Annali di Matematica Pura ed Applicata (1923 -)

, Volume 116, Issue 1, pp 57–86

# Space curves with positive torsion

• John A. Little
Article

## Summary

Space curves may be classified under various kinds of deformation. The following six kinds of deformation have been of special interest; namely first, second and third order homotopy and isotopy. (We say the deformation is k-th order if the first k derivatives remain independent during the deformation.) The first order homotopy classification of space curves may be accomplished using well-known methods of Whitney; there is only one class. The second and third order homotopy classification was done by Feldman[1] and Little[6], respectively. The first order isotopy classification of space curves is knot theory; a subject of its own. The second order isotopy classification has been done by W. F. Pohl (unpublished). Thus, aside from knot theory, the only remaining problem is the third order isotopy problem. In this paper we give a partial answer. Our result is partial because we must restrict the class of curves with which we are dealing; namely to curves with a « twist ». But it may well be that every curve does have a twist, in which case our restricted class of curves would be all curves and the classification would be complete. In addition we construct a curve of positive torsion with any preassigned self-linking number in any preassigned knot class; a question raised by W. F. Pohl.

## Keywords

Special Interest Space Curve Partial Answer Restricted Class Homotopy Classification

## References

1. [1]
E. A. Feldman,Deformations of closed space curves, J. Differential Geometry,2 (1968), pp. 67–75.
2. [2]
W. Fenchel, Über Krummung und Windung geschlossene Raumkurven, Math. Ann.,101 (1929), pp. 238–252.
3. [3]
W. Fenchel, Geschlossene Raumkurven mit vorgeschriebenem Tangentenbeld, Jber. Deutchen Math. Verein.,39 (1930), pp. 183–185.
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H. Guggenheimer,Differential Geometry, McGraw-Hill, New York (1963).
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J. A. Little,Nondegenerate homotopies of curves on the unit 2-sphere, J. Differential Geometry,4 (1970), pp. 339–348.
6. [6]
J. A. Little,Third order nondegenerate homotopics of space curves, J. Differential Geometry,5 (1971), pp. 503–515.
7. [7]
W. F. Pohl,The self-linking number of a closed space curve, Journal of Mathematics and Mechanics,17 (1968), pp. 975–986.