cp's OEIS Frontend

A375357 a(n) is the number of p X p toroidal knot/link mosaics, where p = A000040(n).

%I A375357 #10 Aug 20 2024 12:22:51
%S A375357 110,35237,52006454275147,8149229312286883803155895853,
%T A375357 101957128471911748968541302399445156486848984449235985038696169948167385
%N A375357 a(n) is the number of p X p toroidal knot/link mosaics, where p = A000040(n).
%C A375357 A p X p mosaic is an p X p array of the 11 tiles given by Lomonaco and Kauffman. A period p X p mosaic is an p X p mosaic whose opposite edges are identified. A toroidal p X p mosaic is an equivalence class of period p X p 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 A375357 The Mathematica program below is based on the algorithm given in Theorem 4 of Oh, Hong, Lee, Lee, and Yeon.
%H A375357 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 A375357 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 A375357 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 A375357 <a href="/index/K#knots">Index entries for sequences related to knots</a>
%e A375357 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.
%t A375357 <<OEIS` (* load OEIS-Mathematica package *)
%t A375357 x[0] = o[0] = {{1}}; y[0] = p[0] = {{0}};
%t A375357 x[n_] := ArrayFlatten[{{x[n - 1], p[n - 1]}, {p[n - 1], x[n - 1]}}];
%t A375357 y[n_] := ArrayFlatten[{{y[n - 1], o[n - 1]}, {o[n - 1], y[n - 1]}}];
%t A375357 o[n_] := ArrayFlatten[{{o[n - 1], y[n - 1]}, {y[n - 1], 4 * o[n - 1]}}];
%t A375357 p[n_] := ArrayFlatten[{{p[n - 1], x[n - 1]}, {x[n - 1], 4 * p[n - 1]}}];
%t A375357 t[A_, k_, q_] := 2 + Sum[A[[i + 1, 1 + Mod[i * 2^k, 2^q - 1, 1]]], {i, 0, 2^q - 1}]
%t A375357 f[q_, k_] := -7 + t[x[q] + o[q], k, q];
%t A375357 g[q_] := -7 + Part[OEISImport["A375355","Data"], PolygonalNumber[q], 2] - 2*Sum[f[q, k], {k, 0, (q - 1)/2}];
%t A375357 toroidalcount[q_] := If[q > 2, (1/q^2) * g[q] + (2/q) * Sum[f[q, k], {k, 0, (q - 1)/2}] + 7, 110]
%t A375357 Monitor[Table[toroidalcount[Prime[n]], {n, 1, 5}], Row[{ProgressIndicator[n, {1, 5}], n}, " "]]
%Y A375357 This is a subsequence of the diagonal of A375356.
%Y A375357 Cf. A375355, A375353, A375354, A261400, A374947, A374946.
%K A375357 nonn,more
%O A375357 1,1
%A A375357 _Luc Ta_, Aug 20 2024