Knot Crossings
Two ropes, drawn once and crossing each other 8 times. Every crossing can go either way, over or under, so the one drawing holds 28 = 256 knots. The author drew all 256 by hand on yellow legal paper, numbered 0 to 255, and marked each crossing that goes the other way with a small circle. The circles are the number written in binary, crossing 1 for 1, crossing 2 for 2, crossing 3 for 4, and so on up to crossing 8 for 128. The knots the author recognised are listed in an index: the square knot, the sheet bend, the Carrick bend and its relatives.
Here knot_crossings.py works the 256 out again from the key, and sorts them into knots with the Kauffman bracket, a number-like invariant that stays the same however a knot is redrawn. Without a circle, a crossing goes over for the rope that reaches it first, following the 1st rope and then the 2nd; that is the only reading under which the Carrick bend, number 117, comes out over, under, over, under all the way along both ropes. Reading every crossing the other way gives the mirror image of every drawing, and changes no name.
The index, checked
Every name in the hand-made index lands in a single group, or in a group and its mirror image; the square knot, which is its own mirror image, lands in a group that is its own mirror. No drawing up to 122, the last one indexed, is in a named group without being named. Three pairs of names turn out to be mirror images of one another, and one name, the reformed constrictor, covers two different knots.
The index names 23 drawings, from 24 to 122. Each name, and the groups its drawings fall in: square knot: 24, 28, 56, 60, 98, 122 -> group 7 (one group) sheet bend: 26, 29, 58, 61, 99, 103, 120 -> group 8, 9 (a group and its mirror) whatnot: 73, 101 -> group 10, 13 (2 groups) single carrick bend (unknown): 74, 106 -> group 11, 14 (2 groups) single carrick bend (reformed constrictor): 77, 105, 109 -> group 12, 15 (2 groups) single carrick bend: 114 -> group 16 (one group) the carrick bend: 117 -> group 17 (one group) carrick bend (unknown): 118 -> group 18 (one group) Drawings up to 122 in a named group but not in the index: none Names given to drawings that are mirror images of each other: 73 whatnot <-> 109 single carrick bend (Reformed constrictor) 74 single carrick bend (unknown type) <-> 118 Carrack bend (unknown) 106 single carrack bend (unknown) <-> 114 single carrack bend Alternating drawings (over, under, over ... along both ropes): 117, 138
The groups
Each group holds the drawings with the same Kauffman bracket, counting two drawings as one knot when they differ only by the ends of a rope being turned about each other, as a rope’s two ends can be in the hand. Different brackets prove different knots; the same bracket strongly suggests, but does not prove, the same knot. |lk| is how many times the ropes wind round each other when each is closed on itself. The drawing shown is the group’s first. Names in quotes are written beside the drawings rather than in the index.
| Group | Named | Mirror | |lk| | Drawings | |
|---|---|---|---|---|---|
| 1 | — | itself | 0 | 44: 0 3 4 7 14 17 27 31 32 35 36 39 46 49 59 63 70 80 83 84 87 94 161 168 171 172 175 185 192 196 206 209 216 219 220 223 224 228 238 241 248 251 252 255 | |
| 2 | "overhand knot" (1, drawing) "overhand" (5, drawing) | itself | 1 | 94: 1 2 5 6 8 11 12 15 16 19 20 23 25 30 33 34 37 38 40 43 44 47 48 51 52 55 57 62 64 67 68 71 78 81 82 85 86 88 91 92 95 96 100 110 113 123 127 128 132 142 145 155 159 160 163 164 167 169 170 173 174 177 184 187 188 191 193 198 200 203 204 207 208 211 212 215 217 218 221 222 225 230 232 235 236 239 240 243 244 247 249 250 253 254 | |
| 3 | "overhand" (9, drawing) "overhand" (18, drawing) | 5 | 2 | 22: 9 18 41 50 65 72 75 76 79 89 134 144 147 148 151 158 162 186 201 210 233 242 | |
| 4 | — | 6 | 0 | 12: 10 42 66 90 129 136 139 140 143 153 202 234 | |
| 5 | "overhand" (13, drawing) | 3 | 2 | 22: 13 22 45 54 69 93 97 104 107 108 111 121 166 176 179 180 183 190 205 214 237 246 | |
| 6 | — | 4 | 0 | 12: 21 53 102 112 115 116 119 126 165 189 213 245 | |
| 7 | square knot (24) square knot (28) square knot (56) square knot (60) square knot (98) square knot (122) | itself | 0 | 12: 24 28 56 60 98 122 133 157 195 199 227 231 | |
| 8 | sheet bend (26) sheet bend (58) | 9 | 1 | 8: 26 58 131 135 152 156 194 226 | |
| 9 | sheet bend (29) sheet bend (61) Sheet bend (99) Sheet bend (103) Sheet bend (120) | 8 | 1 | 8: 29 61 99 103 120 124 197 229 | |
| 10 | whatnot (73) | 15 | 3 | 2: 73 146 | |
| 11 | single carrick bend (unknown type) (74) | 18 | 1 | 2: 74 137 | |
| 12 | single carrick bend (Reformed constrictor knot) (77) single carrick bend (Reformed constrictor knot) (105) | itself | 3 | 4: 77 105 150 178 | |
| 13 | whatnot (101) | 19 | 2 | 2: 101 125 | |
| 14 | single carrack bend (unknown) (106) | 16 | 1 | 2: 106 141 | |
| 15 | single carrick bend (Reformed constrictor) (109) | 10 | 3 | 2: 109 182 | |
| 16 | single carrack bend (114) | 14 | 1 | 2: 114 149 | |
| 17 | The Carrack bend (117) | 20 | 1 | 1: 117 | |
| 18 | Carrack bend (unknown) (118) | 11 | 1 | 2: 118 181 | |
| 19 | — | 13 | 2 | 2: 130 154 | |
| 20 | — | 17 | 1 | 1: 138 |
All 256
Drawn from the key, with the crossings the author circled ringed in yellow and the drawings named in the index outlined. The two ropes keep their colours: the 1st in red, the 2nd in blue. The originals are in the scan; the index and the notes beside the drawings are transcribed in original/index.txt and original/drawings.txt.