2·1. Factorisations

Petronio [ Reference Petronio24 ] proved a unique factorisation theorem for closed good 3-orbifolds (M,T) without nonseparating spherical 2-suborbifolds. We are working in a slightly more general context (because our orbifolds may not be closed, we allow good nonseparating spherical 2-suborbifolds, and we allow infinite weight edges). Nevertheless, we adopt Petronio’s terminology and his results carry over to our setting as we now describe.

Suppose that $(M_1, T_1)$ and $(M_2, T_2)$ are distinct nice orbifolds. Let $p_1 \in M_1$ and $p_2 \in M_2$ . We can perform a connected sum of $M_1$ and $M_2$ by removing a regular neighbourhood of $p_1$ and $p_2$ and gluing the resulting 3-manifolds together along the newly created spherical boundary components; after gluing the corresponding sphere is a summing sphere. To extend the sum to the pairs $(M_1, T_1)$ and $(M_2, T_2)$ we place conditions on the points. We require that they are either in the interiors of edges of the same weight or that they are on vertices with incident edges having matching weights. The gluing map is then required to match punctures to punctures of the same weight.

When $p_1$ and $p_2$ are disjoint from $T_1$ and $T_2$ , the sum is a distant sum and we write $(M,T) = (M_1, T_1) \#_0 (M_2, T_2)$ . When each $p_i$ is in the interior of an edge of $T_i$ , it is a connected sum and we write $(M,T) = (M_1, T_1) \#_2 (M_2, T_2)$ . When each $p_i$ is a trivalent vertex, it is a trivalent vertex sum and we write $(M,T) = (M_1, T_1) \#_3 (M_2, T_2)$ . This sum is well-defined, for a particular choice of vertices $p_1, p_2$ and bijection from the ends of edges incident to $p_1$ to those incident to $p_2$ . See [ Reference Wolcott42 , section 4 ]. The pair $\mathbb{S}(0) = (S^3, \varnothing)$ is the identity for $\#_0$ . The pair $\mathbb{S}(2) = (S^3, T)$ where T is the unknot is the identity for $\#_2$ . The pair $\mathbb{S}(3) = (S^3, T)$ where T is a planar (i.e. trivial) $\theta$ -graph, is the identity for $\#_3$ . In each case, we allow T to have whatever weights make sense in context. Note that when $k\lt i \leq j$ , both factors of $\mathbb{S}(i) \#_k \mathbb{S}(j)$ are trivial. This means that some care is needed when discussing prime factorisations.

Conversely, suppose that (M,T) is a nice orbifold. Given a connected spherical 2-suborbifold $S \subset (M,T)$ we may split (M,T) open along S and glue in two 3-balls each containing a graph that is the cone on the points $S \cap T$ . This operation is called surgery along S. Observe that the result is still an orbifold and that if M is connected, each component of $(M,T)|_S$ is incident to one or more scars from the surgery (i.e. the boundaries of the 3-balls we glued in). If S is such a sphere or the disjoint union of such spheres, we denote the result of surgery along (all components of) S by $(M,T)|_S$ . Each component of S produces two scars in $(M,T)|_S$ ; we say those scars are matching. If S separates M, then surgery is the inverse operation to distant sum, connected sum, or trivalent vertex sum.

Since we will be dealing with reducible orbifolds, we need to work with slight generalizations of compressing discs. Suppose that $S \subset (M,T)$ is a surface. An sc-disc for S is a zero or once-punctured disc D with interior disjoint from $T \cup S$ , with boundary in $S \setminus T$ , and which is not isotopic (by an isotopy everywhere transverse to T) into S. If $\partial D$ does not bound a zero or once-punctured disc in S, then D is a compressing disc or cut disc corresponding to whether D is zero or once-punctured. A c-disc is a compressing disc or cut disc. Otherwise, D is a semi-compressing disc or semi-cut disc respectively. In other words, a semi-compressing disc or semi-cut disc is a zero or once-punctured disc with inessential boundary in the surface but which is not parallel into the surface. The weight $\omega(D)$ of an sc-disc D is equal to the weight of the edge of T intersecting it and 1 otherwise. Compressing a surface using an sc-disc D decreases $x_\omega$ by $2/\omega(D)$ . If $\partial M$ admits an sc-disc D, then $(M,T)\setminus D$ is the result of $\partial$ -reducing (M,T).

If S does not admit a c-disc, it is c-incompressible. A c-essential surface is a surface $S \subset (M,T)$ where each component is:

1. (1) a c-incompressible surface

2. (2) not parallel in $M \setminus T$ into $\partial (M\setminus T)$ (i.e. not $\partial$ -parallel); and

3. (3) not an unpunctured sphere bounding a 3-ball in $M \setminus T$ .

Observe that for a surface, being parallel in (M,T) to a component of $\partial M$ is more restrictive than being parallel into $\partial (M\setminus T)$ .

For a system of summing spheres $S \subset (M,T)$ , we call $(M,T)|_S$ , a factorisation of (M,T) using S. The main difference between a factorisation using an efficient system of summing spheres and a prime factorisation is that a factor in an efficient factorisation may contain a nonseparating sphere. By work of Petronio and Hog-Angeloni–Matveev, in the absence of nonseparating spherical 2-orbifolds, up to orbifold homeomorphism for an efficient system of summing spheres S, both S and its factorisation $(M,T)|_S$ are unique; however, these homeomorphisms are not necessarily realisable by an isotopy in (M,T). We note that neither the Petronio, Hog-Angeloni–Matveev results nor the following theorem (which is based on those) is trivial. For instance, as observed in [ Reference Petronio24 , example 1·3], the connected sum of any knot in $S^3$ with $(S^1 \times S^2, S^1 \times \text{(point)})$ is pairwise homeomorphic to $(S^1 \times S^2, S^1 \times \text{(point)})$ . Consequently, $(S^1 \times S^2, S^1 \times \text{(point)})$ does not have a prime factorization. The crux of the existence of an efficient splitting system is deferred to Corollary 4·8 below and is a consequence of Taylor–Tomova’s work on thin position. Based on the discussion in [ Reference Petronio24 , section 3], the ability of thin position to handle nonseparating spherical 2-suborbifolds seems to be an instance where the thin position techniques have an advantage over normal surface techniques. We relegate the proof to the appendix, since it is a slightly more elaborate version of Petronio’s and Hog-Angeloni–Matveev’s proofs.

The Equivariant sphere theorem is an important tool for studying group actions on 3-manifolds. The statement we use is inspired by [ Reference Boileau, Leeb and Porti1 , theorem 3·23] and the subsequent remark.

2·2 Orbifold Heegaard Splittings

As is well known, a Heegaard splitting of a closed 3-manifold is a decomposition of the 3-manifold into the union of two handlebodies glued along their common boundary. Every closed 3-manifold has such a decomposition. 3-manifolds with boundary have a similar decomposition into two compressionbodies glued along their positive boundaries. Zimmermann (e.g. [ Reference Zimmermann44 – Reference Zimmermann46 ]) defined Heegaard splittings of closed 3-orbifolds and used them to study equivariant Heegaard genus. In his decompositions, an orbifold handlebody is an orbifold that is the quotient of a handlebody under a finite group of diffeomorphisms. He gives an alternative description in terms of certain kinds of handle structures [ Reference Zimmermann45 , proposition 1]. We adapt this latter definition to define orbifold compressionbodies. Petronio [ Reference Petronio23 ] also defines handle structures for 3-orbifolds; the definitions differ only on the definition of 2-handles. Zimmermann’s definition has the advantage that handle structures can be turned upside down. Most of our definitions apply to graphs in 3-manifolds more generally, so we state them in that generality if possible.

