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

%I A375356 #11 Sep 07 2024 08:52:56
%S A375356 7,18,110,49,954,35237,171,11591,1662837,308435024,637,155310,
%T A375356 86538181,63440607699,52006454275147
%N A375356 T(m, n) is the number of m X n toroidal knot/link mosaics read by rows, with 1 <= n <= m.
%C A375356 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.
%C A375356 The first five rows of the triangle are from Table 2 of Oh, Hong, Lee, Lee, and Yeon.
%C A375356 Clearly, T(m,n) <= A375355(m,n) for all m,n, with equality iff m = n = 1.
%H A375356 Michael Carlisle and Michael S. Laufer, <a href="https://doi.org/10.48550/arXiv.1206.4227">On upper bounds for toroidal mosaic numbers</a>, Quantum Inf. Process. 12 (2013), no. 9, 2935-2945.
%H A375356 Samuel J. Lomonaco and Louis H. Kauffman, <a href="http://www.csee.umbc.edu/~lomonaco/pubs/psapm561.pdf">Quantum Knots and Mosaics</a>, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
%H A375356 Seungsang Oh, Kyungpyo Hong, Ho Lee, Hwa Jeong Lee, and Mi Jeong Yeon, <a href="http://arxiv.org/abs/1703.04867">Period and toroidal knot mosaics</a>, arXiv: 1703.04867 [math.GT], 2017.
%H A375356 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%e A375356 Triangle begins:
%e A375356     7;
%e A375356    18,    110;
%e A375356    49,    954,    35237;
%e A375356   171,  11591,  1662837,   308435024;
%e A375356   637, 155310, 86538181, 63440607699, 52006454275147;
%e A375356   ...
%e A375356 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.
%e A375356 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.
%Y A375356 The main diagonal T(n,n) contains A375357 as a subsequence.
%Y A375356 Cf. A375355, A375353, A375354, A261400, A374947, A374946.
%K A375356 nonn,tabl,more,hard
%O A375356 1,1
%A A375356 _Luc Ta_, Aug 20 2024