A375355 T(m, n) is the number of m X n period knot/link mosaics read by rows, with 1 <= n <= m.

7, 29, 359, 133, 5519, 316249, 641, 91283, 19946891, 4934695175, 3157, 1549799, 1298065813, 1268810595131, 1300161356831107, 15689, 26576579, 85436799491, 330595705214327, 1353434715973001999, 5644698772550126097593, 78253, 457549079, 5648174618317, 86566215054880187, 1416905739955631598043, 23696846086162116561085541, 399312236302057306354637147077

Offset: 1

Author

Luc Ta, Aug 20 2024

Keywords

• nonn
• tabl

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 period mosaic depicts a knot or link 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 2 of Oh, Hong, Lee, Lee, and Yeon.

T(m, n) >= A375356(m, n) for all m and n, with equality iff m = n = 1.

T(m, 1) = A074600(m) for all m. To see this, proceed by induction on m. In Theorem 2 of Oh, Hong, Lee, Lee, and Yeon, it is clear that tr(X_{m+1}) = 2*tr(X_m) and tr(O_{m+1}) = 5*tr(O_m) for all m. The theorem states that T(m+1, 1) = tr(X_{m+1} + O_{m+1}) = tr(X_{m+1}) + tr(O_{m+1}) = 2*tr(X_m) + 5*tr(O_m), and the claim follows since tr(X_1 + O_1) = 7.

Examples

			Triangle begins:
      7;
     29,      359;
    133,     5519,      316249;
    641,    91283,    19946891,      4934695175;
   3157,  1549799,  1298065813,   1268810595131,    1300161356831107;
    ...
T(1,1) = 7 since 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.
		

Links

• Luc Ta, First 11 rows of the triangle, flattened
• 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

Cf. A375356, A375357, A375353, A375354, A261400, A374947, A374946.

Programs

Mathematica

x[0] = o[0] = {{1}}; y[0] = p[0] = {{0}};
x[n_] := ArrayFlatten[{{x[n - 1], p[n - 1]}, {p[n - 1], x[n - 1]}}];
y[n_] := ArrayFlatten[{{y[n - 1], o[n - 1]}, {o[n - 1], y[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], y[n - 1]}, {y[n - 1], 4 * o[n - 1]}}];
p[n_] := ArrayFlatten[{{p[n - 1], x[n - 1]}, {x[n - 1], 4 * p[n - 1]}}];
periodcount[m_, n_] := Tr[MatrixPower[x[m] + o[m], n]];
Flatten[ParallelTable[periodcount[m, n], {m, 1, 11}, {n, 1, m}]]