A374945 a(n) is the number of knots having a Legendrian representative realizable on a Legendrian n-mosaic.

0, 1, 1, 1, 2, 11

Offset

1,5

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.

Two knots have the same smooth knot type if and only if they are related by an ambient isotopy.

Links

Table of n, a(n) for n=1..6.

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 = 2, there is only a(2) = 1 smooth knot family with Legendrian representatives realizable on a Legendrian 2-mosaic, namely unknots.
For n = 5, every Legendrian 5-mosaic depicts either an unknot or a trefoil. Since unknots and trefoils are not ambient-isotopic, we have a(5) = 2.

Prog

(Python, Rust) //See Margaret Kipe links

Crossrefs

Cf. A364939, A374942, A374943, A374944, A374946, A374947

Sequence in context: A260756 A252707 A288848 * A202952 A201145 A053994

Adjacent sequences: A374942 A374943 A374944 * A374946 A374947 A374948

Keyword

hard,more,nonn

Author

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

Status

approved