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

0, 1, 4, 9, 21, 55

Offset: 1

Author

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

Keywords

• nonn
• 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.

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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