Self-linking of spatial curves without inflections and its applications. (English. Russian original) Zbl 0978.57009
Funct. Anal. Appl. 34, No. 2, 79-85 (2000); translation from Funkts. Anal. Prilozh. 34, No. 2, 1-8 (2000).
Let \(\gamma\) be a smooth closed curve in the standard 3-space. The self-linking number of \(\gamma\) is the linking number of \(\gamma\) with a nearby curve \(\gamma_\varepsilon\) obtained from \(\gamma\) by shifting each point by \(\varepsilon\) in the principal normal direction at that point [G. Călugăreanu, Acad. Républ. Popul. Roum; Rev. Math. Pur. Appl. 4, 5-20 (1959; Zbl 0134.43005)]. The author shows that this topological invariant is a Vassiliev invariant of the first order. Then he gives a formula expressing the self-linking number of a generic curve without inflection points via the signs of the double points of a generic projection of the curve on a plane and the signs of the torsion at the points that are projected into inflection points. Finally, every local invariant of generic curves is proved to be (up to an additive constant) a linear combination of the number of flattening points and the self-linking number.
Reviewer: Alberto Cavicchioli (Modena)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 53A04 | Curves in Euclidean and related spaces |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
Citations:
Zbl 0134.43005References:
| [1] | G. Calugareanu, ”L’integrale de Gauss et I’analyse des noeuds tridimensionnels” Rev. Math. Pures Appl.,4, No. 1, 5–20 (1959). |
| [2] | M. Barner, ”Über die Mindestanzahl stationärer Schmiegebenen bei geschlossen streng-konvexen Raumkurven,” Abh. Math. Sem. Univ. Hamburg,20, 196–215 (1956). · Zbl 0071.15601 |
| [3] | V. I. Arnold, ”On the number of flattening points on space curves,” Institute Mittag-Leffler. Report no.1, (1994–1995), pp. 1–13. |
| [4] | V. D. Sedykh, ”A theorem on four vertices for a spatial curve,” Funkts. Anal. Prilozhen.26, No. 1, 35–41 (1992). · Zbl 0777.53004 |
| [5] | M. C. Romero-Fuster and V. D. Sedykh, ”A lower estimate for the number of zero-torsion points of a space curve,” Beiträge Algebra Geom.,38, No. 1, 183–192 (1978). · Zbl 1008.53001 |
| [6] | I. Rodrigues Costa Sueli, ”On closed twisted curves,” Proc. Amer. Math. Soc.,109, No. 1, 205–214 (1990). · Zbl 0702.53005 |
| [7] | F. Aicardi, ”Topological invariants of knots and framed knots in the solid torus,” C. R. Acad. Sci. Paris Sér. I321, 199–204 (1995). · Zbl 0835.57020 |
| [8] | J. H. White, ”Self-linking and Gauss integral in higher dimensions,” Amer. J. Math.,91, 693–728 (1969). · Zbl 0193.50903 · doi:10.2307/2373348 |
| [9] | F. B. Fuller, ”Decomposition of the linking number of a closed ribbon: a problem from molecular biology,” Proc. Nat. Acad. Sci. U.S.A.,75, No. 8, 3557–3561 (1978). · Zbl 0395.92010 · doi:10.1073/pnas.75.8.3557 |
| [10] | L. H. Kauffman, ”Knots and Physics,” Word Scientific, Singapore, 1991, pp. 488–489. · Zbl 0733.57004 |
| [11] | V. I. Arnold and B. A. Khesin, ”Topological Methods in Hydrodynamics,” In: Applied Mathematical Sciences. Vol. 125, Springer, 1998, pp. 177–179. · Zbl 0902.76001 |
| [12] | V. I. Arnold, ”Topological issues in the theory of asymptotic curves,” Trudy MIAN,225 (1999). · Zbl 1010.37012 |
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