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

110, 35237, 52006454275147, 8149229312286883803155895853, 101957128471911748968541302399445156486848984449235985038696169948167385

Offset: 1

Author

Luc Ta, Aug 20 2024

Keywords

• nonn
• more

Comments

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.

The Mathematica program below is based on the algorithm given in Theorem 4 of Oh, Hong, Lee, Lee, and Yeon.

Examples

			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

This is a subsequence of the diagonal of A375356.

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

Programs

Mathematica

<A375355","Data"], PolygonalNumber[q], 2] - 2*Sum[f[q, k], {k, 0, (q - 1)/2}];
toroidalcount[q_] := If[q > 2, (1/q^2) * g[q] + (2/q) * Sum[f[q, k], {k, 0, (q - 1)/2}] + 7, 110]
Monitor[Table[toroidalcount[Prime[n]], {n, 1, 5}], Row[{ProgressIndicator[n, {1, 5}], n}, " "]]