Band surgeries and crossing changes between fibered links

Abstract

We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes between fibered links, and the resulting changes in monodromy.

Recent advances on the non-coherent band surgery model for site-specific recombination

“…The same research group later showed that the XerCD-dif -FtsK enzymatic complex could unlink replication links tied in vivo in E. coli cells deficient in the decatenase topoIV [33]. Several groups of authors, including our own, studied mathematically the mechanism and pathways of DNA unlinking by site-specific recombination [12,13,83,85]. In [83] we showed that at least 2m steps are needed to unlink a replication link of type T (2, 2n) with one recombination site in each component in the orientation indicated in Figure 3.…”

“…We identified minimal pathways of unlinking under a variety of biologically-motivated assumptions and used the transition probabilities obtained numerically to identify a most probable minimal pathway of DNA unlinking of replication links. In [12], the authors characterize coherent band pathways between knots and two-component links of small crossing number, and in [13], they characterize band surgeries between fibered links in terms of arcs on the fiber surfaces of such links. The above-mentioned studies were all concerned with site-specific recombination modeled as coherent band surgery, and point to the T (2, n) torus knots and links as having heightened importance in pathways of unlinking of replication links.…”

“…While the coherent band surgery model for site-specific recombination has received much attention in the literature, (e.g. [44,12,13,19,83]), non-coherent band surgery is less-understood and is intrinsically difficult from an analytical perspective. This is due to the implicit existence of a non-orientable surface induced by the banding.…”

Distance one lens space fillings and band surgery on the trefoil knot

“…Figure 10. Coherent band surgery is better understood than the non-coherent case (see for example [37,38,13,14,21,64]). If a coherent band surgery decreases the maximal Euler characteristic of an oriented surface without closed components bounding the link, it is known that the band can be isotoped to lie onto a taut Seifert surface [62], [35,Theorem 1.6].…”

“…If the linking number is instead −n/2, Corollary 1.2 follows from the characterization of coherent band surgeries between T (2, n) torus links and certain two-bridge knots in [21,Theorem 3.1]. While both coherent and non-coherent band surgeries have biological relevance, more attention in the literature has been paid to the coherent band surgery model (see for example [37,21,64,38,13,14,67]). This is due in part to the relative difficulty in working with non-orientable surfaces, as is the case with non-coherent band surgery on knots.…”