TANGLE EQUATIONS

We are given fixed rational tangles P and R and knots or links K₁ and K₂. Let O be a tangle such that N(O+P)=K₁and N(O+R)=K₂, where N is the numerator construction on the tangles O+P and O+R, respectively. N(O+P)=K₁and N(O+R)=K₁ form two equations, in which O is treated as a variable and P, R, K₁and K₂ are kept fixed. The number and types of solutions for O in the cases where K₁and K₂ are 4-plats or Montesinos knots are discussed.