Monsters in the hollow: counting Naiki braid patterns using de Bruijn's Monster theorem

Figures & data

Figure 1. Examples of Naiki braids. Braiding and photography by Rosalie Neilson.

Three red and yellow Naiki braids, each shown next to a U.S. penny for size comparison.

Figure 2. (a) The structure of Naiki. Strands going off one side are assumed to wrap around to the other. (b) An abstracted version of the structure, with a fundamental region marked.

A grid of blue and yellow interlaced strands, labeled with numbers, and a grid of blue and yellow diamonds, labeled with numbers, with an 8x8 red diamond superimposed on it.

Figure 3. (a) A stripe pattern in the S direction. (b) A stripe pattern in the Z direction. (c) Regularly spaced spots in the S direction. (d) Regularly spaced spots in the Z direction.

Four grids of blue and yellow interlaced strands, labeled with numbers 1 through 16.

Figure 4. (a) A rotation of the braid around its axis. (b) A helical transformation of the braid fixing the odd-numbered threads.

A grid of blue and yellow diamonds, labeled with numbers 1 through 16, with short red left-pointing arrows superimposed, and a grid of blue and yellow diamonds, labeled with numbers 1 through 16, with short red left-and-down arrows superimposed.

Figure 5. (a) A rotation of the braid around a line perpendicular to its axis. (b) A glide plane reflection of the braid.

A grid of blue and yellow diamonds, labeled with numbers 1 through 16, with concentric half-circle arrows superimposed, and a grid of blue and yellow diamonds, labeled with numbers 1 through 16, with short red arrows zigzagging upwards superimposed.

Table 1. Inventory of patterns given by de Bruijn's Theorem.

Spots:	0	4	8	12	16	20	24	28	32	Total
Patterns:	1	1	5	11	30	52	95	120	91	406

Table 2. Generating polynomial coefficients of $w_{1,a}{w}_{1,8-a}{w}_{2,b}{w}_{2,8-b}$.

Table 3. Inventory of patterns given by the Monster Theorem.