A 1-handle or 2-handle is a pair $(H, T_H)$ pairwise homeomorphic to $(D^2 \times I, p \times I)$ where p is either empty or is the center of $D^2$ . The attaching region of a 1-handle is the preimage of $(D^2 \times \partial I, p \times \partial I)$ . The attaching region of a 2-handle is $((\partial D^2) \times I, \varnothing)$ .

A vp-compressionbody $(C, T_C)$ is the union of finitely many 0-handles and 1-handles so that the following hold:

1. (1) the 0-handles are pairwise disjoint, as are the 1-handles;

2. (2) 1-handles are glued along their attaching regions to the positive boundary of the 0-handles, and are otherwise disjoint from the 0-handles;

3. (3) if $(H, T_H)$ is a 1-handle such that one component (D,p) of its attaching region is glued to a 0-handle $(W, T_W)$ , and if $T_H \neq \varnothing$ , then $p \in T_H \cap \partial_+ W$ ;

4. (4) C is connected.

See Figure 1 for an example. The “vp” stands for “vertex punctured” and is used since drilling out vertices changes trivial ball compressionbodies with vertices into trivial product compressionbodies. If $(C, T_C)$ is a vp-compressionbody, we let $\partial_- C$ be the union of $\partial_- W$ over the 0-handles $(W, T_W)$ and we let $\partial_+ C = \partial C \setminus \partial_- C$ . Edges of $T_C$ disjoint from $\partial_+ C$ are called ghost arcs. Closed loops disjoint from $\partial C$ are core loops. Edges with exactly one endpoint on $\partial_+ C$ are vertical arcs and edges with both endpoints on $\partial_+ C$ are bridge arcs. Dually, vp-compressionbodies may be defined as the union of 2-handles and 3-handles. Equivalently, if $(C, T_C)$ is a connected pair with one component of $\partial C$ designated as $\partial_+ C$ , then $(C, T_C)$ is a vp-compressionbody if and only if there is a collection of pairwise disjoint sc-discs $\Delta$ for $\partial_+ C$ such that the result of $\partial$ -reducing $(C, T_C)$ along $\Delta$ is the union of trivial ball compressionbodies and trivial product compressionbodies. The collection $\Delta$ is a complete collection of sc-discs for $(C, T_C)$ if it is pairwise nonparallel. If $T_C$ has weights such that $(C, T_C)$ is both a vp-compressionbody and an orbifold, then we call $(C, T_C)$ an orbifold compressionbody.

Fig. 1.

An example of a vp-compressionbody $(C, T_C)$ . It has one ghost arc, one core loop, one bridge arc, and three vertical arcs. The horizontal lines represent a closed, possibly disconnected surface F.

For a nice orbifold (M,T), an orbifold Heegaard surface is a transversally oriented separating connected surface $H \subset (M,T)$ such that H cuts (M,T) into two distinct orbifold compressionbodies, glued along their positive boundaries. We define the Heegaard characteristic of (M,T) to be:

\begin{align*}x_\omega(M,T) = \min\limits_H x_\omega(H),\end{align*}

where the minimum is over all orbifold Heegaard surfaces H for (M,T). The invariant $x_\omega(M,T)$ is twice the negative of the “Heegaard number” defined by Mecchia–Zimmerman [ Reference Mecchia and Zimmermann15 ]. Dividing by 2 would also make the comparison with [ Reference Taylor and Tomova37, Reference Taylor and Tomova38 ] easier. However, since orbifold Euler characteristic need not be integral, making that normalisation would unpleasantly complicate some of the calculations in this paper.

The equivariant Heegaard characteristic of W is

\begin{align*}x_\omega(W;G) = \min\limits_H x_\omega(H),\end{align*}

where the minimum is taken over all invariant Heegaard surfaces for H. Zimmermann proved the following when W is closed; the proof extends to the case when W has boundary as we explain. See also [ Reference Futer5 ] for a similar result related to strong involutions on tunnel number 1 knots. The statement is deceptively simple as it implies (and its proof relies on) the Smith Conjecture as well as a thorough understanding of group actions on 3-balls and products. See the remark on page 52 of [ Reference Boileau, Leeb and Porti1 ].

Lemma 2·7 (after Zimmermann [ Reference Zimmermann45 ]). Suppose that W has orbifold quotient (M,T). Every invariant Heegaard surface for W descends to an orbifold Heegaard surface for (M,T) and every orbifold Heegaard surface for (M,T) lifts to an invariant Heegaard surface for W. Consequently,

\begin{align*}x_\omega(W;G) = |G|x_\omega(M,T).\end{align*}

Proof. Suppose that Y is a compressionbody (i.e. orbifold compressionbody with empty graph) and that G is a finite group of orientation-preserving diffeomorphisms of Y. We show the quotient orbifold is an orbifold compressionbody. If $\partial_- Y = \varnothing$ (i.e. Y is a handlebody), then the quotient orbifold is an orbifold compressionbody by [ Reference Zimmermann45 ]. In particular, the quotient of a 3-ball is an orbifold trivial ball compressionbody. Consider the case when $Y = F \times I$ for a closed, connected, oriented surface F. Since each element of G is orientation-preserving, no element interchanges the components of $\partial Y$ . If $Y = S^2 \times I$ , we cap off $\partial_- Y$ with a 3-ball, extend the G-action across the 3-ball, and appeal to the 3-ball case to observe that the quotient orbifold is a trivial product orbifold compressionbody. Suppose that $F \neq S^2$ . Let DY be its double and observe that we can extend the action of G to DY. The 3-manifold DY is a Seifert fiber space with no exceptional fibers, namely $F \times S^1$ . Consider an embedded torus DQ that is the double of an essential spanning annulus Q in Y. By [ Reference Hass9 , theorem 1·5], DQ can be isotoped to DQ $^{\prime}$ so that for each $g \in G$ , gDQ $^{\prime}$ is vertical in DQ and is isotopic to gDQ. Since each $g \in G$ preserves each component of $\partial Y$ , there is an annulus $Q' \subset Y$ , vertical in Y, such that for each $g \in G$ , gQ $^{\prime}$ is isotopic to gQ by a proper isotopy in Y. Applying this observation to each annulus in a collection of spanning annuli in Y cutting Y into 3-balls, we can then conclude that the quotient orbifold of Y by the action of G is a trivial product orbifold, as desired. Finally, suppose that $\partial_+ Y$ is compressible. By the Equivariant Disc Theorem [ Reference William and Yau17 ], Y admits an equivariant essential disc. Boundary-reducing along this disc and inducting on $x_\omega(\partial_+ Y) - x_\omega(\partial_- Y)$ shows that the quotient orbifold is again an orbifold compressionbody. Our lemma then follows from the definition of invariant Heegaard surfaces, orbifold Heegaard surfaces, and the multiplicativity of orbifold characteristic under finite covers.

