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.

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

Author

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

Keywords

• nonn
• tabl
• hard
• more

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.

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

Examples

			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.
		

Links

• 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.

Crossrefs

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