Self-linking of spatial curves without inflections and its applications (Q1580162)

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 [\textit{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.