Proof. Let (M,T) be the quotient orbifold of the action of G on W. Assume the action is free so that $T =\varnothing$ . There exists an efficient system of summing spheres $\overline{S} \subset M$ . By Haken’s Lemma, $\mathfrak{g}(M) = \mathfrak{g}(M|_{\overline{S}})$ . Let S be the pre-image of $\overline{S}$ in W. By Lemma 2·7 (after converting from Euler characteristic to genus), $\mathfrak{g}(W;G) = \mathfrak{g}(W|_S; G)$ .

5·1. Analysing Orbifold Compressionbodies

Definition 5·1. A lens space is a closed 3-manifold of Heegaard genus 1, other than $S^3$ or $S^1 \times S^2$ . A core loop in a solid torus $D^2 \times S^1$ is a curve isotopic to $\{\text{point}\} \times S^1$ . A core loop in a lens space or $S^1 \times S^2$ is a knot isotopic to the core loop of one half of a genus 1 Heegaard splitting. A Hopf link in $S^3$ , $S^1 \times S^2$ , or a lens space is a 2-component link such that there is a Heegaard torus separating the components and so that each component is a core loop for the solid tori on opposite sides of a Heegaard torus. A pillow $(C, T_C)$ is a vp-compressionbody with boundary a 4-punctured sphere that is the result either of joining two (3-ball, arc) trivial ball compressionbodies by an unweighted 1-handle or joining two (3-ball, trivalent graph) compressionbodies by a weighted 1-handle. (See Figure 2.) An orbifold that is a pillow or trivial ball compressionbody is Euclidean if the boundary surface is. A Euclidean double pillow is a pair $(S^3, T)$ with an orbifold bridge surface H such that each of $(S^3, T)\setminus H$ is a Euclidean pillow. (The terminology stems from the fact that Euclidean orbifolds admit a complete metric locally modeled on Euclidean 2 or 3-space.)

Fig. 2.

The two types of pillow.

If $(C, T_C)$ is the disjoint union of orbifold compressionbodies, let $N(C,T_C) = x_\omega(\partial_+ C) - x_\omega(\partial_- C)$ . Our key identity for an multiple orbifold Heegaard surface $\mathcal{H} \in \mathbb H(M,T)$ is:

(1) \begin{equation}2\operatorname{net}x_\omega(M,T) - x_\omega(\partial M) = \sum\limits_{(C, T_C)} N(C, T_C),\end{equation}

where we sum over the vp-compressionbodies $(C, T_C) \sqsubset (M,T) \setminus \mathcal{H}$ . This follows immediately from the fact that each component of $\mathcal{H}^+$ and each component of $\mathcal{H}^-$ appears exactly twice as a boundary component of $(M,T)\setminus \mathcal{H}$ .

Proof. Let $\Delta$ be a complete collection of sc-discs for $(C, T_C)$ such that $\partial$ -reducing $(C, T_C)$ along $\Delta$ results in trivial vp-compressionbodies $(C', T'_C)$ . Each disc of $\Delta$ , leaves 2 “scars” on $\partial_+ C'$ . If E is a scar, let $\omega(E) = 1$ if it is unpunctured and otherwise let $\omega(E)$ be the weight of the puncture. Let $(C_0, T_0) \sqsubset (C', T'_C)$ . Let $N'(C_0, T_0)$ be equal to the sum of $N(C_0, T_0)$ with ${1}/{\omega(E)}$ for all scars E on $\partial_+ C_0$ . Observe that

\begin{align*}N(C, T_C) = \sum\limits_{(C_0, T_0)} N'(C_0, T'_0),\end{align*}

where the sum is taken over all $(C_0, T_0) \sqsubset (C', T')$ .

If $(C_0, T_0)$ is a product compressionbody then $N'(C_0, T_0) \geq N(C_0, T_0) = 0$ with equality if and only if every scar on $\partial_+ C_0$ has weight $\infty$ . Suppose that $(C_0, T_0)$ is a trivial ball compressionbody.

If $\Delta = \varnothing$ , then $(C, T_C) = (C_0, T_0)$ and $N(C, T_C) = -2$ . Otherwise, by the choice of $\Delta$ , $\partial_+ C_0$ contains at least 2 scars, each of weight 1. If it contains exactly 2, then $(C, T_C)$ is (solid torus, $\varnothing$ ). If it has at least 3 scars, then $N'(C_0, T_0) \geq 1$ .

If $\Delta = \varnothing$ , then $(C, T_C) = (C_0, T_0)$ and $N(C, T_C) = -{2}/{k} \geq -1$ . If $N(C, T_C) = 0$ , then $k = \infty$ . If $\Delta \neq \varnothing$ , then $\partial_+ C_0$ contains at least one scar. By our choice of $\Delta$ , either $(C, T_C)$ is (solid torus, core loop) or $\partial_+ C_0$ contains at least 1 scar of weight 1. In which case, $N'(C_0, T_0) \geq 0$ . Equality holds if and only if $k = 2$ , there is exactly one scar and it has weight 1.

Note that $N(C_0, T_0) = x_\omega(\partial_+ C) = x_\omega(v) \lt 0$ where v is the internal vertex of $T_0$ . If $\Delta = \varnothing$ , then we have our result. If $\Delta \neq \varnothing$ , then $\partial_+ C_0$ contains at least one scar and it either has weight 1 or has weight equal to the weight of one of the punctures on $\partial_+ C$ . In which case, $N'(C_0, T_0) \geq 0$ . Equality holds only when there is exactly one scar, it contains a puncture, and the two punctures not contained in the scar both have weight 2.

This concludes our analysis of the individual cases and, in particular, we may assume that $\Delta \neq \varnothing$ and that $(C, T_C)$ is neither (solid torus, $\varnothing$ ) or (solid torus, core loop). By our analysis, each component $(C_0, T_0) \sqsubset (C', T'_C)$ has $N'(C_0, T_0) \geq 0$ . Thus, $N(C, T_C) \geq 0$ . Suppose that $N(C, T_C) = 0$ . Then $N'(C_0, T_0) = 0$ for each component of $(C', T'_C)$ . Consequently, each component is one of:

1. (i) a product compressionbody such that every scar has infinite weight;

2. (ii) a trivial ball compressionbody containing an arc and with a single scar of weight 1;

3. (iii) a trivial ball compressionbody containing a trivalent vertex and with edges of weight (2,2,k) with $k \geq 2$ . It has a single scar of weight k.

The compressionbody $(C, T_C)$ can be reconstructed by attaching possibly weighted 1-handles to the scars on (C $^{\prime}$ , T $^{\prime}$ ). Thus our result holds if $N(C, T_C) \leq 0$ .

We next have two propositions whose proofs are closely related.

The remainder of the section is devoted to the proofs of these propositions. A key bookkeeping device for a vp-compressionbody $(D, T_D)$ is its ghost arc graph. This is the graph $\Gamma$ whose vertices are the components of $\partial_- D$ and the vertices of $T_D$ . The ghost arcs of $T_D$ are the edges. For example, if $(D, T_D)$ has a single ghost arc and it joins distinct components of $\partial_- D$ , then $\Gamma$ is a single edge. The key observation is that if $\partial_+ D$ is a sphere, then $\Gamma$ is acyclic and if $\partial_+ D$ is a torus, then $\Gamma$ contains at most one cycle. If it contains a cycle, then $\partial_- D$ is the union of spheres (that is, it does not contain a torus).

