Band surgeries and crossing changes between fibered links (Q2835336)

This paper characterizes cutting arcs on fiber surfaces such that the cut surfaces are again fibered, and also characterizes the resulting changes in monodromy.NEWLINENEWLINEA link, which is a disjoint union of simple closed curves in a \(3\)-manifold, is said to be fibered if its exterior fibers over a circle so that each fiber is a Seifert surface. Then such a Seifert surface is called a fiber surface. It is known that if a surface is obtained from a fiber surface by plumbing a Hopf band, then the resulting surface is also a fiber surface. Conversely, if a fiber surface contains a Hopf plumbing summand, then the surface obtained by deplumbing the Hopf band is also a fiber surface. We note that deplumbing a Hopf band is equivalent to cutting a surface along a spanning arc of a Hopf band.NEWLINENEWLINEThe authors, in this paper, generalize this operation. The first main theorem gives a necessary and sufficient condition for a properly embedded arc in a fiber surface to produce a new fiber surface after cutting the original fiber surface along the arc (Section 3). They next study coherent band surgeries which increase the Euler characteristic of a link in the \(3\)-sphere. Since coherent band surgeries increasing the Euler characteristic by at least two have been characterized in [\textit{T. Kobayashi}, Osaka J. Math. 26, No. 4, 699--742 (1989; Zbl 0713.57004)], the present authors characterize, by introducing an operation called generalized Hopf banding, coherent band surgeries increasing the Euler characteristic by one (Section 4). In Section 5, they consider arc-loops, which are loops around arcs, in a fiber surface and characterize Dehn surgeries on arc-loops preserving the fiber surface. The authors further study (generalized) crossing changes between fibered links in the \(3\)-sphere. It is noted that if this operation changes the Euler characteristic, it is well-understood (cf. [\textit{M. Scharlemann} and \textit{A. Thompson}, Comment. Math. Helv. 64, No. 4, 527--535 (1989; Zbl 0693.57004)], [\textit{E. Kalfagianni} and \textit{X.-S. Lin}, Pac. J. Math. 228, No. 2, 251--275 (2006; Zbl 1123.57007)], etc.). Hence this paper characterizes the remaining cases in Section 6.