Introduction

A curve in three-space is said to have self-linking numberb n if the curve has linking number n with the push-off curve in the direction of the principal normal vector. The notion was first described by Calugareanu [C1] and rediscovered and extended by Pohl [P1].

What about a curve on the three-sphere? One approach is to project the curve stereographically into three-space and to take the self-linking number of the resulting space curve. This does not work for the simple reason that the self-linking number can be different for projections from different points. This raises the question about the actual self-linking number on the three-sphere and its relationship with the self-linking numbers of the projections into three-space.

To see a curve on the three-sphere rotating in red and white, activate the movie

To see a curve on the three-sphere rotating with strange colors, activate the movie

To see a curve on the three-sphere rotating with nice colors but low resolution, activate the movie

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