Begin by assuming only that (M,T) is a nice orbifold. Let $\mathcal{H} \in \mathbb H(M,T)$ be locally thin. Recall from Corollary 4·8 that $\mathcal{H}^-$ contains an efficient system of summing spheres S. Assume, for the time being, that $S = \varnothing$ ; equivalently, that (M,T) is orbifold-irreducible. Since each component of $\mathcal{H}^-$ is c-essential in (M,T), this also implies that no $S_0 \sqsubset \mathcal{H}^-$ is a sphere with $|S_0 \cap T| \leq 3$ and $x_\omega(S_0) \lt 0$ .

By Lemma 5·2, $(C, T_C)$ is a trivial ball compressionbody. Observe that $\partial_+ C$ is a sphere with 0, 2 or 3 punctures. Let $(D, T_D) \sqsubset (M,T)\setminus \mathcal{H}$ be the other vp-compresionbody having $\partial_+ D = \partial_+ C$ . If $\partial_- D = \varnothing$ , then $M = C \cup D$ and $(D, T_D)$ is also a trivial ball compressionbody. In this case, (M,T) is either $\mathbb{S}(0)$ , $\mathbb{S}(2)$ or $\mathbb{S}(3)$ . Assume there exists $F \sqsubset \partial_- D$ .

Let $\Gamma$ be the ghost arc graph for $(D, T_D)$ . As $\partial_+ D$ is a sphere, $\Gamma$ is acyclic and the components of $\partial_- D$ are all spheres. Since (M,T) is nice, none of them are once-punctured. Since $S = \varnothing$ , F is at least thrice-punctured and has $x_\omega(F) \geq 0$ . If $\Gamma$ contains an isolated vertex, $(D, T_D)$ is a product. Since $\mathcal{H}$ is locally thin, $F = \partial_- D \subset \partial M$ . If $T_C$ contains an interior vertex v, we must have $0 \gt x_\omega(v) = x_\omega(F) \geq 0$ , a contradiction. If $T_C$ does not contain an interior vertex, then F is twice-punctured, contradicting our definition of nice 3-orbifold. Thus, we may assume that $\Gamma$ does not have an isolated vertex. Since no component of $\partial_- D$ is a twice-punctured sphere, each leaf of $\Gamma$ is incident to at least two vertical arcs, so there is at most one leaf. Since $\Gamma$ is acyclic, this is a contradiction. Consequently, (M,T) is one of the exceptional cases in the statement of Proposition 5·3.

By (i), we see that $\operatorname{net}x_\omega(\mathcal{H}) \geq x_\omega(\partial M)/2$ . Let L be the product of all the finite weights on T. Note that for any $\mathcal{J} \in \mathbb H(M,T)$ , the quantity $2L\operatorname{net}x_\omega(\mathcal{J})$ is an integer, as is $2Lx_\omega(\partial M)/2$ . By Theorem 4·7, for any $\mathcal{J} \in \mathbb H(M,T)$ , there exists a locally thin $\mathcal{H} \in \mathbb H(M,T)$ with $\mathcal{J} \to \mathcal{H}$ . By Lemma 4·6, $\operatorname{net}x_\omega(\mathcal{J}) \geq \operatorname{net}x_\omega(\mathcal{H})$ . If (M,T) is one of the exceptional cases from Proposition 5·3, then by the analysis in Case 1, $\mathcal{H}$ is connected and so $L\operatorname{net}x_\omega(\mathcal{J})$ is bounded below by a constant depending only on (M,T). If (M,T) is not one of the exceptional cases from Proposition 5·3, then we see that $2L\operatorname{net}x_\omega(\mathcal{J}) \geq 2Lx_\omega(\partial M)/2 \geq 0$ . Thus, in either case, since the invariant $2L\operatorname{net}x_\omega$ defined on $\mathbb H(M,T)$ is integer-valued and bounded below by a number depending only on it achieves its minimum on a locally thin element of $\mathbb H(M,T)$ . That element also minimizes $\operatorname{net}x_\omega$ . This concludes the analysis when $S = \varnothing$ for the proof of Proposition 5·3.

Now suppose that $S \neq \varnothing$ . Expand S to include all summing spheres in $\mathcal{H}^-$ ; continue to call it S. As we have observed previously,

\begin{align*}\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(\mathcal{H} \setminus S) - x_\omega(S).\end{align*}

Let $(M_0, T_0) \sqsubset (M,T)|_S$ and let $\mathcal{H}_0 = (\mathcal{H}\setminus S) \cap M_0$ . If $\operatorname{net}x_\omega(\mathcal{H}_0) \lt 0$ , then $(M_0, T_0)$ is one of the exceptional cases from Proposition 5·3. Since $\mathcal{H}$ is locally thin, each component of S is essential, so $(M_0, T_0) \neq \mathbb{S}(0)$ . If $(M_0, T_0) = \mathbb{S}(2)$ , then at least one of the components S $^{\prime}$ of S used to sum with $(M_0, T_0)$ must be unpunctured. Thus, if $T_0$ has weight k, we have:

\begin{align*}\operatorname{net}x_\omega(\mathcal{H}_0) - \frac{1}{2} x_\omega(S') = -\frac{2}{k} + 1 \geq 0.\end{align*}

Similarly, if $(M_0, T_0)$ is an $\mathbb{S}(3)$ , then at least one of the components $S' \sqsubset S$ used to sum with $(M_0, T_0)$ must be either unpunctured or twice punctured and with the weight of the punctures equal to the weight c of one of the edges of $T_0$ . In that case, letting a,b be the weights of the other punctures,

\begin{align*}\operatorname{net}x_\omega(\mathcal{H}_0) - \frac{1}{2}x_\omega(S') \geq 1 - \left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) + \frac{1}{2}\cdot \frac{2}{c} \geq 0.\end{align*}

Consequently, $\operatorname{net}x_\omega(\mathcal{H}) \geq x_\omega(\partial M)/2$ , even in this situation. As before, the quantity $L\operatorname{net}x_\omega$ is an integer-valued invariant on $\mathbb H(M,T)$ bounded below by a constant depending only on (M,T) and so, as before, there is a locally thin $\mathcal{H} \in \mathbb H(M,T)$ with $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . This concludes the proof of Proposition 5·3.

Henceforth, suppose that (M,T) is closed and orbifold-irreducible and not one of the exceptional cases from Proposition 5·3. By our previous remarks, this implies that $N(C, T_C) \geq 0$ for every $(C, T_C) \sqsubset (M,T)\setminus \mathcal{H}$ . The dual digraph to $\mathcal{H}$ is acyclic, so it has at least one source and one sink. The sources and sinks are exactly those $(C, T_C) \sqsubset (M,T)\setminus \mathcal{H}$ with $\partial_- C = \varnothing$ . Suppose that $(C, T_C)$ is one such. Note that $N(C, T_C) = x_\omega(\partial_+ C)$ . Let $(D, T_D) \sqsubset (M,T)\setminus \mathcal{H}$ be the other orbifold compressionbody with $\partial_+ D = \partial_+ C = H$ .

