A375357 -id:A375357 - cp's OEIS frontend

A374947 a(n) is the number of suitably connected Legendrian n-Mosaics.

1, 2, 20, 1504, 948032, 5204262912, 254112496082944, 111879597850371293184, 448381477417976615986528256, 16469260582635747355818375736459264, 5571666891811926168753521842383673521864704, 17424018517043252553551626372130243982114254609186816

Offset: 1

Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, Jul 24 2024

nonn

A Legendrian n-mosaic is an n X n array of the 10 tiles given in Figure 5 of Pezzimenti and Pandey. These tiles represent part of a Legendrian curve in the front projection.
A Legendrian n-mosaic is suitably connected iff the connection points of each tile coincide with those of all contiguous tiles. Note that the n-mosaic consisting of all blank tiles is vacuously suitably connected even though it does not represent a link.
This is the main diagonal of A375354. It appears to grow at a quadratic exponential rate, and the ratios a(n)/A261400(n) seem to converge to 0 at a quadratic exponential rate.
For more information, see Sections 4 and 5 of Kipe et al. In particular, see Figures 20 and 21 for explicit best-fit models. - Luc Ta, Oct 27 2024

			For n = 2 there are exactly a(2) = 2 suitably connected Legendrian 2-mosaics, namely the empty mosaic and the Legendrian unknot with maximal Thurston-Bennequin invariant.

Margaret Kipe, Rust
Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, Bounds on the mosaic number of Legendrian knots, arXiv: 2410.08064 [math.GT], 2024.
Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014.
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.
Index entries for sequences related to knots

Cf. A261400, A375354, A374939, A374942, A374943, A374944, A374945, A374946, A375353, A375355, A375356, A375357.

x[0] = o[0] = {{1}};
x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 3*o[n - 1]}}];
legendrianSquare[n_] := If[n > 1, 2*Total[MatrixPower[x[n - 2] + o[n - 2], n - 2], 2], 1];
Flatten[ParallelTable[legendrianSquare[n], {n, 1, 11}]] (* This program is adapted from Theorem 1 of Oh, Hong, Lee, and Lee (see Links, cf. A375354). - Luc Ta, Aug 20 2024 *)

Rust
```
// See Margaret Kipe link
```

a(7)-a(11) from Luc Ta, Aug 20 2024

a(12) from Alois P. Heinz, Aug 20 2024

A261400 Number of n X n knot mosaics.

Original entry on oeis.org

1, 2, 22, 2594, 4183954, 101393411126, 38572794946976686, 234855052870954505606714, 23054099362200397056093750003442, 36564627559441095000442883434988307728126, 937273142571326346553334567317274833729462713413038

Offset: 1

N. J. A. Sloane, Aug 18 2015

nonn

According to Oh, Hong, Lee, and Lee, a(n) grows at a quadratic exponential rate. Moreover, it appears that the ratios A374947(n)/a(n) converge to 0 at a quadratic exponential rate. - Luc Ta, Aug 27 2024

Luc Ta, Table of n, a(n) for n = 1..14
K. Hong, H. Lee, H. J. Lee and S. Oh, Small knot mosaics and partition matrices, J. Phys. A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT].
K. Hong, H. J. Lee, H. Lee and S. Oh, Upper bound on the total number of knot n-mosaics, J. Knot Theory Ramifications, Volume 23, Issue 13, November 2014; arXiv:1303.7044 [math.GT].
Hwa Jeong Lee, Kyungpyo Hong, Ho Lee, and Seungsang Oh, Mosaic number of knots, arXiv: 1301.6041 [math.GT], 2014.
Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
Samuel J. Lomonaco and Louis H. Kauffman, Illustration for a(3) = 22, from "Quantum Knots and Mosaics", 2010, with permission.
Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014.
Index entries for sequences related to knots

Reminiscent of (but of course different from) A200000.
The term 22 is the same 22 that appears in A261399.
a(n) is the main diagonal of A375353.
Cf. A374947, A375354, A375355, A375356, A375357.

x[0] = o[0] = {{1}};
x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 4*o[n - 1]}}];
mosaicsSquare[n_] := If[n > 1, 2*Total[MatrixPower[x[n - 2] + o[n - 2], n - 2], 2], 1];
Flatten[ParallelTable[mosaicsSquare[n], {n, 1, 11}]] (* This program is based on Theorem 1 of Oh, Hong, Lee, and Lee (see Links). - Luc Ta, Aug 13 2024 *)

a(7)-a(11) from Hiroaki Yamanouchi, Aug 19 2015

Typo in a(11) corrected by Luc Ta, Aug 13 2024

A375353 T(m,n) = Number of m X n knot/link mosaics read by rows, with 1<=n<=m.

Original entry on oeis.org

1, 1, 2, 1, 4, 22, 1, 8, 130, 2594, 1, 16, 778, 54226, 4183954, 1, 32, 4666, 1144526, 331745962, 101393411126, 1, 64, 27994, 24204022, 26492828950, 31507552821550, 38572794946976686, 1, 128, 167962, 512057546, 2119630825150, 9841277889785426, 47696523856560453790, 234855052870954505606714

Offset: 1

Luc Ta, Aug 13 2024

An m X n link mosaic is a suitably connected m X n array of the 11 tiles given by Lomonaco and Kauffman. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles. Thus, link mosaics depict projections of a link or a knot onto a plane.

The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee.

			Triangle begins:
  1;
  1,  2;
  1,  4,   22;
  1,  8,  130,    2594;
  1, 16,  778,   54226,   4183954;
  1, 32, 4666, 1144526, 331745962, 101393411126;
  ...
T(2,2) = 2 since the only suitably connected 2 X 2 link mosaics are the empty mosaic and the mosaic depicting an unknot attaining its minimal crossing number.
For all n >= 1, we have T(n,1) = 1 since the only suitably connected mosaic with one column is empty.

Luc Ta, First 11 rows of the triangle, flattened
K. Hong, H. Lee, H. J. Lee and S. Oh, Small knot mosaics and partition matrices, J. Phys. A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT], 2013-2014.
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, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014.
Index entries for sequences related to knots

The main diagonal T(n,n) is A261400.

Cf. A375354, A374947, A374946, A375355, A375356, A375357.

x[0] = o[0] = {{1}};
x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 4*o[n - 1]}}];
mosaics[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1];
Flatten[ParallelTable[mosaics[m, n], {m, 1, 11}, {n, 1, m}]] (* Luc Ta, Aug 13 2024 *)

T(m,2) = A000079(m-1) for all m >= 2 and T(m,3) = A261399(m) for all m >= 3 due to Corollary 2 of Hong, H. Lee, H. J. Lee, and Oh.

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

Original entry on oeis.org

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

Luc Ta, Aug 20 2024

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.

			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.

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

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

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}]]

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

Original entry on oeis.org

7, 18, 110, 49, 954, 35237, 171, 11591, 1662837, 308435024, 637, 155310, 86538181, 63440607699, 52006454275147

Offset: 1

Luc Ta, Aug 20 2024

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.

			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.

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

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

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

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A374947 a(n) is the number of suitably connected Legendrian n-Mosaics.

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A261400 Number of n X n knot mosaics.

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A375353 T(m,n) = Number of m X n knot/link mosaics read by rows, with 1<=n<=m.

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

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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.

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Examples

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