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The 34 Convex Heptahedra and their Characteristic
Polynomials
Followed by free programs you can use as you untangle
the 257 convex octahedra.
x (x^{2}+x1) (x^{3}x^{2}9x4) 
x (x+1) (x+2) (x^{3}3x^{2}4x+4) 
(x1)^{2} (x+1)^{2} (x+2) (x^{2}2x6) 
(x+2) (x^{2}2) (x^{4}2x^{3}6x^{2}+2x+2) 
x^{7}12x^{5}10x^{4}+23x^{3}+20x^{2}9x6 
x (x1) (x+1) (x^{4}11x^{2}8x+8) 
x (x^{2}2)(x^{4}10x^{2}8x+2) 
x^{2} (x+2)^{2} (x^{3}4x^{2}+6) 
(x^{2}+x1)^{2} (x^{3}2x^{2}7x+6) 
x (x+2) (x^{5}2x^{4}8x^{3}+8x^{2}+8x6) 
(x1) (x+1)^{2} (x^{2}3x1) (x^{3}+2x^{2}4x7) 
(x1) (x+1) (x+2) (x^{5}2x^{4}8x^{3}+8x^{2}+11x+2) 
(x^{4}x^{3}8x^{2}+5) (x^{4}+x^{3}4x^{2}4x+1) 
x^{8}13x^{6}8x^{5}+42x^{4}+34x^{3}35x^{2}26x+4 
(x^{3}+2x^{2}x1) (x^{5}2x^{4}8x^{3}+9x^{2}+11x9) 
(x^{3}+x^{2}2x1) (x^{5}x^{4}10x^{3}+3x^{2}+18x4) 
(x1)(x+2)(x^{2}2)(x^{4}x^{3}8x^{2}+4) 
x^{8}13x^{6}6x^{5}+40x^{4}+20x^{3}38x^{2}14x+9 
(x+1) (x^{7}x^{6}12x^{5}+6x^{4}+32x^{3}10x^{2}14x+4) 
(x^{4}+x^{3}4x^{2}+1) (x^{4}x^{3}8x^{2}+5) 
(x1) (x+1) (x^{2}+x1) (x^{4}x^{3}10x^{2}+5x+9) 
(x+2) (x^{3}3x1) (x^{5}2x^{4}7x^{3}+9x^{2}+6x2) 
(x+1) (x^{3}+x^{2}2x1) (x^{5}2x^{4}9x^{3}+13x^{2}+15x8) 
(x^{3}+x^{2}2x1) (x^{6}x^{5}11x^{4}+4x^{3}+32x^{2}x+22) 
(x^{3}+2x^{2}x1) (x^{6}2x^{5}9x^{4}+11x^{3}+24x^{2}12x14) 
x (x2) (x1) (x+1)^{2} (x+2) (x^{3}x^{2}8x+4) 
(x^{2}x1) (x^{2}+x1) (x^{2}+3x+1) (x^{3}3x^{2}3x+8) 
x^{9}14x^{7}4x^{6}+55x^{5}+18x^{4}70x^{3}20x^{2}+25x+8 
(x^{3}+x^{2}2x1) (x^{6}x^{5}11x^{4}+8x^{3}+20x^{2}7x8) 
(x3) (x1) (x+2)^{2} (x^{3}3x1)^{2} 
(x3) (x^{2}x1) (x^{2}+x1) (x^{2}+3x+1) (x^{3}4x1) 
x (x3) (x1) (x+1) (x+2) (x^{5}+x^{4}7x^{3}5x^{2}+10x+4) 
(x3) (x^{3}+x^{2}2x1)^{2} (x^{3}+x^{2}5x1) 
(x3)(x1)(x^{2}x1)^{2}(x^{2}+3x+1)^{2} 


I've done an interesting study using Matrix calculator and Factoris . There are 7 convex hexahedra, and 34 convex heptahedra. It is fairly easy to figure out the characteristic polynomial of each one when such tools are available. I've started with a somewhat irregular sixsided figure. Note that points A and C are connected. Also, note that A and F are not connected. To make an adjacency matrix, one merely makes a grid of ones and zeros. Ones indicate a connection, or an adjacency. Zeroes indicate no connection. Since the unlying graph has six points (A B C D E and F), it has a 6x6 adjacency matrix. You can copy the matrix in the third box, and paste it into the Matrix calculator . Then indicate you want the characteristic polynomial. You'll get x^{6}  10x^{4}  8x^{3} + 9x^{2} + 4x  1. If you relabel the graph in a different way (perhaps A C E F B D) and create the new adjacency matrix corresponding to this new labeling, you'll still get the same polynomial.

0 1 1 0 1 0
1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 
0,1,1,1,1,1,0,0,0,0,0,0
1,0,1,0,1,0,1,0,0,0,1,0
1,1,0,1,0,0,0,0,0,1,1,0
1,0,1,0,0,1,0,0,1,1,0,0
1,1,0,0,0,1,1,1,0,0,0,0
1,0,0,1,1,0,0,1,1,0,0,0
0,1,0,0,1,0,0,1,0,0,1,1
0,0,0,0,1,1,1,0,1,0,0,1
0,0,0,1,0,1,0,1,0,1,0,1
0,0,1,1,0,0,0,0,1,0,1,1
0,1,1,0,0,0,1,0,0,1,0,1
0,0,0,0,0,0,1,1,1,1,1,0
An adjacency matrix of the icosahedron
x (x + 2) (x^{5}  2x^{4}  12x^{3}  6x^{2} + 10x + 6) 
(x + 1)(x^{5 } x^{4 } 9x^{3 } 3x^{2 }+ 10x + 4) 