A196111 Number of isomorphism classes of simple quandles of order n.

1, 1, 1, 3, 0, 5, 2, 3, 1, 9, 1, 11, 0, 2, 3, 15, 0, 17, 2, 2, 0, 21, 1, 10, 0, 8, 2, 27, 1, 29, 6, 0, 0, 0, 3, 35, 0, 0, 2, 39, 3, 41, 0, 3, 0, 45

Offset

2,4

Comments

A quandle is simple if it has more than one element, and if it has no homomorphic images other than itself or the singleton quandle. Since a simple quandle with more than two elements is connected, we have a(n) <= A181771(n), for n > 2, with equality if n is prime.

Some authors consider the quandle with one element to be simple and some do not.

Links

Table of n, a(n) for n=2..47.

W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013

David Joyce, Simple Quandles, J. Algebra 79(2) 1982, 307-318.

Leandro Vendramin, On the classification of quandles of low order, arXiv:1105.5341v1 [math.GT].

Leandro Vendramin and Matías Graña, Rig, a GAP package for racks and quandles.

Wikipedia, Racks and quandles

Formula

a(p) = A181771(p) = p - 2, for prime p > 2.

Example

a(2) = 1 since the quandle of order 2 is trivially simple (though not connected).

Prog

(GAP) (using the Rig package)
LoadPackage("rig");
IsSimpleQuandle:=function(q)
local g, N, gg, n;
if IsFaithful(q) = false then return false; fi;
g:=InnerGroup(q);;
if Size(Center(g))>1 then return false; fi;
N:=NormalSubgroups(g);;
gg:=DerivedSubgroup(g);;
for n in N do
  if Size(n) = 1 then continue; fi;
  if IsSubset(gg, n) and Size(n)<Size(gg) then return false;  fi;
od;
return true;
end;;
a:=[1, 1];;
for n in [3..35] do
  a[n]:=0;
  for i in [1..NrSmallQuandles(n)] do
    if IsSimpleQuandle(SmallQuandle(n, i)) then
      a[n]:=a[n]+1;
    fi;
  od;
od;
List([1..35], u->a[u]); # W. Edwin Clark, Dec 06 2011

Crossrefs

Cf. A181769, A181771.

See also Index to OEIS under quandles.

Sequence in context: A324103 A130054 A236146 * A261628 A007431 A215447

Adjacent sequences: A196108 A196109 A196110 * A196112 A196113 A196114

Keyword

nonn,hard,more

Author

James McCarron, Oct 27 2011

Extensions

a(21) corrected by W. Edwin Clark, Dec 06 2011

a(32)-a(35) added by W. Edwin Clark, Dec 06 2011

a(36)-a(47) added by W. Edwin Clark, Dec 28 2014

Status

approved