# tutte_polynomial (sage, matroid theory) gives the wrong result

I have been using Sage in my research recently.

I am not sure which part is wrong but the following code gives the wrong results. The code is long, but I think you may only need to read the last four rows to understand my question.

All the rows before that generate a matrix `D`

where the columns are the cycles of the complete graph `K_5`

(each cycle is a vector in `Z^10`

, 10 is the number of edges.)

Then I produce a linear matroid from `D`

and calculate `T(2, 0)`

using `tutte_polynomial()`

. The weird thing is that the result
is an odd number, which is impossible because the Tutte polynomial
has no constant term. The more weird thing is that if I repeat the
order again, I get a different result.

If I run the code for the graph `K_4`

, nothing is wrong,
which is why I need to give the complete code.
The code is run by CoCalc.

```
import numpy as np
from sage.matroids.advanced import *
graph = graphs.CompleteGraph(5)
g = graph.to_directed()
allcycles = g.all_simple_cycles()
nv = g.order() # nv is number of vertices
ne = nv*(nv-1)/2 # ne is the number of edges
ncycle = (len(allcycles)-g.size()/2)/2 # ncycle is number of simple cycles
def cycle(array): # array is a cycle in vertices notation eg. [0,1,2,3,0]
l = len(array) - 1 # l is the length of the cycle
if l <= 2:
return False
C = np.zeros(nv*(nv-1)/2) # C is to be a cycle in Z^E
for X in range(l):
i = array[X] + 1
j = array[X+1] + 1
if i < j:
C[(2*nv-i)*(i-1)/2+j-i-1] = 1
else:
C[(2*nv-j)*(j-1)/2+i-j-1] = -1
return(C)
D = np.zeros((ne,ncycle*2)) # D is all directed cycles
X = 0
for list in allcycles:
array = np.array(list)
if len(array) > 3:
D[:, X] = cycle(array).transpose()
X = X + 1
for X in range(ncycle): # reduce D so that D is all cycles with one orientation
positivecolumn = D[:, X]
Y = X + 1
while any(D[:, Y] + positivecolumn):
Y = Y + 1
D = np.delete(D, Y, 1)
print(D)
M = Matroid(Matrix(D))
print("M.tutte_polynomial(2, 0) =", M.tutte_polynomial(2, 0))
print("M.tutte_polynomial(2, 0) =", M.tutte_polynomial(2, 0))
```

The results are

```
[[ 1. 1. 1. 0. 0. 0. 0. 0. 0. 0. 1. 1. 1. 1. 1. 1. 0. 0.
0. 0. 0. 0. 0. 0. 0. 1. 1. 1. 1. 1. 1. 0. 0. 0. 0. 0. 0.]
[-1. 0. 0. 1. 1. 0. 0. 0. 0. 0. 0. 0. -1. 0. -1. 0. 1. 1.
1. 1. 0. 0. 0. 0. 0. 0. 0. 0. -1. 0. -1. 1. 1. 1. 1. 0. 0.]
[ 0. -1. 0. -1. 0. 1. 0. 0. 0. 0. -1. 0. 0. 0. 0. -1. -1. 0.
0. -1. 1. 1. 0. 0. 0. 0. -1. 0. 0. -1. 0. 0. -1. 0. -1. 1. 1.]
[ 0. 0. -1. 0. -1. -1. 0. 0. 0. 0. 0. -1. 0. -1. 0. 0. 0. -1.
-1. 0. -1. -1. 0. 0. 0. -1. 0. -1. 0. 0. 0. -1. 0. -1. 0. -1. -1.]
[ 1. 0. 0. 0. 0. 0. 1. 1. 0. 0. 1. 1. 0. 0. 0. 0. -1. -1.
0. 0. 0. 0. 1. 1. 0. 1. 1. 0. 0. 0. 0. -1. -1. 0. 0. 1. -1.]
[ 0. 1. 0. 0. 0. 0. -1. 0. 1. 0. 0. 0. 1. 1. 0. 0. 1. 0.
0. 0. -1. 0. 0. -1. 1. 0. 0. 1. 1. 0. 0. 1. 0. -1. 1. -1. 0.]
[ 0. 0. 1. 0. 0. 0. 0. -1. -1. 0. 0. 0. 0. 0. 1. 1. 0. 1.
0. 0. 1. 0. -1. 0. -1. 0. 0. 0. 0. 1. 1. 0. 1. 1. -1. 0. 1.]
[ 0. 0. 0. 1. 0. 0. 1. 0. 0. 1. 1. 0. -1. 0. 0. 0. 0. 0.
1. 0. 0. -1. 1. 0. -1. 1. 0. -1. 0. 1. -1. 0. 0. 1. 0. 0. -1.]
[ 0. 0. 0. 0. 1. 0. 0. 1. 0. -1. 0. 1. 0. 0. -1. 0. 0. 0.
0. 1. 0. 1. 0. 1. 1. 0. 1. 1. -1. -1. 0. 0. 0. 0. 1. 1. 0.]
[ 0. 0. 0. 0. 0. 1. 0. 0. 1. 1. 0. 0. 0. 1. 0. -1. 0. 0.
1. -1. 0. 0. 1. -1. 0. 1. -1. 0. 1. 0. -1. 1. -1. 0. 0. 0. 0.]]
M.tutte_polynomial(2, 0) = 46229
M.tutte_polynomial(2, 0) = 52640
```

Welcome to Ask Sage! Thank you for your question.

Thank you for editing my question so that it looks neat. This is a wonderful place.