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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.

The Lickorish-Dehn theorem says that every orientation-preserving homeomorphism of a compact connected orientable surface, fixing its boundary pointwise, is isotopic relative to the boundary to a finite product of Dehn twists and their inverses about simple closed curves. Boundary-parallel curves are permitted. For a closed surface of genus $g\ge 1$, a finite collection of curves suffices to generate its mapping class group. If boundary components are allowed to be permuted, twists alone do not express those permutations; the boundary condition is part of the precise statement.

The fundamental theorem of Dehn surgery asserts that every closed connected orientable three-manifold is integer surgery on a finite framed link in $S^3$. To prove it, first choose a Heegaard splitting of $M$ of genus $g$. Existence of such splittings is a standard background result: a triangulation, with spanning trees collapsed in its primal and dual one-skeletons, gives complementary regular neighborhoods of finite graphs, hence two handlebodies. Choose a standard genus-$g$ splitting of $S^3$, formed by stabilizing its genus-zero splitting. Identify the two pairs of handlebodies. If $f_0$ is the sphere's gluing map and $f$ is the gluing map for $M$, then $f_0^{-1}f$ is an orientation-preserving self-homeomorphism of the common surface. The Lickorish-Dehn theorem writes it, up to isotopy, as a product ${\tau }_{c_1}^{{ϵ}_1}\cdots {\tau }_{c_m}^{{ϵ}_m}$, ${ϵ}_i=\pm 1$.

Here is the local surgery mechanism. Place $c$ on an intermediate level of a product collar of the surface. Its tubular neighborhood has longitude $\lambda$ given by that surface and meridian $\mu$ around the normal disk. Remove the neighborhood and fill with meridian slope $\mu \pm \lambda$. Cut the remaining annular neighborhood of $c$ along a transverse arc. Going around the new meridian now makes one extra positive or negative turn in the longitude direction. On gluing the cut annulus back, the two collar levels are therefore identified by $(\theta ,s)\mapsto (\theta \pm \rho (s),s)$, precisely the Dehn twist from part (a). Outside this annulus the identification is unchanged. This proves unit surface-framed surgery realizes a Dehn twist; its surgery coefficient is $+1$ or $-1$ relative to the surface framing, with sign fixed by the chosen normal convention.

Put the curves $c_i$ on distinct collar levels and order those levels to realize the indicated composition. They are disjoint as curves in the three-manifold even if their surface projections intersect. Performing the prescribed surgeries replaces $f_0$ by $f$, up to isotopy. An isotopy of a gluing map extends across a collar, so the filled manifold is homeomorphic to $M$.

Finally each surface framing differs from the preferred longitude framing in $S^3$ by an integer number of twists. Thus its coefficient $\pm 1$ becomes an integer coefficient in the usual knot longitude convention. These finitely many disjoint curves form the required framed link $L$:The ingredients used were Heegaard-splitting existence, the stated twist-generation theorem, and the explicit local unit-surgery construction.

The second assertion is true. Separate the points $p_i$ in the circle direction by a small isotopy if necessary. Cutting $S^1\times S$ at fibers just before and after $p_i\times {\gamma }_i$ exposes two copies of $S$. The surface framing is precisely the stated push-off $p_i^{\prime }\times {\gamma }_i$: the two normal directions give homotopic nonzero normal fields along the curve.

For a single fiber-curve surgery, a cut and regluing of these copies by a Dehn twist changes the curve identified with the transverse meridian by one copy of the surface longitude. In the usual convention that negative surface-framed surgery produces a positive Dehn twist, the filling slope $\mu +\lambda$ implements ${\tau }_{{\gamma }_i}^{-1}$. One can see the coefficient locally by tracing a transverse arc across the twist annulus: it acquires one reverse turn along ${\gamma }_i$, and reversing this cut-and-glue identifies the compressible slope as $\mu +\lambda$. Reversing the circle parameter reverses the monodromy convention, without changing the existence of the fiber bundle.

Every filled block is therefore a product of $S$ with an interval, with a modified endpoint identification. Their cyclic assembly is the mapping torus of the product of these inverse Dehn twists, in the order of the $p_i$ along the oriented circle. It has the original closed topological surface $S$ as fiber, hence fibers over the circle. Intersections between the ${\gamma }_i$ do not obstruct this argument: their twists occur in different fibers, and their ordered product need not commute.

Use the meridian of a knot $\mu$ and Seifert longitude $\lambda$ of the unknot on its boundary torus. The unknot's knot exterior is a solid torus whose disk-bounding curve is $\lambda$. A genus-one Heegaard diagram is thereforeHere the horizontal coordinate is $\mu$ and the vertical coordinate is $\lambda$. The drawing uses a small translate of $\alpha$ to avoid intersections on the edge of the square; opposite edges are identified. The algebraic intersection number of curves on an oriented surface has absolute value five.

Starting with $\Sigma \times [0,1]$, attach a three-dimensional two-handle along $\alpha \times \{0\}$, using its surface framing, and cap the resulting sphere with a three-handle. This gives the solid torus on that side. Attach a two-handle along $\beta \times \{1\}$, again using the surface framing, and cap its sphere with a three-handle. The second solid torus has its disk-bounding curve identified with $5\mu +3\lambda$, so the resulting oriented lens space is precisely the specified $5/3$ Dehn filling.

Put a simple closed curve in a surface collar and use its surface framing. Filling its deleted tubular neighborhood along $\mu \pm \lambda$ inserts one full positive or negative twist of the annulus between the collar levels. Thus surgery of coefficient $\pm 1$ changes the gluing by a Dehn twist or its inverse. The exact sign depends on the orientations of the seam and normal direction.