Observe that $x_\omega(H) \geq -\chi(H) + |H \cap T|/2$ . Equality holds only if every puncture on H has weight 2. Consequently, if $1/6 \gt x_\omega(H)$ , then H is a sphere with $|H \cap T| \leq 4$ . If $|H \cap T| \leq 3$ , then by our analysis above (M,T) is one of the exceptional cases from Proposition 5·3. Consider, therefore, the case that $|H \cap T| = 4$ . If at least one puncture does not have weight 2, then $x_\omega(H) \geq 1/6$ , so assume that each puncture has weight 2. If $\partial_- D = \varnothing$ , then $\mathcal{H}$ divides (M,T) into two Euclidean pillows. Suppose $\partial_- D \neq \varnothing$ and let $\Gamma$ be the ghost arc graph for $(D, T_D)$ as above. It is acyclic. Each component of $\partial_- D$ is a sphere with at least three punctures, since (M,T) is orbifold-irreducible and nice. Also $\partial_- D \subset \partial M$ since M is closed. An isolated vertex of $\Gamma$ is a sphere incident to at least three vertical arcs and a leaf is a sphere incident to at least two vertical arcs. Since H has four punctures, if $\Gamma$ has an isolated vertex, that vertex is the entirety of $\Gamma$ and it is incident to four vertical arcs. This implies $(D, T_D)$ is a product and contradicts local thinness of $\mathcal{H}$ . Thus, $\Gamma$ has two leaves, each incident to two vertical arcs. At least one of those leaves F is a component of $\partial_- D$ (the other may be a vertex of T). Since (M,T) is orbifold-irreducible, $x_\omega(F) \geq 0$ . Consequently, at least two of the arcs incident to F have weight at least 3. At least one of those is a vertical arc, contradicting the fact that each puncture of H has weight 2. Consequently, $N(C, T_C) = x_\omega(H) \geq 1/6$ .

Since the dual digraph to $\mathcal{H}$ has at least one source and one sink either (M,T) is one of the exceptional cases of Proposition 5·3, or $\mathcal{H}$ is a four punctured sphere dividing (M,T) into two Euclidean pillows, or $2\operatorname{net}x_\omega(\mathcal{H}) \geq 2\cdot (1/6)$ . This concludes the proof of Proposition 5·4.

Figure 3 shows that our bound of 1/6 is asymptotically sharp.

Fig. 3.

An example of an orbifold with underlying 3-manifold $S^3$ . The thick circles represent thick spheres and the thin circle is a thin sphere of a multiple vp-bridge surface $\mathcal{H}$ . Arbitrary gluing maps preserving the punctures pointwise can be used along the thick spheres. For $a \in \mathbb{N}^\infty_2$ we have $\operatorname{net}x_\omega(\mathcal{H}) = {1}/{6} + {1}/{a}$ and the orbifold characteristic of the thin sphere is ${1}/{6} - {1}/{a}$ . Thus, for $a \geq 6$ , $\mathcal{H}^-$ does not contain a spherical orbifold. As $a \to \infty$ , we approach 1/6.

5·2. Equivariant Heegaard Genus and the Order of the Group

Theorem 5·5. Suppose that W is closed and connected and that G does not act freely. If $S \subset W$ is an invariant system of summing spheres, then

\begin{align*}\mathfrak{g}(W;G) \geq |G|(\nu/12 - \mu/2) + |S| + 1,\end{align*}

where $\nu$ is the number of orbits of the components of $(W|_S)$ that are not $S^3$ or lens spaces and $\mu$ is the number of orbits of components of $W|_S$ that are homeomorphic to $S^3$ .

Proof. Let (M,T) be the quotient orbifold and note that no edge of T has infinite weight. Let $\overline{S}$ be the image of S and note that each component of $(M,T)|_{\overline{S}}$ is orbifold-irreducible by the definition of “system of summing spheres” and Theorem 2·5. By Theorem 2·4, $\overline{S}$ contains an efficient subset. Let $S_0 \subset S$ be the preimage of a component of $\overline{S}$ that is not in our chosen efficient subset. Since W is closed, each component of $W|_S$ that is not a component of $W|_{S_0}$ is a homeomorphic to $S^3$ . Thus, passing from S to $S\setminus S_0$ increases the right-hand side of our inequality by $|G|/2 - 1$ . Since G does not act freely, it is not the trivial group. We conclude that it is enough to prove our result when $\overline{S}$ is efficient. Furthermore, by Theorem 2·4, we may prove it when $\overline{S}$ is any efficient system of summing spheres for (M,T), not merely the given one.

Recall that $x_\omega(W;G) \geq |G|\operatorname{net}x_\omega(M,T)$ . By Theorem 4·7 and Proposition 5·3, there is a locally thin $\mathcal{H} \in \mathbb H(M,T)$ such that $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . Furthermore, $\mathcal{H}^-$ contains an efficient set of summing spheres (Corollary 4·8). Call them $\overline{S}$ . Let S be the preimage of $\overline{S}$ in W. By Theorem 4·9, we have

\begin{align*}\operatorname{net}x_\omega(M,T) = \operatorname{net}x_\omega((M,T)|_{\overline{S}}) - x_\omega(\overline{S}).\end{align*}

Suppose some $(M_i, T_i)\sqsubset (M,T)|_{\overline{S}}$ is an $\mathbb{S}(0)$ . Since $\overline{S}$ is efficient, the matching scar to every scar in $(M_i, T_i)$ is also in $(M_i, T_i)$ . It follows that $(M,T)|_{\overline{S}} = (M_i, T_i)$ . In which case, $T = \varnothing$ , and G acts freely, contrary to hypothesis. Henceforth, we may assume no $(M_i, T_i) \sqsubset (M,T)|_S$ is an $\mathbb{S}(0)$ . Suppose that $(M_i, T_i)$ is an $\mathbb{S}(2)$ . Then $x_\omega(M_i, T_i) \geq -1$ and each component of the pre-image of $(M_i, T_i)$ in $W|_S$ , is a copy of $S^3$ . Similarly, if $(M_i, T_i)$ is an $\mathbb{S}(3)$ , then $x_\omega(M_i, T_i) \geq -1/2$ and each component of the pre-image of $(M_i, T_i)$ in $W|_S$ is also a copy of $S^3$ . Conversely, by the classification of finite groups of diffeomorphisms of $S^3$ , if $W_i \sqsubset W|_S$ is a copy of $S^3$ , then its image in $(M,T)|_S$ is an $\mathbb{S}(k)$ for some $k \in \{0,2,3\}$ . Consequently, $\mu$ both the number of orbits of $S^3$ components of $W|_S$ and the number of $(M_i, T_i)$ that are $\mathbb{S}(2)$ or $\mathbb{S}(3)$ .

If $(M_i, T_i)$ is a ( $S^3$ , Hopf link), (lens space, core loop), (lens space, Hopf link) or a Euclidean double pillow, then $x_\omega(M_i, T_i) = 0$ . If a component $W_i$ of $W|_S$ covers $(M_i, T_i)$ , then $W_i$ admits an invariant Heegaard torus, but no Heegaard sphere. Indeed, any $W_i$ that is a lens space has $x_\omega(W_i) \geq 0$ . Amalgamating a generalised Heegaard splitting of a 3-manifold produces a Heegaard surface and does not change $x_\omega$ . Thus, $\operatorname{net}x_\omega(W_i; G) \geq 0$ , whenever $W_i$ is a lens space.

If $(M_i, T_i)$ is neither an $\mathbb{S}(k)$ for $k \in \{0,2,3\}$ nor a ( $S^3$ , Hopf link), (lens space, core loop), (lens space, Hopf link), or a Euclidean double pillow, then by Proposition 5·4, $\operatorname{net}x_\omega(M_i, T_i) \geq 1/6$ .

