An n-string tangle (B3,T) is a 3-ball B3 containing n properly embedded arcs T={ti} and it is called rational if there is a homeomorphism of pairs from (B3,T) to (D,P)×I where D is the unit disk, P is any set of n points in the interior of D and I is the unit interval. In this article we partially generalize the well known classification of 2-string rational tangles to 3-string rational tangles. Using the results obtained we analyze knotted and linked products of site-specific recombination mediated by the Gin DNA invertase and give a possible solution to its action. Gin is an enzyme that carries out processive recombination.

An n-string tangle (B3,T) is a 3-ball B3 which contains n properly embedded arcs T={ti}, it is called rational if there is a homeomorphism of pairs from (B3,T) to (D,P)×I where D is the unit disk, P is any set of n points in the interior of D and I is the unit interval. In this article we extend the classification of the 3-braid group, [Formula: see text], obtained by using Kauffman bracket polynomial, to other families of rational 3-tangles.

The topological analysis of enzymes, an active research topic, has allowed the application of the tangle model of Ernst and Sumners to deduce the action mechanism of several enzymes, modeled as 2-string tangles. By first deriving some results in the theory of 3-braids, in this paper we analyze knotted and linked products of site-specific recombination mediated by the Gin DNA invertase, an enzyme that involves 3-string tangles. Provided that the 3-tangles involved are 3-braids, we determine four families of solutions to its action, two families for each of the directly and inversely repeated site cases. For each case, one of the given solutions had not previously been reported in the related literature. These solutions were found using a computer algorithm, based on our theoretical results, which allows one to solve tangle equations under the assumption that the product of two or more rounds of recombinations is known.

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