A374947 a(n) is the number of suitably connected Legendrian n-Mosaics.

1, 2, 20, 1504, 948032, 5204262912, 254112496082944, 111879597850371293184, 448381477417976615986528256, 16469260582635747355818375736459264, 5571666891811926168753521842383673521864704, 17424018517043252553551626372130243982114254609186816

Offset: 1

Author

Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, Jul 24 2024

Keywords

• nonn

Comments

A Legendrian n-mosaic is an n 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.

A Legendrian n-mosaic is suitably connected iff the connection points of each tile coincide with those of all contiguous tiles. Note that the n-mosaic consisting of all blank tiles is vacuously suitably connected even though it does not represent a link.

This is the main diagonal of A375354. It appears to grow at a quadratic exponential rate, and the ratios a(n)/A261400(n) seem to converge to 0 at a quadratic exponential rate.

For more information, see Sections 4 and 5 of Kipe et al. In particular, see Figures 20 and 21 for explicit best-fit models. - Luc Ta, Oct 27 2024

Examples

			For n = 2 there are exactly a(2) = 2 suitably connected Legendrian 2-mosaics, namely the empty mosaic and the Legendrian unknot with maximal Thurston-Bennequin invariant.
		

Links

• Margaret Kipe, Rust
• 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.
• Index entries for sequences related to knots

Crossrefs

Cf. A261400, A375354, A374939, A374942, A374943, A374944, A374945, A374946, A375353, A375355, A375356, A375357.

Programs

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]}}];
legendrianSquare[n_] := If[n > 1, 2*Total[MatrixPower[x[n - 2] + o[n - 2], n - 2], 2], 1];
Flatten[ParallelTable[legendrianSquare[n], {n, 1, 11}]] (* This program is adapted from Theorem 1 of Oh, Hong, Lee, and Lee (see Links, cf. A375354). - Luc Ta, Aug 20 2024 *)

Rust

// See Margaret Kipe link

Extensions

a(7)-a(11) from Luc Ta, Aug 20 2024

a(12) from Alois P. Heinz, Aug 20 2024