Consequently,

\begin{align*}x_\omega(W;G) \geq |G|\operatorname{net}x_\omega(M,T) \geq |G|(\nu/6 - \mu - x_\omega(\overline{S})) = |G|(\nu/6 - \mu) - x_\omega(S).\end{align*}

Converting to genus, we have

\begin{align*}\mathfrak{g}(W;G) \geq |G|(\nu/12 - \mu/2) + |S| + 1 \end{align*}

Example 5·6. Consider a lens space M containing an unknot T such that there is a 2-sphere $\overline{S}$ in M bounding a 3-ball in M containing T. Give T weight 2 and let W be the 3-manifold such that there is an orientation preserving involution of W whose quotient produces the orbifold (M,T). Observe that W is homeomorphic to the connected sum of M with itself. This is depicted in Figure 4.

Fig. 4.

The covering of the orbifold (M,T) by $(W, \varnothing)$ in Example 5·6. The manifold M is a lens space and $W = M \times M$ . The surface S is the union of two disjoint spheres; it is a double cover of the unpunctured sphere $\overline{S}$ . The knot T is an unknot contained in 3-ball bounded by the sphere S. The horizontal lines represent bridge surfaces for (M,T) and $(W, \varnothing)$ , with $\overline{H}$ being a twice punctured torus and H being a genus 2 surface.

Let S be the preimage of $\overline{S}$ in W. Since $\overline{S} \cap T = \varnothing$ , the surface S is the union of two spheres and $W|_S$ is the disjoint union of two lens spaces (each homeomorphic to M) and a copy of $S^3$ . Thus, $|G|\nu/12 - |G|\mu/2 + |S| + 1 = 2$ . Notice that (M,T) has an orbifold Heegaard surface $\overline{H}$ with $x_\omega(\overline{H}) = 1$ . The surface $\overline{H}$ is a torus intersecting T twice. The preimage of $\overline{H}$ in W is an invariant Heegaard surface $H \subset W$ of genus 2. Thus, in this case, our inequality is sharp.

Example 5·7. Let $M_1 = S^3$ and let $T_1$ be the spatial graph constructed as follows, and depicted in Figure 5. Choose a 2-bridge knot K and and attach both an upper and lower tunnel to obtain $T_1$ . The graph $T_1$ has four edges that intersect a bridge sphere $H_1$ and two edges (the upper and lower tunnels) that are disjoint from $H_1$ . Give one of the edges intersecting $H_1$ a weight of 3 and give all the other edges weight 2. The subgraph $T_0$ of $T_1$ with edges of weight 2 is a trivial $\theta$ -curve in $S^3$ . The double-branched cover of $S^3$ over a cycle in $T_0$ is again $S^3$ and the third edge of $T_0$ lifts to an unknot. Taking another double-branched cover over that unknot again produces $S^3$ . Thus, $S^3$ is a 4-fold orbifold cover over $(M_1, T_0)$ . The edge of weight 3 in T lifts to a knot $\kappa$ in the cover. The sphere $H_1$ lifts to a bridge sphere for $(S^3, \kappa)$ such that $|H_1 \cap \kappa| = 4$ . In particular, $\kappa$ is either the unknot or a 2-bridge knot. Since K is knotted, $\kappa$ is a 2-bridge knot.

Fig. 5.

The singular set $T_1$ and the bridge sphere $H_1$ for the orbifold $(M_1, T_1)$ in Example 5·7

Let $W_1$ be the 3-fold branched cover of $S^3$ over $\kappa$ . Thus, $(M_1, T_1)$ is the quotient of $W_1$ by a group of diffeomorphisms of order 12 (isomorphic to the product of two groups of order 2 and one of order 3). As long as $\kappa$ is not a torus knot or the figure-eight knot, $W_1$ is hyperbolic [ Reference Daryl Cooper2 , chapter 1]. Let (M,T) be the distant sum of $(M_1, T_1)$ with $(M_2, \varnothing)$ , a lens space having empty singular set. Let $\overline{S}$ be the summing sphere. The action of G on $W_1$ extends to an action of (a group isomorphic to) G on the connected sum W of $W_1$ with 12 copies of $M_2$ . The sphere $\overline{S}$ lifts to 12 copies of $S^2$ . The right hand-side of the inequality in Theorem 5·5 is then 14. The orbifold (M, T) admits an orbifold Heegaard surface with orbifold characteristic 13/6. (It is a torus having three punctures of weight 2 and one of weight 3.) This lifts to an invariant Heegaard surface of W having genus 14. So our lower bound is sharp in this case as well.

5·3. Comparatively Small Factors

A 3-orbifold (M,T) is comparatively small if each c-essential surface $F \subset (M, T)$ has $x_\omega(F) \gt x_\omega(M,T)$ .

Proof. By definition, $x_\omega(M,T) \geq \operatorname{net}x_\omega(M,T)$ . We now show $x_\omega(M,T) \leq \operatorname{net}x_\omega(M,T)$ . Let $\mathcal{H} \in \mathbb H(M,T)$ be locally thin, with $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . We claim $\mathcal{H}^- = \varnothing$ .

Suppose, for a contradiction, that $\mathcal{H}^- \neq \varnothing$ . The dual digraph to $\mathcal{H}$ has at least one source and at least one sink. Each such source and sink corresponds to a $(C, T_C) \sqsubset (M, T) \setminus \mathcal{H}$ with $\partial_- C \subset \partial M$ . Let $(C, T_C)$ be one such and let $(D, T_D) \sqsubset (M, T) \setminus \mathcal{H}$ be distinct from $(C, T_C)$ but have $\partial_+ D =\partial_+ C$ . Then $\mathcal{H}^- \cap \partial_- D \neq \varnothing$ ; let F be a component. Since $\mathcal{H}$ is locally thin, F is c-essential in (M, T) (Theorem 4·7). Since (M, T) is comparatively small, $x_\omega(F) \gt x_\omega(M, T)$ . By Lemma 5·2,

\begin{align*}N(C, T_C) + x_\omega(\partial_- C) = x_\omega(\partial_+ C) = x_\omega(\partial_+ D) \geq x_\omega(F) \gt x_\omega(M, T) \geq \operatorname{net}x_\omega(M, T).\end{align*}

As there are at least two such $(C, T_C)$ , by (1) and the niceness of (M, T), we conclude $\operatorname{net}x_\omega(M,T) \gt \operatorname{net}x_\omega(M, T)$ , a contradiction. Thus, $\mathcal{H}^-= \varnothing$ . Consequently, $\mathcal{H}$ is an orbifold Heegaard surface for (M,T) and so $\operatorname{net}x_\omega(\mathcal{H}) \geq x_\omega(M,T)$ .

Theorem 5·9. Suppose that (M,T) is a nice 3-orbifold. Assume also that for some (hence, any) efficient system of summing spheres S each component of $(M,T)|_S$ is comparatively small. Then

\begin{align*}x_\omega(M,T) \geq \operatorname{net}x_\omega(M,T) \geq x_\omega((M,T)|_S)- x_\omega(S).\end{align*}

Proof. If (M,T) is orbifold-irreducible, the result is vacuously true. Suppose that (M,T) is orbifold-reducible. By Theorem 2·4, each component of $(M,T)|_S$ is comparatively small for any efficient system of summing spheres S. By definition, $x_\omega(M,T) \geq \operatorname{net}x_\omega(M,T)$ . By Theorem 4·9, $\operatorname{net}x_\omega(M,T) \geq \operatorname{net}x_\omega((M,T)|_S) - x_\omega(S)$ for an efficent system of summing spheres S. By Lemma 5·8, $\operatorname{net}x_\omega((M,T)|_S) = x_\omega((M,T)|_S)$ , and the result follows.

