A051764 Number of torus knots with n crossings.

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 1, 3, 3

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• D. Bar-Natan, 36 Torus Knots(with up to 36 crossings)
• Jim Hoste, Morwen Thistlethwaite, Jeff Weeks, The First 1,701,936 Knots, Math. Intell., 20, 33-48, Fall 1998.
• Andrei Malyutin, On the question of genericity of hyperbolic knots, arXiv preprint arXiv:1612.03368 [math.GT], 2016.
• Kunio Murasugi, On the braid index of alternating links, Trans. Amer. Math. Soc. 326 (1991), 237-260.
• R. G. Scharein, Torus knots and links by crossing number
• Eric Weisstein's World of Mathematics, Hyperbolic Knot
• Eric Weisstein's World of Mathematics, Knot
• Eric Weisstein's World of Mathematics, Torus Knot

Programs

Maple

a:= n-> nops(select(k-> nAlois P. Heinz, Apr 25 2012

Mathematica

a[n_] := (r = Reduce[Sqrt[n] < k <= n && GCD[k, 1 + n/k] == 1, k, Integers]; Which[r === False, 0, r[[0]] === Equal, 1, True, Length[r]]); Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jan 16 2013 *)

PARI

a(n)=my(t=sqrtint(n));sumdiv(n,k,k>t && gcd(k,n/k+1)==1) \\ Charles R Greathouse IV, Apr 26 2012

Formula

a(n) = cardinality of the set {k| sqrt(n) < k <= n and gcd(k, 1+n/k) = 1}; see Murasugi article. - Hermann Gruber, Mar 05 2003