Quadratic equation - Condensation

Let's apply the condensation theorem 1 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



All we have to do is to choose the right a,b,c,d from the ABA graph:



Now we use the theorem:

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

  • W(G-a-b) = W(AB)*W(A)*y
  • W(G-c-d) = W(AB)*W(A)*y

  • W(G-a-d) = W(A)^2*z^(nb/4+1)*y^(wb/2-1+2)*x^(hb/2)
  • W(G-b-c) = W(A)^2*z^(nb/4+1)*y^(wb/2-1+2)*x^(hb/2)

    Therefore ,
    W(ABA)*W(B)*W(A)^2*y^2 = W(AB)^2*W(A)^2*y^2 + W(A)^4*z^(nb/2+2)*y^(wb+2)*x^(hb)

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

              W(ABA)*W(B) = W(A)^2*z^(nb/2+2)*y^(wb)*x^(hb) + W(AB)^2

    QED

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