Theorem 5·10. Suppose that when $S \subset W$ is an invariant system of summing spheres, then every component of $W|_S$ is equivariantly comparatively small. Then:

\begin{align*}\mathfrak{g}(W;G) \geq \mathfrak{g}(W|_S;G).\end{align*}

Proof. Without loss of generality, we may assume that W is reducible. Let (M,T) be the quotient orbifold. Observe that it is nice and orbifold-reducible. Let $\overline{S}$ be an efficient system of summing spheres for (M,T). We claim that each component of $(M,T)|_S$ is comparatively small. To see this, suppose that $(M_i, T_i) \sqsubset (M,T)|_S$ and that $\overline{F} \subset (M_i, T_i)$ is a c-essential surface. Let $W_i$ be a component of the preimage of $(M_i, T_i)$ . Since $(M_i, T_i)$ is orbifold-irreducible, $\overline{F}$ is not a sphere. Thus, no component of the preimage of $\overline{F}$ to W is a sphere. Let F be the preimage of $\overline{F}$ in $W|_S$ , with $F_i = F \cap W_i$ . Let $G_i \subset G$ be the stabiliser of $W_i$ . We have $x_\omega(\overline{F}) =x_\omega(F_i)/|G_i|$ .

By the equivariant loop theorem [ Reference William and Yau17 ], F is incompressible. Suppose that some $F' \sqsubset F$ is $\partial$ -parallel in W. Let $W' \sqsubset W \setminus F'$ be homeomorphic to $F' \times I$ . We may assume that F $^{\prime}$ was chosen so that the interior of W $^{\prime}$ is disjoint from F. The image of W $^{\prime}$ in (M,T) is the quotient of a trivial product compressionbody (namely W $^{\prime}$ ) by a finite group of diffeomorphisms (the stabiliser of F $^{\prime}$ ). By Lemma 2·7, it a product trivial compressionbody. Thus, $\overline{F}$ bounds a trivial product compressionbody with a component of $\partial W$ and so $\overline{F}$ is not c-essential, a contradiction. Thus, F is essential in $W|_S$ . Since $W_i$ is equivariantly comparatively small, $x_\omega(F_i) \gt x_\omega(W_i)$ . Thus,

\begin{align*}x_\omega(\overline{F}) = x_\omega(F_i)/|G_i| \gt x_\omega(W_i)/|G_i| = x_\omega(M_i, T_i).\end{align*}

Thus, each component of $(M,T)|_{\overline{S}}$ is comparatively small. Our result follows from Theorem 5·9 after multiplying by $|G|$ .

Fig. 6.

Creating a removable arc.

6·1. Examples of super additivity

Theorem 6·4. For $k \geq 2$ , there exist reducible W having a finite cyclic group of diffeomorphisms G of order k, and an invariant essential sphere S dividing W into two irreducible manifolds, such that

\begin{align*}\mathfrak{g}(W;G) = \mathfrak{g}(W|_S; G) + k - 1.\end{align*}

Furthermore, $\mathfrak{g}(W;G)$ can be arbitrarily high when G is a cyclic group of fixed order.

Associated to each Heegaard surface H of genus at least 2 of a compact 3–manifold X, is a nonnegative integer invariant d(H), called Hempel distance [ Reference Hempel12 ]. When $d(H) = 0$ , the Heegaard surface H is reducible. When $d(H) = 1$ , the Heegaard surface H can be untelescoped to a generalised Heegaard surface with multiple components. Essential surfaces or strongly irreducible Heegaard surfaces can often be used to provide upper bounds on Hempel distance. Furthermore, if every thick component of a multiple Heegaard surface has high enough Hempel distance, the generalised Heegaard surface is locally thin. We apply this philosophy in our context.

By [ Reference Minsky, Moriah and Schleimer18 ], for each $t \in \mathbb{N} $ and $N \in \mathbb{N} $ , there exists a knot $K \subset S^3$ such that $S^3 \setminus K$ admits a Heegaard surface of genus $t + 1$ and Hempel distance $d(H) \geq N$ . Fix $t, N, k \in \mathbb{N} $ . For $i = 1,2$ , choose knots $K_i$ in $S^3$ such that each has a Heegaard surface $H_i$ of its exterior of genus $t + 1$ (and $x_\omega(H_i) = 2t$ ) and Hempel distance at least $\zeta = 2(4t + N+ 2(1-1/k) + 2)/(1 - 1/k) + 3$ . We use the following, drawn from work of Scharlemann, Scharlemann–Tomova and Tomova. The statement for F is due to Scharlemann [ Reference Scharlemann30 , theorem 3·1] and is a generalisation of [ Reference Hartshorn8 ]. The case when J is disjoint from $K_i$ is the main result of [ Reference Scharlemann and Tomova32 ] and when J intersects $K_i$ , it is the main result of [ Reference Tomova41 ]. We rephrase their results, using our terminology.

Let $K = K_1 \# K_2$ . Consider $(S^3, K)$ as an orbifold where K is given weight $k \in \mathbb{N}^\infty_2$ . Note that $(S^3, K)$ has a multiple orbifold Heegaard surface $\mathcal{H}$ with $\mathcal{H}^+ = H_1 \cup H_2$ and $S = \mathcal{H}^-$ the twice-punctured summing sphere realising K as a connected sum of $K_1$ and $K_2$ . (We also need to give $\mathcal{H}$ one of the two orientations making it an oriented multiple vp-bridge surface.) For the record, we have $\operatorname{net}x_\omega(\mathcal{H}) = 4t + 2/k$ . By the definitions of Hempel distance and the partial order $\to$ (both of which we have omitted), $\mathcal{H}$ is locally thin.

Proof. Let $\mathcal{J} \in \mathbb H(S^3, K)$ be locally thin and have $\operatorname{net}x_\omega(\mathcal{J}) \leq \operatorname{net}x_\omega(S^3, K) + N + 2(1 - 1/k)$ . Since $(S^3, K)$ is orbifold-reducible, $\mathcal{J}^-$ contains an efficient system of summing spheres for $(S^3, K)$ . By Theorem 6·5 (or, indeed, by [ Reference Hempel12 ]), both $K_1$ and $K_2$ are prime knots. Consequently, there is a unique essential summing sphere for $(S^3, K)$ , up to isotopy. Isotope $\mathcal{J}$ so that $S \sqsubset \mathcal{J}^-$ . Suppose that there exists $F \sqsubset \mathcal{J}^- \setminus S$ . Choose F so that it bounds a 3-submanifold $X \subset S^3$ with interior disjoint from $\mathcal{J}^-$ (i.e. F is outermost). Let $J = \mathcal{J}^+ \cap X$ and let $(C, T_C) \sqsubset (S^3, K) \setminus \mathcal{J}$ be the orbifold compressionbody with $J = \partial_+ C$ and $\partial_- C = \varnothing$ . Without loss of generality, we may assume that F is on the same side of S as $K_1$ . If F is essential in $(M,T)\setminus S$ , then by Theorem 6·5,

\begin{align*}\begin{array}{rcl}2(4t+N + 2(1-1/k) + 2) &=& (1 - 1/k)(\zeta - 3) \\ \\[-8pt] &<& (1 - 1/k)(-\chi(F) + |F \cap K_1|) \\ \\[-8pt]&\leq& -\chi(F) + (1 - 1/k)|F \cap K_1| \\ \\[-8pt]&=& x_\omega(F).\end{array}\end{align*}

