# Codimension 4 cyclic polytopes -- Constructing minimal resolutions for codimension 4 cyclic polytopes

The following code prints the combinatorial unprojection structure of a codimension 4 cyclic polytope.

Δ(0,4) -> Δ(2,6) -> Δ(4,8) -> Δ(6,10)

 `i1 : R=QQ[x_1..x_10];` ```i2 : cc=cycRes(6,R,verbose=>1); delta(2,{z, x_2, x_3, x_4, x_5}) + delta(0,{z, x_2, x_3, x_4}) -> delta(2,{z, x_2, x_3, x_4, x_5, x_6}) delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7}) + delta(2,{z, x_2, x_3, x_4, x_5, x_6}) -> delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7, x_8}) delta(6,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9}) + delta(4,{z, x_2, x_3, x_4, x_5, x_6, x_7, x_8}) -> delta(6,{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10})``` ```i3 : betti cc 0 1 2 3 4 o3 = total: 1 25 48 25 1 0: 1 . . . . 1: . . . . . 2: . . . . . 3: . 25 48 25 . 4: . . . . . 5: . . . . . 6: . . . . 1 o3 : BettiTally```

We compute the differentials:

 `i4 : R=QQ[x_1..x_4];` `i5 : cc=cycRes(0,R);` ```i6 : print cc.dd_1 | x_1 x_3 x_4 x_2 |``` ```i7 : print cc.dd_2 {1} | x_3 x_4 x_2 0 0 0 | {1} | -x_1 0 0 0 x_2 -x_4 | {1} | 0 -x_1 0 -x_2 0 x_3 | {1} | 0 0 -x_1 x_4 -x_3 0 |```

 `i8 : R=QQ[x_1..x_6];` `i9 : cc=cycRes(2,R);` ```i10 : print cc.dd_1 | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 -x_1x_5 x_3x_6 x_4x_6 x_2x_6 |``` ```i11 : print cc.dd_2 {2} | 0 x_1 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 0 0 | {2} | -x_1 0 x_2 0 0 -x_1 0 0 0 0 0 0 x_6 0 0 0 | {2} | 0 -x_2 0 x_3 0 0 -x_5 0 0 0 0 0 0 x_6 0 0 | {2} | 0 0 -x_3 0 x_4 0 0 -x_1 0 0 0 0 0 0 x_6 0 | {2} | x_5 0 0 -x_4 0 0 0 0 0 0 0 0 0 0 0 x_6 | {2} | 0 0 0 0 0 -x_3 -x_4 -x_2 0 0 0 0 0 0 0 0 | {2} | 0 0 0 0 0 0 0 0 0 -x_2 x_4 0 -x_5 0 0 -x_1 | {2} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 -x_1 0 0 | {2} | 0 0 0 0 0 0 0 0 -x_4 x_3 0 -x_4 0 0 -x_5 0 |```

 `i12 : R=QQ[x_1..x_8];` `i13 : cc=cycRes(4,R);` ```i14 : print cc.dd_1 | x_2x_4x_6 x_3x_5x_7 x_1x_4x_6 x_2x_5x_7 x_1x_3x_6 x_2x_4x_7 x_1x_3x_5 x_2x_4x_8 x_3x_5x_8 -x_1x_4x_7 x_2x_5x_8 -x_1x_3x_7 x_1x_5x_7 x_3x_6x_8 x_4x_6x_8 x_2x_6x_8 |``` ```i15 : print cc.dd_2 {3} | 0 x_1 0 0 0 0 -x_7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 0 | {3} | -x_1 0 x_2 0 0 0 0 -x_1 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 0 0 | {3} | 0 -x_2 0 x_3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 0 0 | {3} | 0 0 -x_3 0 x_4 0 0 0 0 0 0 0 0 0 -x_1 0 0 0 0 0 0 0 0 0 0 0 x_8 0 0 0 | {3} | 0 0 0 -x_4 0 x_5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_7 0 0 0 0 x_8 0 0 | {3} | 0 0 0 0 -x_5 0 x_6 0 x_1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 0 | {3} | x_7 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_8 | {3} | 0 0 0 0 0 0 0 0 0 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 -x_6 0 0 0 0 -x_7 0 | {3} | 0 0 0 0 0 0 0 0 0 -x_2 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 0 -x_1 | {3} | 0 0 0 0 0 0 0 0 x_2 0 -x_3 0 0 x_5 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 x_3 0 -x_4 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 -x_7 0 0 0 | {3} | 0 0 0 0 0 0 0 -x_5 0 0 x_4 0 0 0 0 0 0 0 0 0 0 0 -x_6 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 x_3 x_4 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 -x_4 0 x_5 0 0 0 0 0 0 0 -x_1 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_2 0 x_3 0 0 0 0 0 0 0 -x_1 0 0 0 0 | {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_4 -x_3 0 x_4 0 0 x_5 0 0 0 0 0 0 0 0 |```