Abstract
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.
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