A374947 -id:A374947 - cp's OEIS frontend

A374946 a(n) is the number of suitably connected Legendrian n-mosaics that form a Legendrian knot.

0, 1, 17, 793, 275557, 831699598

Offset: 1

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

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 if the connection points of each tile coincide with those of the contiguous tiles.

			For n = 2 there is exactly a(2) = 1 Legendrian 2-mosaic forming the front projection of a Legendrian knot, namely the Legendrian unknot with maximal Thurston-Bennequin invariant.

Margaret Kipe, Python
Margaret Kipe, Rust
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.

Cf. A374947, A374945, A374944, A374943, A374942, A374939.

A374942 T(|tb|,r) is the 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.

Original entry on oeis.org

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

Offset: 1

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

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.
For more entries (but with incomplete rows), see Figure C.1 of Kipe et al. - Luc Ta, Oct 27 2024

			T(1,0)=2 because the mosaic number of the Legendrian unknot with tb=-1 and r=0 is 2. T(3,-2)=3 because the mosaic number of the Legendrian unknot with tb=-3 and r=-2 is 3.

Margaret Kipe, Python
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.
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.

Cf. A374939, A374943, A374944, A374945, A374946, A374947.

A374943 a(n) is the number of distinct Legendrian unknots with nonnegative rotation numbers that can be realized on a Legendrian n-mosaic.

Original entry on oeis.org

0, 1, 4, 9, 21, 55

Offset: 1

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

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.

			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.

Y. Eliashberg and M. Fraser, Topologically trivial Legendrian knots, Journal of Symplectic Geometry, 7 (2009), 77-127.
Margaret Kipe, Python
Margaret Kipe, Rust
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.

Cf. A374939, A374942, A374944, A374945, A374946, A374947.

A374944 a(n) is the maximum over the minimum crossing numbers of all Legendrian knots that can be realized on a Legendrian n-mosaic.

Original entry on oeis.org

0, 0, 0, 0, 3, 8

Offset: 1

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

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.

			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.

Margaret Kipe, Python
Margaret Kipe, Rust
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.

Cf. A374939, A374942, A374943, A374945, A374946, A374947.

A374945 a(n) is the number of knots having a Legendrian representative realizable on a Legendrian n-mosaic.

Original entry on oeis.org

0, 1, 1, 1, 2, 11

Offset: 1

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

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.

			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.

Margaret Kipe, Python
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.
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.

Cf. A374939, A374942, A374943, A374944, A374946, A374947

A374939 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.

Original entry on oeis.org

0, 1, 4, 9, 40, 328

Offset: 1

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

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.

			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.

Margaret Kipe, Python
Margaret Kipe, Rust
S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.

Cf. A374942, A374943, A374944, A374945, A374946, A374947.

A383146 Number of medial GL-racks of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 4, 13, 61, 298, 2087, 16941, 187160

Offset: 0

Luc Ta, Apr 17 2025

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.
GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; cf. Cattabriga and Nasybullov, Section 3.2.
A rack or quandle X is medial (also called abelian) if the map X x X -> X defined by (x,y) -> y(x) is a rack homomorphism. Equivalently, the identity (xy)(uv)=(xu)(yv) holds for all elements x, y, u, and v in X.

Alessia Cattabriga and Timur Nasybullov, Virtual quandle for links in lens spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A. Matemáticas, 112 (2018), no. 3, 657-669.
Biswadeep Karmakar, Deepanshi Saraf, and Mahender Singh, Generalised Legendrian racks of Legendrian links, arXiv: 2301.06854 [math.GT], 2023.
Naoki Kimura, Bi-Legendrian rack colorings of Legendrian knots, Journal of Knot Theory and its Ramifications, 32 (2023), no. 4, Paper No. 2350029.
Lực Ta, Generalized Legendrian racks: Classification, tensors, and knot coloring invariants, arXiv: 2504.12671 [math.GT], 2025.
Lực Ta, GL-Rack Classification, GitHub, 2025.

Cf. A383145.
Sequences related to medial racks and quandles: A383144, A165200, A242044, A226193, A242275, A243931, A257351.
Other sequences related to racks and quandles: A181769, A181770, A181771, A176077, A178432, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A198147, A225744, A226172, A226174.
Sequences related to Legendrian knots: A374939, A374942, A374943, A374944, A374945, A374946, A374947.

