Topology in Molecular Biology

6 Writhe Versus the Number of Crossings 125

tangle (a)(b)(c)...(r) having a fixed number of crossings onr−1rows,PWr
presents a linear behavior versusn, the minimal crossing number of the knot
or link, with a slope of± 4 /7or± 10 /7. One can also consider families of ratio-
nal tangles (a)(b)(c)...(r), where several ofa,b,c,...change in a coordinated
fashion, such thatPWris still a linear function ofn.Itisalsopossibleto
compute the shift between two lines having the same slope. We thus have
at our disposal a formalism allowing to predict a number of data usually ob-
tained numerically, and which might help to shed some light on the “quantum
mystery of knots” [1].

Acknowledgments

C. C. is a Research Associate of the Belgian FNRS (National Fund for Sci-
entific Research). This work was supported in part by Swiss National Science
Foundation Grant 31-61636.00 to A. S.

References

1. K. Devlin, New Scientist, 40–42 (2001)


2. V. Katritch, J. Bednar, D. Michoud, R.G. Scharein, J. Dubochet, A. Stasiak,
   Nature 384 142–145 (1996)


3. E.J. Janse van Rensburg, E. Orlandini, D.W. Sumners, M.C. Tesi, S.G. Whit-
   tington, Phys. A: Math. Gen. 26 L981–L98 (1993)


4. E.J. Janse van Rensburg, D.W. Sumners, S.G. Whitington, inIdeal Knots,ed.
   by A. Stasiak, V. Katritch, L.H. Kauffman, (World Scientific, Singapore, 1998),
   pp. 70–87


5. A. Stasiak, J. Dubochet, V. Katritch, P. Pieranski, in Ideal Knots,ed.by
   A. Stasiak, V. Katritch, L.H. Kauffman (World Scientific, Singapore, 1998),
   pp. 1–19


6. P. Pieranski, inIdeal Knots, A. Stasiak, V. Katritch, L.H. Kauffman (World
   Scientific, Singapore, 1998), pp. 20–41


7. A. Stasiak, inProceedings of the Delphi Conference on Knots, ed. by C. Gordon,
   V.F.R. Jones, L. Kauffman, S. Lambropoulou, J.H. Przytycki (World Scientific,
   Singapore, 2000), pp. 477–500


8. C. Cerf, A. Stasiak, Proc. Natl Acad. Sci. USA 97 3795–3798 (2000)


9. C. Cerf, J. Knot Theory Ramif. 6 621–632 (1997)


10. P. Pieranski, S. Przybyl, Eur. Phys. J. E 6 117 (2001)


11. J.-Y. Huang, and P.-Y. Lai, Phys. Rev. E 63 021506 (2001)


12. J.H. Conway, inComputational Problems in Abstract Algebra(Pergamon Press,
    Oxford, 1970), pp. 329–358


13. C. Cerf, Topology Atlas Invited Contributions 3 1–32 (1998).
    http://at.yorku.ca/t/a/i/c/31.htm