Systems of 3-braid equations

Abstract

"The tangle model is a useful topological tool in the study of the mechanism of action of certain enzymes on DNA molecules. In particular, the model proves helpful to determine the topological structure of the DNA molecules resulting from those reactions. Roughly speaking, the tangle model consists in solving a system of three equations in which the unknowns on the left-hand side of each equation are tangles, whereas the known data on the right-hand sides are 2-bridge knots in many cases. Initially [6], the model was successfully applied to study the Tn3 enzyme acting on rational 2-tangles, for which a complete classification exists [4]. By contrast, the Gin enzyme is known [8] to act on 3-tangles and, since no complete classification is known for general 3-tangles, the tangle model was used to study the mechanism of action of Gin under the assumption [2] that the 3-tangles involved were in fact 3-braids, a particular class of 3-tangles. Some questions derived from the application of the tangle model are of mathematical interest in themselves, e.g., given a system of equations that admits a solution, what kinds of 2 -bridge knots may appear on the right-hand sides of the equations so that a (nonempty) solution is guaranteed to exist? In this paper, we address and solve this question by showing that, while a system of two equations always admits a solution for any selection of 2 -bridge knots, adding a third equation reduces the number of possible knots to only 6, 9 or 18, the exact value depending on the relationships satisfied by the knots in the first two equations. If a fourth equation is adjoined, however, exactly one 2 -bridge knot may appear in its the right-hand side for the system to admit a solution. Furthermore, a new simple method that exploits an unexpected cyclic behavior of the solutions is presented and used to construct the proofs. The method relies on the continued fractions associated with 2 -bridge knots and their behavior under the concatenation of 3-braids."