A394661 Triangle read by rows: T(n, k) is the number of prime knots with n crossings and three-genus k, for n >= 3 and 1 <= k <= floor((n-1)/2).

1, 1, 1, 1, 1, 2, 2, 4, 1, 2, 10, 9, 4, 22, 22, 1, 2, 44, 93, 26, 6, 96, 289, 160, 1

Offset

3,6

Comments

The three-genus (Seifert genus) of a knot is the minimum genus of an orientable Seifert surface bounded by the knot.

For prime knots through 11 crossings, every value k with 1 <= k <= floor((n-1)/2) is realized. The maximum value k = floor((n-1)/2) is achieved by the (2, 2j+1) torus knot at n = 2j+1.

The Perko pair (10_161 = 10_162; see Perko 1974) is excluded; counts are over distinct unoriented prime knots up to mirror image, following the KnotInfo enumeration. (Two unoriented knots that are mirror images of each other are counted once.)

Row sums equal A002863, the number of prime knots with n crossings: 1, 1, 2, 3, 7, 21, 49, 165, 552 for n = 3..11.

Column k = 1 (genus-one prime knots) for n = 3..11: 1, 1, 1, 1, 2, 2, 4, 2, 6.

Links

Table of n, a(n) for n=3..27.

Charles Livingston and Allison H. Moore, KnotInfo: Table of Knot Invariants.

Sebastian Oehms, database_knotinfo: KnotInfo and LinkInfo databases as Python package, GitHub.

K. A. Perko, Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262-266.

Alex Towell, Joint distributions of knot invariants for prime knots through 11 crossings, Zenodo, 2026.

Wikipedia, Crossing number (knot theory).

Wikipedia, Perko pair.

Wikipedia, Seifert surface.

Example

Triangle begins:
  n=3:  1;
  n=4:  1;
  n=5:  1, 1;
  n=6:  1, 2;
  n=7:  2, 4, 1;
  n=8:  2, 10, 9;
  n=9:  4, 22, 22, 1;
  n=10: 2, 44, 93, 26;
  n=11: 6, 96, 289, 160, 1.
Row n=7: of the 7 prime knots with 7 crossings, 2 have genus 1 (7_2, 7_4), 4 have genus 2 (7_3, 7_5, 7_6, 7_7), and 1 has genus 3 (the (2,7) torus knot 7_1).

Crossrefs

Cf. A002863 (row sums: prime knots by crossing number).

Cf. A002864 (alternating prime knots), A051763 (nonalternating prime knots).

Cf. A318051 (|signature| distribution by crossing).

Sequence in context: A214740 A064848 A333335 * A212791 A175001 A205843

Adjacent sequences: A394658 A394659 A394660 * A394662 A394663 A394664

Keyword

nonn,tabf,hard,more

Author

Alex Towell, May 01 2026

Status

approved