GAP
```
# see Ta, GitHub link
```

A383145 Number of GL-racks of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 4, 13, 62, 308, 2132, 17268, 189373

Offset: 0

Luc Ta, Apr 17 2025

Generalized Legendrian racks, also called GL-racks or bi-Legendrian racks, are racks equipped with a rack automorphism that commutes with all inner rack automorphisms; see Ta, Definition 3.1 and Proposition 3.12. They are used to distinguish Legendrian links in contact three-space.

GL-racks are precisely virtual racks in which all inner rack automorphisms are virtual rack automorphisms; see Cattabriga and Nasybullov, Section 3.2.

Alessia Cattabriga and Timur Nasybullov, Virtual quandle for links in lens spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A. Matemáticas, 112 (2018), no. 3, 657-669.
Biswadeep Karmakar, Deepanshi Saraf, and Mahender Singh, Generalised Legendrian racks of Legendrian links, arXiv: 2301.06854 [math.GT], 2023.
Naoki Kimura, Bi-Legendrian rack colorings of Legendrian knots, Journal of Knot Theory and its Ramifications, 32 (2023), no. 4, Paper No. 2350029.
Lực Ta, Generalized Legendrian racks: Classification, tensors, and knot coloring invariants, arXiv: 2504.12671 [math.GT], 2025.
Lực Ta, GL-Rack Classification, GitHub, 2025.

Cf. A383146.
Sequences relating to racks and quandles: A181769, A181770, A383144, A181771, A176077, A178432, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174.
Sequences related to Legendrian knots: A374939, A374942, A374943, A374944, A374945, A374946, A374947.

GAP
```
# see Ta, GitHub link
```

A383831 Number of medial Legendrian quandles of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 2, 5, 14, 48, 219, 1207, 9042

Offset: 0

Luc Ta, May 16 2025

A rack or quandle is medial if it satisfies the identity (xy)(uv) = (xu)(yv).
A Legendrian quandle is a pair (X,u) where X is a quandle and u is an involutory automorphism of X such that u(yx)=y(u(x)); see Ta, "Generalized Legendrian...," Corollary 3.13.
a(n) is also the number of medial racks X such that the kink map X -> X defined by x -> x(x) is an involution; see Ta, "Equivalences of...," Theorem 1.1.

Jose Ceniceros, Mohamed Elhamdadi, and Sam Nelson, Legendrian rack invariants of Legendrian knots, Communications of the Korean Mathematical Society, 36 (2021), no. 3, 623-639.
Lực Ta, Equivalences of racks, Legendrian racks, and symmetric racks, arXiv:2505.08090 [math.GT], 2025.
Lực Ta, Generalized Legendrian racks: Classification, tensors, and knot coloring invariants, arXiv:2504.12671 [math.GT], 2025.
Lực Ta, GL-Rack Classification, GitHub, 2025.

Cf. A383828-A383830, A383145, A383146, A181769, A181770, A178432.
Sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174.
Sequences related to Legendrian knots: A374939, A374942, A374943, A374944, A374945, A374946, A374947.

GAP
```
# See Ta, GitHub link
```

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

cp's OEIS Frontend

A374946 a(n) is the number of suitably connected Legendrian n-mosaics that form a Legendrian knot.

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A374942 T(|tb|,r) is the 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.

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A374943 a(n) is the number of distinct Legendrian unknots with nonnegative rotation numbers that can be realized on a Legendrian n-mosaic.

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A374944 a(n) is the maximum over the minimum crossing numbers of all Legendrian knots that can be realized on a Legendrian n-mosaic.

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A374945 a(n) is the number of knots having a Legendrian representative realizable on a Legendrian n-mosaic.

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A374939 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.

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A383146 Number of medial GL-racks of order n, up to isomorphism.

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A383145 Number of GL-racks of order n, up to isomorphism.

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Programs

GAP

A383831 Number of medial Legendrian quandles of order n, up to isomorphism.

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GAP

A261400 Number of n X n knot mosaics.

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