Link Relations in the w-Alexander Envelope
August-18-10
8:29 AM
wAlexanderLinkRelations.nb
a[1, 1] x[2],  a[1, 2] x[2],  a[2, 1] x[2],  a[2, 2] x[2],  x[1] x[2], x[2]^2
a[1, 1]^2 a[2, 1] x[1]
a[1, 1] a[2, 1] x[1]
a[2,1]x[1]^2 is a relation too, and according to Mathematica it comes from the raw relations in {a[1, 1] x[1] x[2], -a[2, 1] x[1]^2 + a[1, 1] x[1] x[2]}
Further degree 2 relations: {a[2, 1] x[1], a[2, 2] x[1] - 2 a[1, 2] x[2], a[1, 1] x[2],  a[2, 1] x[2], a[2, 2] x[2], x[1] x[2], x[2]^2}
{s0, s1} = {1, 1}; Study[8]
{2,1}->{6,1,5}
{2,2}->{21,7,14}
{2,3}->{56,26,30}
{2,4}->{126,71,55}
{2,5}->{252,161,91}
{2,6}->{462,322,140}
{2,7}->{792,588,204}
{2,8}->{1287,1002,285}
{{{6, 1, 5}, {21, 7, 14}, {56, 26, 30}, {126, 71, 55}, {252, 161, 91}, {462, 322, 140}, {792, 588, 204}, {1287, 1002, 285}}, {5, 14, 30, 55, 91, 140, 204, 285}, 1/6 (1 + m) (2 + m) (3 + 2 m)}
{s0, s1} = {-1, -1}; Study[8]
{2,1}->{6,1,5}
{2,2}->{21,7,14}
{2,3}->{56,24,32}
{2,4}->{126,63,63}
{2,5}->{252,141,111}
{2,6}->{462,282,180}
{2,7}->{792,518,274}
{2,8}->{1287,890,397}
{{{6, 1, 5}, {21, 7, 14}, {56, 24, 32}, {126, 63, 63}, {252, 141, 111}, {462, 282, 180}, {792, 518, 274}, {1287, 890, 397}}, {5, 14, 32, 63, 111, 180, 274, 397}, 1/6 (6 + 17 m + 3 m^2 + 4 m^3)}