baty H: Magnetic topology during the reconnection process in a kinked coronal loop. Astronomy and Astrophysics, 360, 345–350 (2000)

    ADS  Google Scholar 

  6. [C59] Călugăreanu G: Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants. Czechoslovak Math J, 11, 588–625 (1959)

    Google Scholar 

  7. [C05] Cantarella J: On comparing the writhe of a smooth curve to the writhe of an inscribed polygon. SIAM J. of Numerical Analysis, 42, 1846–1861 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. [CD00] Chmutov S V & Duzhin S V: The Kontsevich Integral. Acta Appl. Math., 66, 155–190 (2000)

    CrossRef  MathSciNet  Google Scholar 

  9. [DH05] Dennis M R & Hannay J H: Geometry of Călugăreanu 's theorem. Proc. Roy. Soc. A, 461, 3245–3254 (2005)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  10. [F78] Fuller F B: Decomposition of the linking of a ribbon: a problem from molecular biology. Proc. Natl. Acad. Sci. USA, 75, 3557–3561 (1978)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  11. [GVV03] Ghrist R W, Van den Berg J B, & Vandervorst R C: Morse theory on spaces of braids and Lagrangian dynamics. Inventiones Mathematicae, 152, 369–432 (2003)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  12. [K93] Kontsevich M: Vassiliev's knot invariants. Adv. Soviet Math., 16, 137–150 (1993)

    MathSciNet  Google Scholar 

  13. [KHS06] Kristiansen K D, Helgesen G, Skjeltorp A T: Braid theory and Zipf-Mandelbrot relation used in microparticle dynamics. European Physical J., B 51, 363–371 (2006)

    ADS  Google Scholar 

  14. [LK97] Longcope D W & Klapper I: Dynamics of a thin twisted flux tube. Astrophysical J., 488, 443–453 (1997)

    CrossRef  ADS  Google Scholar 

  15. [MR92] Moffatt H K & Ricca R L: Helicity and the Călugăreanu invariant. Proc. Roy. Soc. A, 439, 411–429 (1992)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  16. [O94] Orlandini E, Test M C, Whittington S G, Sumners D W, & Janse van Rensburg E J: The writhe of a self-avoiding walk. J. Physics A: Mathematical and General, 27, L333–L338 (1994)

    CrossRef  ADS  Google Scholar 

  17. [R05] Ricca R L: Inflexional disequilibrium of magnetic flux-tubes. Fluid Dynamics Research, 36, 319–332 (2005)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  18. [RM03] Rossetto V & Maggs A C: Writhing geometry of Open DNA. J. Chem. Phys., 118, 9864–9874 (2003)

    CrossRef  ADS  Google Scholar 

  19. [RK96] Rust D M & Kumar A: Evidence for helically kinked magnetic flux ropes in solar eruptions. Astrophys. J. Lett., 464, L199–L202 (1996)

    CrossRef  ADS  Google Scholar 

  20. [S05] Starostin E L: On the writhing number of a non-closed curve. In: Calvo J, Millett K, Rawdon E, & Stasiak A (eds) Physical and Numerical Models in Knot Theory Including Applications to the Life Sciences. Series on Knots and Everything, World Scientific Publishing, Singapore 525–545 (2005)

    CrossRef  Google Scholar 

  21. [TK05] Török T & Kliem B: Confined and ejective eruptions of kink-unstable flux ropes. Astrophysical J., 630, L97–L100 (2005)

    CrossRef  ADS  Google Scholar 

  22. [VT00] van der Heijden G H M & Thompson J M T: Helical and localised buckling in twisted rods: A unified analysis of the symmetric case. Nonlinear Dynamics, 21, 71–99 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. [VM97] Vologodskii A V & Marko J F: Extension of torsionally stressed DNA by external force. Biophys. J., 73, 123–132 (1997)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mitchell A. Berger .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Berger, M.A. (2009). Topological Quantities: Calculating Winding, Writhing, Linking, and Higher Order Invariants. In: Ricca, R. (eds) Lectures on Topological Fluid Mechanics. Lecture Notes in Mathematics(), vol 1973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00837-5_2

Download citation