INVARIANT SEIFERT SURFACES FOR STRONGLY INVERTIBLE
KNOTSMIKAMI HIRASAWA, RYOTA HIURA, AND MAKOTO SAKUMA
Abstract. We study invariant Seifert surfaces for strongly invertible knots, andprove that the gap between the equivariant genus (the minimum of the genera
of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genusof the underlying knot can be arbitrary large. This forms a sharp contrast with
Edmonds’ theorem that every periodic knot admits an invariant minimal genus
Seifert surface. We also prove variants of Edmonds’ theorem, which are useful instudying invariant Seifert surfaces for strongly invertible knots.

Dedicated to Professor Norbert A’Campo on his 80th birthday
1. Introduction
A smooth knot K in S 3 is said to be (cyclically) periodic with period n if there
is a periodic diﬀeomorphism f of S 3 of period n which leaves K invariant andﬁxes a simple loop in the knot complement S 3 \ K. Since the seminal work by
Trotter [46] and Murasugi [37], periodic knots have been studied extensively. In
particular, Edmonds and Livingston [20] proved that every periodic knot admits aninvariant incompressible Seifert surface, and this was enhanced by Edmonds [19] tothe existence of an invariant minimal genus Seifert surface. (He applied this result toprove Fox’s conjecture that a given nontrivial knot has only ﬁnitely many periods.)
It is natural to ask if the same result holds for strongly invertible knots. Recall
that a smooth knot K in S 3 is said to be strongly invertible if there is a smooth
involution h of S 3 which leaves K invariant and ﬁxes a simple loop intersecting
K in two points. The involution h is called a strong inversion of K. As in [42],we use the term strongly invertible knot to mean a pair (K, h) of a knot K and astrong inversion h of K, and regard two strongly invertible knots (K, h) and (K ′ , h′ )equivalent if there is an orientation-preserving diﬀeomorphism φ of S 3 mapping K
to K ′ such that h′ = φhφ−1 .
Note that if S is an h-invariant Seifert surface for a strongly invertible knot
(K, h) then Fix(h) ∩ S is equal to one of the two subarcs of Fix(h) ∼= S 1 bounded2020 Mathematics Subject Classiﬁcation. Primary 57K10, Secondary 57M60.
The ﬁrst author is supported by JSPS KAKENHI JP18K03296. The third author is supported
by JSPS KAKENHI JP20K03614 and by Osaka Central Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).
1

by Fix(h) ∩ K ∼= S 0 . So the problem of whether K admits an invariant minimal
genus Seifert surface depends on the choice of the subarc of Fix(h), in addition
to the choice of the strong inversion h. By a marked strongly invertible knot, wemean a triple (K, h, δ) where (K, h) is a strongly invertible knot and δ is a subarc
of Fix(h) bounded by Fix(h) ∩ K. Two marked strongly invertible knots (K, h, δ)and (K ′ , h′ , δ ′ ) are regarded to be equivalent if there is an orientation-preserving
diﬀeomorphism φ of S 3 mapping K to K ′ such that h′ = φhφ−1 and δ ′ = φ(δ).
Deﬁnition 1.1. By an invariant Seifert surface (or a Seifert surface in brief) fora marked strongly invertible knot (K, h, δ), we mean a Seifert surface S for K suchthat h(S) = S and Fix(h) ∩ S = δ. The equivariant genus (or the genus in brief)g(K, h, δ) of (K, h, δ) is deﬁned to be the minimum of the genera of Seifert surfacesfor (K, h, δ). A Seifert surface for (K, h, δ) is said to be of minimal genus if its genus
is equal to g(K, h, δ).
Every marked strongly invertible knot admits an invariant Seifert surface (Proposition 3.1), and so its equivariant genus is well-deﬁned. However, in general, theequivariant genus is bigger than the (usual) genus. In fact, the following theorem
is proved in the second author’s master thesis [26] supervised by the third author
with support by the ﬁrst author.
Theorem 1.2. For any integer n ≥ 0, there exits a marked strongly invertible knot
(K, h, δ) such that g(K, h, δ) − g(K) = n.
This theorem follows from a formula of the equivariant genera for certain marked
strongly invertible knots that arise from 2-bridge knots. See Examples 3.4 and 4.3
for special simple cases. However, there remained various marked strongly invertible
2-bridge knots whose equivariant genera were undetermined.
This paper and its sequel [24] are motivated by the desire to determine the equivariant genera of all marked strongly invertible 2-bridge knots.
In this paper, we prove the following two variants of Edmonds’ theorem on periodic knots, which are useful in studying invariant Seifert surfaces for general strongly
invertible knots: these theorems are used in [24] to give a uniﬁed determination of
the equivariant genera of all marked strongly invertible 2-bridge knots.
Theorem 1.3. Let (K, h) be a strongly invertible knot. Then there is a minimal
genus Seifert surface F for K such that F and h(F ) have disjoint interiors.
Theorem 1.4. Let (K, h, δ) be a marked strongly invertible knot, and let F be aminimal genus Seifert surfaces for K such that F and h(F ) have disjoint interiors. Then there is a minimal genus Seifert surface S for (K, h, δ) whose interior is
disjoint from the interiors of F and h(F ).
For ﬁbered knots, we have the following stronger conclusion.
2

