Knotted and linked products of recombination on \(T(2,n)\#T(2,m)\) substrates (Q2877785)

This work is an extension of the theory developed by one of the authors in [\textit{D. Buck} and \textit{E. Flapan}, J. Phys. A, Math. Theor. 40, No. 41, 12377--12395 (2007; Zbl 1189.57003)]. By enzyme actions on circular DNA molecules (called substrates) knots and links (called products) can occur as the result of experiments. These are called site-specific recombination experiments where topoisomerase enzymes change the topology of DNA. The difficulty in understanding this mechanism is that there is no way to observe the details of an enzyme action while it's taking place, and hence the mechanism cannot be studied directly. To solve this problem mathematical models have been created. The first was the tangle model of recombination by \textit{C. Ernst} and \textit{D. W. Sumners} [Math. Proc. Camb. Philos. Soc. 108, No. 3, 489--515 (1990; Zbl 0727.57005)]. The tangle model allows for a mathematical complexity of the enzyme action that is not needed biologically. In the paper by Buck and Flapan [loc. cit.] a simpler model was developed to make it easier for molecular biologists to identify the knotted or linked products of recombination. In particular, the simplification rests on additional assumptions that limit the enzyme action to essentially involve only one crossing and that limit the complexity of the DNA molecule away from the enzyme action to be the boundary of a relatively simple surface. The actual assumptions are technical and lengthy and cannot be stated in a short review. Using these assumptions Buck and Flapan showed that all knots and links that could be produced by site-specific recombination with substrates that were (possibly trivial) \(T(2,n)\) torus links were in a single family. In the paper under review here the model of Buck and Flapan is modified and extended to include substrates which are the connected sum of two torus links of the form \(T(2,n)\#T(2,m)\). The authors use this new model to prove that all knots and links that can be produced by site-specific recombination on such substrates are contained in one of two relatively simple knot families, both of which belong to the class of Conway algebraic knots and include some non-prime products.