Numbering of the vertices of the regular 24-cell: 1 (2, 0, 0, 0) 2 (1, 1, 1, 1) 3 (1, 1, 1, -1) 4 (1, 1, -1, 1) 5 (1, 1, -1, -1) 6 (1, -1, 1, 1) 7 (1, -1, 1, -1) 8 (1, -1, -1, 1) 9 (1, -1, -1, -1) 10 (0, 2, 0, 0) 11 (0, 0, 2, 0) 12 (0, 0, 0, 2) 13 (0, 0, 0, -2) 14 (0, 0, -2, 0) 15 (0, -2, 0, 0) 16 (-1, 1, 1, 1) 17 (-1, 1, 1, -1) 18 (-1, 1, -1, 1) 19 (-1, 1, -1, -1) 20 (-1, -1, 1, 1) 21 (-1, -1, 1, -1) 22 (-1, -1, -1, 1) 23 (-1, -1, -1, -1) 24 (-2, 0, 0, 0) ----------------------------------------------------------------------------------------------------------------------- NONTRIVIAL KNOTS AS SUBGRAPHS OF THE GRAPH OF THE REGULAR 24-CELL Inspired by the admittedly rather simple knots appearing among the induced cycles of maximal length in the graph of the regular 600-cell, I have searched for knots in the form of non-induced cycles in the graph of the regular 24-cell. First, a couple of definitions. A cycle type is a graph isomorphism class of connected 2-regular subgraphs of the graph of the regular 24-cell, typically represented by the lexicographically first element (using the above numbering). Two cycle types are considered equivalent if an element from one can be transformed into an element from the the other through a finite number of steps, each consisting of replacing a single vertex and its two edges in the cycle by a vertex and two edges not in the original cycle, such that the modified graph is still a cycle. Such a step can not change the knot type. An equivalence class containing a cycle type for which each cycle contains 3 or 4 consecutive vertices such that the first and the last of these are adjacent in the original graph, is omitted, leaving only objects that can not be "tightened", in a sense. Only one representative of each cycle type is listed, but all cycle types from the same equivalence class are represented. This yields the following output. Length 24: 1 2 10 17 13 9 14 4 12 6 7 21 23 22 18 16 11 3 5 19 24 20 15 8 Three-twist knot (prime knot with 5 crossings) 1 2 10 17 13 9 15 20 24 19 5 3 7 6 12 4 14 23 21 11 16 18 22 8 Figure-eight knot (prime knot with 4 crossings) 1 2 10 17 21 7 6 8 14 18 16 11 3 13 23 22 12 4 5 19 24 20 15 9 Three-twist knot 1 2 10 17 21 20 12 4 5 3 11 16 24 23 14 8 6 7 13 19 18 22 15 9 3_1 # 3_1 (composite knot with 6 crossings) 1 2 10 17 21 23 14 8 6 7 13 19 24 20 12 4 5 3 11 16 18 22 15 9 6_3 (prime knot with 6 crossings) Length 23: Two equivalence classes with two cycle types in each: 1 2 10 17 21 7 9 14 18 16 11 3 13 23 15 6/20 12 4 5 19 24 22 8 Three-twist knot 1 2 10 17 21 15 8 4 5 3 11 20 22 14 19 13 7 6 12 16/18 24 23 9 8_19; torus knot with coefficients (4, 3) (prime knot with 8 crossings) Length 21: There is an equivalence class with 16 cycle types. Since the lexicographically earliest representative of each cycle type is listed, it is not immediately apparent which ones are one step away from which. 1 2 10 17 13 7 6 12 16 24 21 15 8 4 5 3 11 20 22 14 9 Cinquefoil knot (prime knot with 5 crossings) 1 2 10 17 13 7 6 12 16 24 21 15 8 4 5 3 11 20 22 23 9 1 2 10 17 13 7 6 12 16 24 23 15 8 4 5 3 11 20 22 14 9 1 2 10 17 13 7 6 12 18 19 23 15 8 4 5 3 11 20 22 14 9 1 2 10 17 13 7 6 12 18 24 21 15 8 4 5 3 11 20 22 14 9 1 2 10 17 13 7 6 12 18 24 21 15 8 4 5 3 11 20 22 23 9 1 2 10 17 13 7 6 12 18 24 21 15 8 14 5 3 11 20 22 23 9 1 2 10 17 13 7 6 12 18 24 23 15 8 4 5 3 11 20 22 14 9 1 2 10 17 21 7 6 12 16 24 23 14 5 3 11 20 22 18 19 13 9 1 2 10 17 21 7 6 12 18 19 23 15 8 4 5 3 11 20 22 14 9 1 2 10 17 21 15 8 4 5 3 11 16 18 19 13 7 6 12 22 23 9 1 2 10 17 21 15 8 4 5 3 11 16 18 19 13 7 6 20 24 23 9 1 2 10 17 21 15 8 4 5 3 11 20 24 19 13 7 6 12 22 23 9 1 2 10 17 21 15 8 14 5 3 11 16 18 19 13 7 6 12 22 23 9 1 2 10 17 21 15 8 14 5 3 11 16 18 19 13 7 6 20 24 23 9 1 2 10 17 21 15 8 14 19 13 7 6 12 4 5 3 11 20 22 23 9 In addition, we have these two cycle types: 1 2 10 17 21 15 8 14 19 24 20 12 4 5 3 11 16 18 22 23 9 Three-twist knot 1 2 10 17 21 20 12 18 19 23 15 8 4 5 3 11 16 24 22 14 9 Cinquefoil knot Length 18: 1 2 10 17 21 15 8 14 18 16 11 3 5 4 12 22 23 9 Figure-eight knot Length 15: 3 equivalence classes with 14, 1 and 3 cycle types respectively. First equivalence class: 1 2 10 17 13 7 6 8 4 5 3 11 20 15 9 Trefoil knot (prime knot with 3 crossings) 1 2 10 17 13 7 6 8 4 5 3 11 21 15 9 1 2 10 17 13 7 6 8 14 5 3 11 20 15 9 1 2 10 17 13 7 6 8 14 5 3 11 21 23 9 1 2 10 17 13 7 6 12 4 5 3 11 20 15 8 1 2 10 17 13 7 6 12 4 5 3 11 20 22 8 1 2 10 17 13 7 6 12 4 5 3 11 21 15 8 1 2 10 17 13 7 6 12 4 5 3 11 21 23 9 1 2 10 17 21 7 6 8 14 5 3 11 20 15 9 1 2 10 17 21 7 6 12 4 5 3 11 20 15 9 1 2 10 17 21 15 6 12 4 5 3 11 20 22 8 1 2 10 17 21 20 12 4 5 3 11 16 18 14 9 1 2 10 19 13 7 6 8 14 5 3 11 20 15 9 1 2 10 19 13 7 6 8 14 5 3 11 21 23 9 Second equivalence class: 1 2 10 17 21 15 8 4 5 3 11 16 18 14 9 Trefoil knot Third equivalence class: 1 2 10 17 21 15 8 4 5 3 11 16 24 23 9 Trefoil knot 1 2 10 17 21 15 8 4 5 3 11 20 22 23 9 1 2 10 17 21 15 8 14 5 3 11 20 24 23 9