A375356 T(m, n) is the number of m X n toroidal knot/link mosaics read by rows, with 1 <= n <= m.

7, 18, 110, 49, 954, 35237, 171, 11591, 1662837, 308435024, 637, 155310, 86538181, 63440607699, 52006454275147

Offset

1,1

Comments

An m X n mosaic is an m X n array of the 11 tiles given by Lomonaco and Kauffman. A period m X n mosaic is an m X n mosaic whose opposite edges are identified. A toroidal m X n mosaic is an equivalence class of period m X n mosaics up to finite sequences of cyclic rotations of rows and columns. A toroidal mosaic depicts the projection of a knot or link on the surface of a torus iff the connection points of each tile coincide with those of the contiguous tiles and with those of the tiles on identified edges.

The first five rows of the triangle are from Table 2 of Oh, Hong, Lee, Lee, and Yeon.

Clearly, T(m,n) <= A375355(m,n) for all m,n, with equality iff m = n = 1.

Links

Table of n, a(n) for n=1..15.

Michael Carlisle and Michael S. Laufer, On upper bounds for toroidal mosaic numbers, Quantum Inf. Process. 12 (2013), no. 9, 2935-2945.

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, Hwa Jeong Lee, and Mi Jeong Yeon, Period and toroidal knot mosaics, arXiv: 1703.04867 [math.GT], 2017.

Index entries for sequences related to knots

Example

Triangle begins:
    7;
   18,    110;
   49,    954,    35237;
  171,  11591,  1662837,   308435024;
  637, 155310, 86538181, 63440607699, 52006454275147;
  ...
The only period 1 X 1 link mosaics are given by the tiles T_0 and T_5 through T_10 of Lomonaco and Kauffman. None of these mosaics are cyclic rotations of rows and columns of the others (since there are no rows or columns to permute in the first place). Therefore, T(1,1) = 7.
An exhaustive list of all 110 distinct 2 X 2 toroidal link mosaics is given collectively by Appendix A of Carlisle and Laufer and Figure 4 of Oh, Hong, Lee, Lee, and Yeon.

Crossrefs

The main diagonal T(n,n) contains A375357 as a subsequence.

Cf. A375355, A375353, A375354, A261400, A374947, A374946.

Sequence in context: A203381 A207158 A304992 * A282092 A197938 A207153

Adjacent sequences: A375353 A375354 A375355 * A375357 A375358 A375359

Keyword

nonn,tabl,more,hard

Author

Luc Ta, Aug 20 2024

Status

approved