Lettered Baskets II

Compact as it is, the Gauss code for a knot projection omits certain information about the knot and its projection that might be needed:

1. If we are interested in the knot that made the projection (knots of different types can make the same projection, and, in particular, the unknot can mimic any knot projection) we will certainly want to know, as we creep around the knot projection in one-way traffic: is the cross-traffic at each intersection above or below us?

2. Whether or not we are interested in the knot, we may wish to know this much more about the projection: is the cross-traffic at each intersection coming from the left or from the right?

Notice, of course, that on the two visits to a given intersection we will arrive at contrary answers to these two questions: if we see cross-traffic above us, they see cross traffic below them; if we see cross-traffic coming from the right, they see cross-traffic coming from the left.

Around 1960, knot theorist Kunio Murasugi became interested in special alternating knot projections; these are knot projections where complete answers to both the above questions burden the Gauss code with very little additional information. In particular, for these projections the complete answers to the above questions are:

1. Strictly alterating above/below.

2. Strictly alternating right/left.

In fact, if we adopt a convention that only codes that begin "above-right" are valid (a given knot projection has numerous equivalent codes to choose from) the additional information gets built into the Gauss code in a way that is invisible to the user.

The locked-crossing technology is indifferent to the above/below information, so we only need a convention that cross-traffic at the first intersection is from the right. See the photo above of the first crossing, 'A', in a lettered basket.