Lettered Baskets: an easy example

This is a 9-crossing lettered basket that stands out as a beginner's project because its Tait graph is 2-connected (meaning no monogon hills or dales, which are fussy to make) and it has a rewarding symmetry when finished. The code, Be Fa Af Gc Cg Hb Eh Id Di, uses the letters A-i, but J is also needed to make the closing splice back to A. The tape is 75-lb Kraft paper, 0.80" wide. At this scale, A-J uses about 48" of paper.

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.

Lettered Baskets I

We know, from notes published posthumously in 1900, that, by the 1820's, the great mathematician Carl Friedrich Gauss knew of a clever way to encode the projection of a knot on the sphere.

A spatial knot is an embedding of a circle into 3D space in a possibly entangled way. What is called a knot in everyday speech would be a good embodiment of a spatial knot if we could seamlessly splice the two tag ends together, making of it a circle of rope. The centerline of a spatial knot becomes a knot projection on the sphere (barring rare accidents of projection such as two strands appearing to be tangent to one another) when placed in a spherical coordinate system and all values of the radial component are rounded to 1.

Thus a knot projection is a general closed curve on the sphere with a finite number of double points where distant portions of the curve have an intersection.

In "Zur Geometria Situs," what we would call topology today, Gauss was uninterested in all the ways the surface of the sphere could be stretched like a sheet of rubber, because these distortions do not change the relative situation of the projection's arcs and double points, and can be considered equivalent.

Above is a simplified drawing of the knot projection Gauss provides as an example, together with his encoding of the projection on the right (the numbers are reproduced exactly as they appear in the 1900 publication of his notes.) It is easy to see what Gauss has done. Starting at the position, and in the direction, of the red arrow I added to the diagram, he has gone around the closed curve numbering each double point, or crossing, as it is encountered {1, 2, 3 ...}. By the time the full circuit is completed, each crossing has been assigned two numbers, one odd and one even. He reports these two-number sets in the order they are completed, conjoining the two numbers with a decimal point while placing the greater number to the left of the decimal point. The last convention has the advantage that the number pairs are reported in the same ordering they would have if read as decimal numbers.

We will use this version of the Gauss encoding to encode lettered baskets, transliterating the counting numbers {1, 2, 3, ...51, 52} into a doubled, uppercase/lowercase alphabet {A, a, B, b ...Z, z}. Thus odd numbers become uppercase letters and even numbers become lowercase. However, we deviate from Gauss' decimal convention in that we do not put the larger number on the left, rather we put the uppercase letter on the left (i.e., we do not put the larger number on the left, we put the odd number on the left.) Thus we report the two-letter pairs in an order that can be seen as alphabetical provided we look only at the z-most letter in each pair.

Clearly, if there are no more than 26 crossings in the projection, we can interconvert between the lettered basket coding and the Gauss coding, but attempting to build the code as a lettered basket will run into problems unless the knot projection is, as we will see below and using Murasugi's term, special.

In the image below, Gauss' coding in leftmost column, its direct transliteration in middle column, and, finally, permuted to place uppercase letters first in each pair in rightmost column. The transliteration key is along the bottom. For the reason mentioned above, the resulting letter code for Gauss' example, Ad Ea Cf Dg Fh Ib Bi Je Hj Kc Gk, does not describe a lettered basket.

Coding corrugated baskets

Corrugated baskets present two different coding problems: how one might mark assembly information directly on a custom-punched tape, and how one might write down (on a piece of paper) assembly information for a tape pre-marked with a standard consecutive numbering. The latter is the topic here.

In the standard consecutive numbering, orienting the tape from the smaller to the larger numbers, each segment of the tape (i.e., a segment between crossings) should be marked near its terminal end as shown above, so that the number naturally becomes a sub-address for the nearest crossing.

All the crossings in a corrugated basket look the same. Rotated so that the converging quadrant is at the bottom, as shown above, they all look like the diagram. The crossing has a binomial address (represented here generically by 0,1): an even sub-address on the left and an odd sub-address on the right. (Those positions might just as well be reversed, but, to make things easier for the basket maker, we will always choose a basepoint for the counting that makes the above diagram correct.)

The oriented geodesic path is always a boundary between white and a tinted region. At each crossing, white switches sides; “black” also switches sides but its tint changes as well. For example, in the diagram, the 0 strand, before this crossing, has white on the right, and the dark tint on the left. After this crossing, white is on the left and the light tint is on the right. Also, the sub-address will have incremented by one and so changed parity.

If we are going to be walking along the path a lot in the oriented direction, it might be good to have the right leg shorter than the left because there is always going to be either a dale on the left or a hill on the right.

0 and 1 in the diagram merely symbolize even and odd sub-addresses, their relative magnitude is unknown: either could be the bigger sub-address and thus the later strand. If, say, the even sub-address is bigger, then we are colliding with the left side of the earlier strand and are potentially closing a dale region, conversely, if the odd address is bigger, then we are colliding with the right side of the earlier strand, and are potentially closing a hill region.

If we always write the full address in "even-odd" order, it is easy to remember on which side of the smaller sub-address the bigger sub-address is approaching.

The ouroboros splice

When finishing a unicursal basket a splice is needed to connect the ribbon's head to its tail. The method advocated here is simple, and is sturdy enough for cardstock baskets. The ribbon is cut just long enough at both ends to doubly-cover the first crossing and the last crossing as shown above. The cutting is not done at 90°, but at the angle determined by the two outer notch positions, as seen in the photo below. The first crossing is unlocked, allowing the ribbon’s tail to be threaded under both the last and first crossing. Those crossings are then locked with a doubled thickness of ribbon underneath. Made this way, the splice is scarcely noticeable in the finished basket.

Locked crossing profiles for cardstock

I have been using the above notch pattern for 2cm-wide 65-lb cardstock weaving elements cut on a Cricut Maker 3 (courtesy of the DC Public Libray's FabLab.) The 7.5° alternately plus or minus rotation of the crossed diagonals makes locked crossings that are 75°/105°. These can be used to corrugate crossings that are nominally 90°/90° (tabby weave) as deficit/excess, as well as crossings that are nominally 60°/120° (kagome weave) as excess/deficit.

Notch detail for locked crossings in aluminum

The notch profile above works for aluminum sheet metal having a springy temper, such as is widely sold to the building trades as flashing (9 mil thickness,) trim stock (18 mil thickness,) and gutter stock (27 mil thickness.)

For example, with 9 mil (0.009") aluminum flashing a width of 1.5", tangent circles of 0.125" diameter, and outer rounds of 0.5" diameter are suitable for weaving with locked crossings.

If the weavers are to cross at an angle other than 90-degrees, the diagonals of the crossing rhombus remain perpendicular to each other, but rotate together to align with the crossing's planes of symmetry. The notches translate to maintain tangency.