A376155 Number of prime knots with 10 or fewer crossings whose mosaic number is n.

0, 1, 0, 1, 6, 96, 146

Offset: 1

Author

Luc Ta, Sep 12 2024

Keywords

• nonn
• fini
• full

Comments

An n X n mosaic is an n X n array of the 11 tiles given by Lomonaco and Kauffman. The mosaic number of a knot K is the smallest integer n such that K is realizable on an n X n knot mosaic.

Here, we count the unknot as a prime knot.

Examples

			There are exactly 6 prime knots that are realizable on a 5 X 5 knot mosaic but not realizable on a 4 X 4 knot mosaic. Namely, these knots are 4_1, 5_1, 5_2, 6_1, 6_2, and 7_4 (see Table 1 of Lee et al.). Hence, a(5) = 6.
		

Links

• Aaron Heap, Douglas Baldwin, James Canning, and Greg Vinal, Tabulating knot mosaics: Crossing number 10 or less, arXiv: 2303.12138 [math.GT], 2023.
• Hwa Jeong Lee, Ludwig Lewis, Joseph Paat, and Amanda Peiffer, Knot mosaic tabulation, Involve, Vol. 11 (2018), pp. 13-26.
• Samuel J. Lomonaco and Louis H. Kauffman, Quantum Knots and Mosaics, Proc. Sympos. Applied Math., Amer. Math. Soc., Vol. 68 (2010), pp. 177-208.
• Index entries for sequences related to knots

Crossrefs

Cf. A002863, A374942, A374943, A374944, A374945, A261400, A375353.