A394661 - OEIS

%I #57 Jun 12 2026 16:10:29

%S 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

%N 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).

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

%C 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.

%C 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.)

%C 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.

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

%H Charles Livingston and Allison H. Moore, <a href="https://knotinfo.org">KnotInfo: Table of Knot Invariants</a>.

%H Sebastian Oehms, <a href="https://github.com/soehms/database_knotinfo">database_knotinfo: KnotInfo and LinkInfo databases as Python package</a>, GitHub.

%H K. A. Perko, Jr., <a href="https://doi.org/10.2307/2039466">On the classification of knots</a>, Proc. Amer. Math. Soc. 45 (1974), 262-266.

%H Alex Towell, <a href="https://doi.org/10.5281/zenodo.19962842">Joint distributions of knot invariants for prime knots through 11 crossings</a>, Zenodo, 2026.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Crossing_number_(knot_theory)">Crossing number (knot theory)</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Perko_pair">Perko pair</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Seifert_surface">Seifert surface</a>.

%e Triangle begins:

%e   n=3:  1;

%e   n=4:  1;

%e   n=5:  1, 1;

%e   n=6:  1, 2;

%e   n=7:  2, 4, 1;

%e   n=8:  2, 10, 9;

%e   n=9:  4, 22, 22, 1;

%e   n=10: 2, 44, 93, 26;

%e   n=11: 6, 96, 289, 160, 1.

%e 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).

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

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

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

%K nonn,tabf,hard,more

%O 3,6

%A _Alex Towell_, May 01 2026