Linear equation - Condensation

Let's apply the condensation theorem 2 to our Snakes. Consider the substitution, from the triple (A,B,AB) to (A,ABA,AB).
We have to prove that W(ABA)*W(B) = W(A)^2*z^(nb/2+2)*y^(wb)*x^(hb) + W(AB)^2



First rotate the bottom AB in the ABA snake, which is symmetric and then choose a,b,c,d:



Now we use the theorem:

  • W(G-a-d) = W(B) * x * W(A)^2 * (x^2+y^2+z^2)                (A-unit-A has the same matchings as A^2*(x^2+y^2+z^2)
  • W(b-c) = W(BA) * y * W(A)

  • W(G) = W(ABA)
  • W(G-a-b-c-d) = W(A)^2 * W(B) *  y * x 

  • W(G-a-c) = W(A) * W(B) * z^(na/4+1)*y^(wa/2)*x^(ha/2+1)
  • W(G-b-d) = W(A) * W(B) * z^(na/4+1)*y^(wa/2+1)*x^(ha/2)

    Therefore ,
    W(B) * W(AB) * (x^2+y^2+z^2) * W(A)^3 * x * y = W(ABA) * W(B) * W(A)^2 * x * y + W(A)^2 * W(B)^2 * z^(na/2+2)*y^(wa+1)*x^(ha+1)

    We just have to cancel W(B) * W(A)^2 * x * y on both sides, to get:

              W(AB) * W(A) * (x^2+y^2+z^2) = W(ABA) + W(B) * z^(na/2+2)*y^(wa)*x^(ha)

    QED

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