 
Summary: In[1]:= H*Checking Murakami's Third Relation for the Multivariable Alexander Polynomial*L
H*We have three strands colored from left to right color a and color b and color
c. Coloring assignments from left to right correspond with the bottom of the
strand. We must determine sigma 1 and 2 that correspond to each crossing in
the R3 relation. First let's label some submatrices that will appear as the
various sigma 1's and 2's that come up as we check the Murakami relations.*L
In[2]:= Clear@a, b, cD
H*From old notation a=y, p=b, and c=g.*L
In[3]:= Uab = 88b  1 ê b, a<, 81 ê a, 0<<;
H*This is an upper triangular submatrix for when strand a crosses over strand b*L
Ubc = 88c  1 ê c, b<, 81 ê b, 0<<;
H*This is an upper triangular submatrix for when strand b crosses over strand c*L
Uac = 88c  1 ê c, a<, 81 ê a, 0<<;
H*This is an upper triangular submatrix for when strand c crosses over strand a*L
H*These are the U matrices we needed to check R3 for the multivariable version.*L
In[6]:= Lab = 880, 1 ê a<, 8a, b  1 ê b<<;
H*This is a lower triangular submatrix for when strand a crosses over strand b*L
Lbc = 880, 1 ê b<, 8b, c  1 ê c<<;
H*This is a lower triangular submatrix for when strand b crosses over strand c*L
Lac = 880, 1 ê a<, 8a, c  1 ê c<<;
