A375353 T(m,n) = Number of m X n knot/link mosaics read by rows, with 1<=n<=m.

1, 1, 2, 1, 4, 22, 1, 8, 130, 2594, 1, 16, 778, 54226, 4183954, 1, 32, 4666, 1144526, 331745962, 101393411126, 1, 64, 27994, 24204022, 26492828950, 31507552821550, 38572794946976686, 1, 128, 167962, 512057546, 2119630825150, 9841277889785426, 47696523856560453790, 234855052870954505606714

Offset: 1

Author

Luc Ta, Aug 13 2024

Keywords

• nonn
• tabl

Comments

An m X n link mosaic is a suitably connected m X n array of the 11 tiles given by Lomonaco and Kauffman. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles. Thus, link mosaics depict projections of a link or a knot onto a plane.

The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee.

Examples

			Triangle begins:
  1;
  1,  2;
  1,  4,   22;
  1,  8,  130,    2594;
  1, 16,  778,   54226,   4183954;
  1, 32, 4666, 1144526, 331745962, 101393411126;
  ...
T(2,2) = 2 since the only suitably connected 2 X 2 link mosaics are the empty mosaic and the mosaic depicting an unknot attaining its minimal crossing number.
For all n >= 1, we have T(n,1) = 1 since the only suitably connected mosaic with one column is empty.
		

Links

• Luc Ta, First 11 rows of the triangle, flattened
• K. Hong, H. Lee, H. J. Lee and S. Oh, Small knot mosaics and partition matrices, J. Phys. A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT], 2013-2014.
• Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
• Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014.
• Index entries for sequences related to knots

Crossrefs

The main diagonal T(n,n) is A261400.

Cf. A375354, A374947, A374946, 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], 4*o[n - 1]}}];
mosaics[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1];
Flatten[ParallelTable[mosaics[m, n], {m, 1, 11}, {n, 1, m}]] (* Luc Ta, Aug 13 2024 *)

Formula

T(m,2) = A000079(m-1) for all m >= 2 and T(m,3) = A261399(m) for all m >= 3 due to Corollary 2 of Hong, H. Lee, H. J. Lee, and Oh.