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

0,2

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.

Links

Table of n, a(n) for n=0..38.

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.

Example

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

Crossrefs

First occurrence of each n in A051764.

Sequence in context: A364585 A260326 A204086 * A171761 A229277 A218313

Adjacent sequences: A379239 A379240 A379241 * A379243 A379244 A379245

Keyword

nonn,new

Author

Alex Klotz, Dec 18 2024

Extensions

More terms from Alois P. Heinz, Dec 29 2024

Status

approved