A374942 -id:A374942 - 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

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

Original entry on oeis.org

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.

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
```

A383828 Number of involutory racks of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 2, 5, 13, 42, 180, 906, 6317

Offset: 0

Luc Ta, May 11 2025

A rack is involutory if it satisfies the identity y(yx) = x. In particular, involutory quandles are called kei.
a(n) is also the number of Legendrian kei (i.e., kei equipped with Legendrian structures) up to order n up to isomorphism; see Ta, Theorem 1.1.
a(n) is also the number of symmetric kei (i.e., kei equipped with good involutions) up to order n up to isomorphism; see Ta, "Equivalences of...," Corollary 1.3.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, pages 101-108.

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, GL-Rack Classification, GitHub, 2025.

Cf. A383829-A383831, 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
```

cp's OEIS Frontend

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

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

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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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GAP

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

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