OFFSET
1,3
COMMENTS
An m X n Legendrian mosaic is an m X n array of the 10 tiles given in Figure 5 of Pezzimenti and Pandey. These tiles represent part of a Legendrian curve in the front projection. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles.
The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee, adapted to the Legendrian setting: since Legendrian mosaic tiles omit the crossing tile T_9 used in general knot mosaics, the bottom-right submatrix of O_(k+1) is 3*O_k rather than 4*O_k. See Theorem 6 of Kipe et al.
LINKS
Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, Bounds on the mosaic number of Legendrian knots, arXiv: 2410.08064 [math.GT], 2024.
Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014.
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.
EXAMPLE
Triangle begins:
1;
1, 2;
1, 4, 20;
1, 8, 104, 1504;
1, 16, 544, 22208, 948032;
1, 32, 2848, 329216, 40930304, 5204262912;
...
T(2,2) = 2 since the only suitably connected 2 X 2 Legendrian mosaics are the empty mosaic and the mosaic depicting the Legendrian unknot with maximal Thurston-Bennequin invariant.
For all n >= 1, we have T(n,1) = 1 since the only suitably connected Legendrian mosaic with one column is empty.
MATHEMATICA
x[0] = o[0] = {{1}};
x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 3*o[n - 1]}}];
legendrian[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1];
Flatten[ParallelTable[legendrian[m, n], {m, 1, 11}, {n, 1, m}]] (* Luc Ta, Aug 13 2024 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved