Displaying 1-10 of 11 results found.
a(n) is the number of suitably connected Legendrian n-Mosaics.
+10
13
1, 2, 20, 1504, 948032, 5204262912, 254112496082944, 111879597850371293184, 448381477417976615986528256, 16469260582635747355818375736459264, 5571666891811926168753521842383673521864704, 17424018517043252553551626372130243982114254609186816
COMMENTS
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. - Luc Ta, Aug 20 2024
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(Rust) // See Margaret Kipe link
CROSSREFS
Cf. A261400, A375354, A374939, A374942, A374943, A374944, A374945, A374946, A375353, A375355, A375356, A375357.
EXTENSIONS
a(7)-a(11) from Luc Ta, Aug 20 2024
a(|tb|,r)= Mosaic number of the Legendrian unknot, read by rows of the mountain range organized by Thurston-Bennequin number and rotation number, where 1-|tb|<=r<=|tb|-1.
+10
7
2, 3, 3, 3, 3, 3, 5, 4, 4, 5, 6, 4, 4, 4, 6, 6, 5, 4, 4, 5, 6, 6, 6, 5, 4, 5, 6, 6, 7, 6, 5, 5, 5, 5, 6, 7, 7, 6, 6, 5, 5, 5, 6, 6, 7
COMMENTS
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.
The mosaic number of a Legendrian knot L is the smallest integer n such that L is realizable on a Legendrian n-mosaic.
Note that the Thurston-Bennequin number of a Legendrian unknot is always negative, so we take the absolute value in this sequence.
EXAMPLE
a(1,0)=2 because the mosaic number of the Legendrian unknot with tb=-1 and r=0 is 2. a(3,-2)=3 because the mosaic number of the Legendrian unknot with tb=-3 and r=-2 is 3.
PROG
(Python, Rust) //See Margaret Kipe links
a(n) is the number of distinct Legendrian unknots with nonnegative rotation numbers that can be realized on a Legendrian n-mosaic.
+10
7
COMMENTS
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.
By Theorem 1.5 of Eliashberg and Fraser, two Legendrian unknots are equivalent if and only if they share the same Thurston-Bennequin invariant and rotation number.
EXAMPLE
For n = 3 there are exactly a(3) = 4 distinct Legendrian unknots that can be realized on a Legendrian 3-mosaic, namely those whose Thurston-Bennequin invariants are -1, -2, -3, and -3 and whose rotation numbers are 0, 1, 0, and 2, respectively.
PROG
(Python, Rust) //See Margaret Kipe links
a(n) is the maximum over the minimum crossing numbers of all Legendrian knots that can be realized on a Legendrian n-mosaic.
+10
7
COMMENTS
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.
EXAMPLE
For n = 5, the only Legendrian knots that can be realized on a Legendrian 5-mosaic are positive and negative Legendrian trefoils, which have a minimal crossing number of 3, and Legendrian unknots, which have a minimal crossing number of 0. Therefore, a(5) = 3.
PROG
(Python, Rust) //See Margaret Kipe links
a(n) is the number of knots having a Legendrian representative realizable on a Legendrian n-mosaic.
+10
7
COMMENTS
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.
Two knots have the same smooth knot type if and only if they are related by an ambient isotopy.
EXAMPLE
For n = 2, there is only a(2) = 1 smooth knot family with Legendrian representatives realizable on a Legendrian 2-mosaic, namely unknots.
For n = 5, every Legendrian 5-mosaic depicts either an unknot or a trefoil. Since unknots and trefoils are not ambient-isotopic, we have a(5) = 2.
PROG
(Python, Rust) //See Margaret Kipe links
T(m,n) = Number of m X n knot/link mosaics read by rows, with 1<=n<=m.
+10
7
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
COMMENTS
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.
LINKS
Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
FORMULA
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.
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
The main diagonal T(n,n) is A261400.
T(m,n) is the number of suitably connected m X n Legendrian mosaics read by rows, with 1<=n<=m.
+10
7
1, 1, 2, 1, 4, 20, 1, 8, 104, 1504, 1, 16, 544, 22208, 948032, 1, 32, 2848, 329216, 40930304, 5204262912, 1, 64, 14912, 4883968, 1772261888, 666548862976, 254112496082944, 1, 128, 78080, 72464384, 76795762688, 85575149027328, 97392800416399360, 111879597850371293184
COMMENTS
An m X n Legendrian mosaic is an m 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. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles.
The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee, adapted to the Legendrian setting: since Legendrian mosaic tiles omit the crossing tile T_9 used in general knot mosaics, the bottom-right submatrix of O_(k+1) is 3*O_k rather than 4*O_k.
T(m,2) = A375353(m,2) = A000079(m-1) for all m >= 2 since neither classical nor Legendrian link mosaics with only 2 columns or rows can use T_9 tiles.
EXAMPLE
Triangle begins:
1;
1, 2;
1, 4, 20;
1, 8, 104, 1504;
1, 16, 544, 22208, 948032;
1, 32, 2848, 329216, 40930304, 5204262912;
...
T(2,2) = 2 since the only suitably connected 2 X 2 Legendrian mosaics are the empty mosaic and the mosaic depicting the Legendrian unknot with maximal Thurston-Bennequin invariant.
For all n >= 1, we have T(n, 1) = 1 since the only suitably connected Legendrian mosaic with one column is empty.
MATHEMATICA
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]}}];
legendrian[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1];
Flatten[ParallelTable[legendrian[m, n], {m, 1, 11}, {n, 1, m}]] (* Luc Ta, Aug 13 2024 *)
CROSSREFS
The main diagonal T(n,n) is A374947.
a(n) is the number of distinct Legendrian knots, up to smooth knot type and classical invariants, with nonnegative rotation numbers that can be realized on a Legendrian n-mosaic.
+10
6
COMMENTS
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.
The classical invariants of Legendrian knots are the Thurston-Bennequin invariant and the rotation number.
EXAMPLE
For n = 3 there are exactly a(3) = 4 distinct Legendrian knots with nonnegative rotation numbers that can be realized on a Legendrian 3-mosaic, namely the four Legendrian unknots whose Thurston-Bennequin invariants are -1, -2, -3, and -3 and whose rotation numbers are 0, 1, 0, and 2, respectively.
PROG
(Python, Rust) //See Margaret Kipe links
T(m, n) is the number of m X n period knot/link mosaics read by rows, with 1 <= n <= m.
+10
6
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
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.
LINKS
Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
EXAMPLE
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.
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}]]
T(m, n) is the number of m X n toroidal knot/link mosaics read by rows, with 1 <= n <= m.
+10
5
7, 18, 110, 49, 954, 35237, 171, 11591, 1662837, 308435024, 637, 155310, 86538181, 63440607699, 52006454275147
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 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.
LINKS
Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
EXAMPLE
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.
CROSSREFS
The main diagonal T(n,n) contains A375357 as a subsequence.
Search completed in 0.011 seconds
| 