A379242 Minimum crossing number at which there are n torus knots.

1, 3, 15, 63, 189, 432, 792, 1232, 1584, 2880, 4320, 5040, 6336, 7920, 12096, 15120, 19008, 22176, 30240, 33264, 43200, 47520, 44352, 65520, 75600, 108000, 90720, 120960, 168480, 131040, 151200, 181440, 252000, 196560, 221760, 237600, 362880, 403200, 302400

Offset: 0

Author

Alex Klotz, Dec 18 2024

Keywords

• nonn

Comments

Minimum number that can be factored N different ways into p*(q-1) for coprime p and q with p>q. e.g. 63=63*(2-1)=9*(8-1)=21*(4-1); 63 is the smallest crossing number with three torus knots. Odd numbers will admit an alternating (p,2) torus knot with p crossings, all others with q>2 are non-alternating. Based on definition of torus knot and data from A051764.

Examples

			3 = 3*(2-1), 15 = 15*(2-1) = 5*(4-1), 63 = 63*(2-1) = 9*(8-1) = 21*(4-1).
		

Links

• Alexander R. Klotz and Caleb J. Anderson, Ropelength and writhe quantization of 12-crossing knots, arXiv:2305.17204 [math.GT], 2023; Experimental Mathematics (2024): 1-8.

Crossrefs

First occurrence of each n in A051764.

Extensions

More terms from Alois P. Heinz, Dec 29 2024