Band Surgeries between Knots and Links with Small Crossing Numbers

Abstract

In DNA site-specific recombination, an enzyme attaches to a pair of DNA sites, and recombines the sites to different ends. During site-specific recombination, the topology of circular DNA can change, forming knots and links. We model Xer recombination as band surgery. By regarding Xer recombinations as band surgeries and applying mathematical results on band surgery, we confirm experimental results of Xer recombination acting on circular DNA. In particular, our motivation is the unlinking of DNA catenanes by Xer-dif-FtsK recombinations reported by Grainge et al.,6) and Xer recombination at the psi-site on DNA catenanes with 2k crossings which yields DNA knots with 2k + 1 crossings reported by Bath, Sherratt and Colloms.1) The main result of this paper is summarized in Table II of band surgeries between knots with 7 and fewer crossings and catenanes with 8 and fewer crossings. In §2, we relate band surgeries to site-specific recombinations. In §3, we give the table which characterize band surgeries. In §4, we give a table for band surgery between knots. For knots with 7 and fewer crossings, we use the classical notation as in the book by Rolfsen (see Appendix C in 14)). For a knot or link K, we denote by K! the mirror image of K throughout this paper. We use several invariants of knots and links to construct Table II (see Theorems 3.4, 3.5, 3.7, 3.9, 3.11, and 3.13).