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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.

%I A374942 #23 Oct 27 2024 12:12:25
%S A374942 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,
%T A374942 6,7,7,6,6,5,5,5,6,6,7
%N 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.
%C A374942 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 A374942 The mosaic number of a Legendrian knot L is the smallest integer n such that L is realizable on a Legendrian n-mosaic.
%C A374942 Note that the Thurston-Bennequin number of a Legendrian unknot is always negative, so we take the absolute value in this sequence.
%C A374942 For more entries (but with incomplete rows), see Figure C.1 of Kipe et al. - _Luc Ta_, Oct 27 2024
%H A374942 Margaret Kipe, <a href="/A374942/a374942.py.txt">Python</a>
%H A374942 Margaret Kipe, <a href="/A374942/a374942.rs.txt">Rust</a>
%H A374942 Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, <a href="http://arxiv.org/abs/2410.08064">Bounds on the mosaic number of Legendrian knots</a>, arXiv: 2410.08064 [math.GT], 2024.
%H A374942 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 A374942 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.
%o A374942 (Python, Rust) //See Margaret Kipe links
%Y A374942 Cf. A374939, A374943, A374944, A374945, A374946, A374947.
%K A374942 nonn,tabl,hard,more
%O A374942 1,1
%A A374942 _Margaret Kipe_, _Samantha Pezzimenti_, _Leif Schaumann_, _Luc Ta_, _Wing Hong Tony Wong_, Jul 24 2024