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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*.