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

Author

Luc Ta, Aug 20 2024

Keywords

• nonn
• tabl
• more
• hard

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.

Examples

			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.
		

Links

• 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

Crossrefs

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

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