By Lemma 5·2, $N(C, T_C) = x_\omega(H) \geq x_\omega(F)$ and so, by (1),

\begin{align*}\begin{array}{rcl}\operatorname{net}x_\omega(S^3, K) + N + 2(1 - 1/k) &\geq& \operatorname{net}x_\omega(\mathcal{J}) \\ \\[-8pt]&>& 4t + N + 2(1 - 1/k) + 2 \\ \\[-8pt]&=& \operatorname{net}x_\omega(\mathcal{H}) + N + 2(1 - 1/k) - 2/k + 2\\ \\[-8pt]&\geq& \operatorname{net}x_\omega(S^3, K) + N + 4(1- 1/k). \\\end{array}\end{align*}

This is a contradiction. Thus, F must be inessential in $(S^3, K)\setminus S$ . Since F is c-essential in $(S^3, K)$ , it must be isotopic to S. Perform the isotopy to make F coincide with S. After the isotopy, X is the side of $S=F$ containing $K_1$ . By Theorem 6·5, either $-\chi(J) + |J \cap T| \gt \zeta - 3$ or $J \to H_i$ . In the former case, we have:

\begin{align*}N(C, T_C) &= x_\omega(J) = -\chi(J) + (1 - 1/k)|J \cap T| \gt (1 - 1/k)(\zeta -3)\\[3pt] &= 2(4t + N+ 2(1-1/k) + 2).\end{align*}

The same arithmetic as before establishes a contradiction. Thus, $J \to H_i$ . However, $\mathcal{J}$ is locally thin, so in fact, J is isotopic to $H_i$ (ignoring orientations).

We have shown that each outermost $F \sqsubset \mathcal{J}^-$ is isotopic to S. Furthermore, if $\mathcal{J}^-$ is connected, then $\mathcal{J}$ is isotopic to $\mathcal{H}$ , ignoring orientations. Suppose that $\mathcal{J}^-$ is disconnected. Since every surface in $S^3$ separates, there are at least two outermost $F_1, F_2 \sqsubset \mathcal{J}^-$ . They cobound $(Y, T_Y) \subset (S^3, K)$ that is a product compressionbody homeomorphic to $(S^2 \times I, \{p_1, p_2\} \times I)$ , where $p_1, p_2 \in S^2$ . Since each component of $\mathcal{J}^-$ is c-essential, each component of $\mathcal{J}^-$ must be parallel to each of $F_1$ and $F_2$ in $(Y, T_Y)$ . Suppose that $F_i, F_{i+1} \sqsubset \mathcal{J}^-$ cobound a submanifold $(Y', T'_Y) \sqsubset (Y, T_Y)\setminus \mathcal{J}^-$ . Note that $(Y', T'_Y)$ is homeomorphic to $(Y, T_Y)$ . Let $J' = \mathcal{J}^+ \cap Y'$ . Since $\mathcal{J}$ is locally thin, J $^{\prime}$ must be a twice-punctured sphere bounding a trivial ball compressionbody B(J $^{\prime}$ ) to one side. We conclude that

\begin{align*}\operatorname{net}x_\omega(\mathcal{J}) &= x_\omega(H_1) + x_\omega(H_2) + \frac{-2}{k}(|\mathcal{J}^-|-1) - \frac{-2}{k}|\mathcal{J}^-| = x_\omega(H_1) + x_\omega(H_2) + \frac{2}{k}\\ &= \operatorname{net}x_\omega(\mathcal{H}).\end{align*}

Thus, $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(S^3, K)$ . Furthermore, after deleting all the components of $\mathcal{J} \cap Y$ except $F_1$ , we obtain a multiple vp-bridge surface isotopic to $\mathcal{H}$ , ignoring orientations.

We will also need the following:

We can now calculate the Heegaard characteristic of $(S^3, K)$ .

Proof. There are four orbifold compressionbody components of $(S^3, K)\setminus \mathcal{H}$ . The two not adjacent to $S = \mathcal{H}^-$ are disjoint from K and each of the other two contain a single ghost arc, both of whose endpoints are on S. Choose one of them and use it to create a removable arc. Call the new multiple orbifold Heegaard surface $\mathcal{H}'$ with two thick surfaces and thin surface S. Note that the thick surfaces are amalgable. Amalgamate them, converting $\mathcal{H}'$ into a connected $H' \in \mathbb H(M,T)$ with

\begin{align*}\operatorname{net}x_\omega(H') = \operatorname{net}x_\omega(\mathcal{H}') = \operatorname{net}x_\omega(M,T) + 2(1 - 1/k) = 4t + 2.\end{align*}

Let J be an orbifold Heegaard surface for $(S^3, K)$ such that $x_\omega(J) = x_\omega(S^3, K)$ . There is a locally thin $\mathcal{J} \in \mathbb H(S^3, K)$ such that $J \to \mathcal{J}$ . By Lemma 6·6,

\begin{align*}\operatorname{net}x_\omega(M,T) + 2(1-1/k)= x_\omega(H') \geq x_\omega(J) \geq \operatorname{net}x_\omega(\mathcal{J}) \geq \operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T).\end{align*}

By Lemma 6·6, $\operatorname{net}x_\omega(\mathcal{J}) = \operatorname{net}x_\omega(\mathcal{H})$ and after deleting pairs of twice-punctured spheres from $\mathcal{J}^-$ and $\mathcal{J}^+$ , $\mathcal{J}$ is isotopic to $\mathcal{H}$ (ignoring orientations). As in Lemma 4·5, the thinning moves that create $\mathcal{J}$ from J potentially consist of four types of moves. Since $S^3$ is closed and K is a knot, of the moves listed in Type (I), we never need to perform a $\partial$ -destabilisation, meridional $\partial$ -destabilisation or meridional ghost $\partial$ -destabilisation. Performing a ghost $\partial$ -destabilisation, involves compression along a separating compressing disc and the discarding of a torus boundary component. (That torus is isotopic to the frontier of a regular neighbourhood of K.) Such a move decreases negative orbifold Euler characteristic of a thick surface by $2 \geq 2(1 - 1/k)$ . Destabilisation also decreases the negative orbifold Euler characteristic of a thick surface by 2. Meridional destabilisation decreases it by $2/k$ . The moves in Type (II) decrease it by $2(1-1/k)$ . Consolidation and untelescoping leave $\operatorname{net}x_\omega$ unchanged.

We see, therefore, that in the thinning sequence producing $\mathcal{J}$ from J, there can be at most one move that is a ghost $\partial$ -destabilisation, destabilisation, unperturbation, or undoing a removable arc. All other moves are either meridional destabilisation, untelescoping, or consolidation. Ghost $\partial$ -stabilisation, destabilisation, meridional destabilisation, untelescoping, and consolidation do not decrease $\operatorname{net}\iota$ . Unperturbing and undoing a removable arc decrease $\operatorname{net}\iota$ by 2.