Fiber-curve surgery
= Fiber-curve surgery
Let a <knot> lie in a fiber of a <mapping torus> of an oriented <topological surface>. Surgery with coefficient $-1$ relative to its <surface framing> replaces the monodromy by its composition with a <right-handed Dehn twist>. Coefficient $+1$ gives the inverse twist. The composition side depends on the choice of seam and time orientation; the existence of the new <fiber bundle> does not.