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%I #14 Aug 09 2024 08:02:27
%S 0,1,4,9,21,55
%N a(n) is the number of distinct Legendrian unknots with nonnegative rotation numbers that can be realized on a Legendrian n-mosaic.
%C 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.
%C 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.
%H Y. Eliashberg and M. Fraser, <a href="https://doi.org/10.48550/arXiv.0801.2553">Topologically trivial Legendrian knots</a>, Journal of Symplectic Geometry, 7 (2009), 77-127.
%H Margaret Kipe, <a href="/A374943/a374943.py.txt">Python</a>
%H Margaret Kipe, <a href="/A374943/a374943.rs.txt">Rust</a>
%H S. Pezzimenti and A. Pandey, <a href="https://doi.org/10.1142/S021821652250002X">Geography of Legendrian knot mosaics</a>, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.
%e 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.
%o (Python, Rust) //See Margaret Kipe links
%Y Cf. A374939, A374942, A374944, A374945, A374946, A374947.
%K nonn,hard,more
%O 1,3
%A _Margaret Kipe_, _Samantha Pezzimenti_, _Leif Schaumann_, _Luc Ta_, _Wing Hong Tony Wong_, Jul 24 2024