OFFSET
1,3
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.
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.
EXAMPLE
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.
PROG
(Python, Rust) //See Margaret Kipe links
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
STATUS
approved