The Behavior of the Total Twist and Self-Linking Number of a Closed Space Curve under Inversions
by Thomas F. Banchoff and James H. White

Given that x:C -> E^3 is a smooth imbedding of a closed space curve, the total twist of a a unit normal vector field is a measurement of how much the normal plane turns as it moves along the curve. It can be shown that although the total twist is dependent on the particular vector field, its reduction mod Z, denoted Tw(x)~ is independent of the field. The first part of this paper proves that if x is an imbedded space curve and Ix is its image under an inversion through a sphere, the Tw(x)~ + Tw(Ix)~ = 0.

It is then shown as a corollary that if x and Ix both have nowhere vanishing curvatures, then the normalized total torsion of x mod Z is identical to the negative of the normalized total torsion of Ix mod Z. Similar results hold for conformal transformations of E^3.

The remainder of the article discussed the self-linking number of x, ie. the integer SL(x) which describes the linking number of x moved a small distance in the direction of its principal normal vector field. We prove the the self-linking number is the sum of the normalized total torsion and the Gauss integral, G(x), of x. In the second section, a deformation argument is used to prove the under an inversion, G(x) + G(Ix) = 0, which implies that SL(x) + SL(Ic) is the sum of the respective normalized torsions of x and Ix.

The main theorem, presented in section 3, is:

Theorem 4. If I is an inversion such that x and Ix have nowhere vanishing curvature, then SL(x) + SL(Ix) is equal to the winding numbers of the locus of osculating circles to x about the center of the sphere of inversion.

In the last section, the behavior of plane curves under inversion is examined, and the we give the construction of a curve, C, an inversion, I, such that SL(C)=0 and SL(IC)=a for any integer a.