In an orientable \(3\)-manifold \(M\), take an annulus \(A\) meeting a knot transversely. In general, twisting the knot along the annulus \(A\) yields an infinite family of knots. If \(A\) lies in a genus \(g\) Heegaard surface of \(M\), then the minimal bridge number of the family of knots among all genus \(g\) Heegaard surfaces is bounded. Also, if the boundary of \(A\) bounds another annulus whose framing is the same as \(A\) and which is disjoint from the knot, the minimal bridge number is bounded again. The result claims that under a certain condition, these two situations are the only possibilities for the minimal bridge number to be bounded. When \(M\) has genus at most two, the situation is examined in detail.
As an application, the paper discusses the infinite family of knots in the \(3\)-sphere given in [M. Teragaito, Int. Math. Res. Not. 2007, No. 9, Article ID rnm028, 16 p. (2007; Zbl 1138.57012)], \(4\)-surgery on which yields the same small Seifert fibered manifold. It is shown that the bridge number (in the \(3\)-sphere) of the family is unbounded, and the genus two bridge number (in the Seifert fibered manifold) is also unbounded. This phenomenon is in sharp contrast to non-integral surgery studied in [K. L. Baker et al., Algebr. Geom. Topol. 13, No. 5, 2471–2634 (2013; Zbl 1294.57013); Trans. Am. Math. Soc. 367, No. 8, 5753–5830 (2015; Zbl 1329.57017)].