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Orientations and Semi-Orientations

To orient a set of lines means to give a direction to each line in the set. There are possible orientations of a set of lines. A semi-orientation of a set of lines is a pair of opposite orientations (Figure 2).

**Figure 2:** Semi-orientation of a pair of lines.
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Any pair of nonperpendicular lines has a canonical semi-orientation which is determined by the relative position of the two lines. Namely, we choose an arbitrary orientation of one of the lines, and then we determine the orientation of the second line by rotating the first line in the most economical way (i.e., with the smallest angle of rotation) so as to make it parallel to the second line (see Figure 3). We then give the second line the orientation pointing in the same direction as the (now parallel) first line (Figure 4). Thus, choosing an orientation of one of the lines determines an orientation of the pair. If we choose the opposite orientation of the first line, then we obtain the opposite orientation of the pair. If we were to use the other line to start with, we would obtain the same pair of opposite orientations. These two opposite orientations are what we meant by the canonical semi-orientation of the pair of nonperpendicular lines.

**Figure 3:**
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**Figure 4:**
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An isotropy during which the angle between the lines remains fixed takes the canonical semi-orientation to the canonical semi-orientation. This suggests the idea of considering another type of isotopy--isotopies of semi-oriented pairs of skew lines. Here we allow the angle and distance between the lines to change, but we require that the semi-orientation be preserved. Such an isotopy occupies an intermediate position between an arbitrary isotopy and an isotopy during which the distance and angle (where we suppose that the angle is $\not=90^\circ$ ) remain fixed. That is, if there is no semi-oriented isotopy between two semi-oriented pairs of lines, then there is certainly no isotopy between them which preserves the distance and angle. What can stand in the way of an isotopy of semi-oriented pairs of lines?

Next: The Linking Number Up: Configurations of Skew Lines Previous: Two Lines

Oleg Viro 2000-12-29