The Method Of Solving Tangle Equations

This paper tries to solve the problem is how to solve tangle equations. The left side of thetangle equations are the numerator construction of the summand of the unknown tangles, theright side of the tangle equations are the known knots or links. These tangle equations comefrom the tangle model of the DNA site-specific recombination. During the DNAsite-specific recombination, DNA’s double-stranded break and recombine different ends underthe function of the topoisomerase, then form new DNA molecule. Biologists obtain thefollowing mathematical model: N （O+iR）=K_i（i=0,1,2,3）, where O is a rational tangle orthe summand of two rational tangles, and R is a rational tangle, in addition, O and R areunknown tangles, and iR denotes the tangle sum of i copies of R, and N is thenumerator construction of the tangle, andK_iare the known knots or links. Then our task isworking out the unknown tangles O and R in the above mathematical model, in otherwords, finding out what is the DNA before and after recombination.In order to simplify the calculation, this paper gives the vector representation of tanglesby the constructions of the tangles. There is a canonical vector representation for every tangle,and every canonical vector representation corresponds to a rational number by continuedfraction. The paper connects rational tangles and rational knots or links（2-bridges） throughrational numbers, then find out the solution of the tangle equationsN （O）=K₀,N （O+R）=K₁（O is a rational tangle or the sum of two rational tangles）. However, forequations N （O+iR）=K_i（i=0,1,2,3）, we consider the crossing numbers of K_i（0≤i≤3）,and get the general solution from it.