Proposition 1.5. Let (K, h, δ) be a marked strongly invertible knot such that K isa ﬁbered knot. Then it admits an h-invariant minimal genus Seifert surface, and sog(K, h, δ) = g(K).
Motivated by isolated singularities of complex hypersurfaces, A’Campo [1] formulated a way to construct ﬁbered links in the 3-sphere from divides, i.e., immersionsof copies of 1-manifolds in a disk. He proved that the link obtained from a connected
divide is strongly invertible and ﬁbered. In [1, Section 3], we can ﬁnd a beautiful
description of a pair of invariant ﬁber surfaces for the link. This nicely illustrates
Proposition 1.5. (See [23] for visualization of these links and their ﬁber surfaces.)
Furthermore, Couture [15] introduced the more general notion of ordered Morsesigned divides, and proved that every strongly invertible link is isotopic to the linkof an ordered Morse signed divide. As A’Campo suggested to the authors, it would
be interesting to study invariant Seifert surfaces from this view point.
Theorem 1.3 is also motivated by our interest in the Kakimizu complex M S(K) of
the knot K and in the action of the symmetry group Sym(S 3 , K) = π0 Diﬀ(S 3 , K)
on M S(K). The complex was introduced by Kakimizu [27] as the ﬂag simplicial
complex whose vertices correspond to the (isotopy classes of) minimal genus Seifert
surfaces for K and edges to pairs of such surfaces with disjoint interiors. The
following corollary of Theorem 1.3 may be regarded as a reﬁnement of a special caseof a theorem proved by Przytycki and Schultens [40, Theorem 1.2].
Corollary 1.6. Let (K, h) be a strongly invertible knot, and let h∗ be the automorphism of M S(K) induced from of the strong inversion h. Then, h∗ either ﬁxes avertex of M S(K) or interchanges a pair of adjacent vertices of M S(K) (and so ﬁxes
the mid point of an edge).
For a 2-bridge knot K, the structure of M S(K) is described by [44, Theorem 3.3];
in particular, the underlying space |M S(K)| is identiﬁed with a linear quotient of acube. The actions of strong inversions on M S(K) will be described in [24].
This paper is organized as follows. In Section 2, we recall basic facts aboutstrongly invertible knots. In Section 3, we give two constructions of invariant Seifert
surfaces and prove a basic proposition (Proposition 3.2) concerning the quotientof an invariant Seifert surface by the strong inversion. In Section 4, we give an
argument for determining the equivariant genera (Proposition 4.1 and Corollary4.2), and prove Theorem 1.2 by using that argument (Example 4.3). In Section 5,
we prove our main theorems, Theorems 1.3 and 1.4. In the ﬁnal section, Section 6,
we brieﬂy review old and new studies of equivariant 4-genera of symmetric knots.
Acknowledgement. M.H. and M.S learned the theory of divides through A’Campo’s
lecture in Osaka in 1999, and R.H. learned it through the lecture by A’Campo’s former student Masaharu Ishikawa in Hiroshima in 2015. The theory of divides anddiscussions with A’Campo have been sources of inspiration for the authors. The
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authors would like to thank Professor Norbert A’Campo for invaluable discussions
and encouragements.
2. Basic facts concerning strongly invertible knots
Recall that a knot K in S 3 is said to be invertible if there is an orientationpreserving diﬀeomorphism h that maps K to itself reversing orientation. The existence of non-invertible knots was proved by Trotter [47] using 2-dimensional hyperbolic geometry. (His proof is based on the fact that the pretzel knots admit
the structure of Seifert ﬁberd orbifolds, where the base orbifolds are generically
hyperbolic.) If h can be chosen to be an involution then K is said to be strongly
invertible, and h is called a strong inversion. Though strong invertiblity of course
implies invertiblity, the converse does not hold, as shown by Whitten [50]. However, for hyperbolic knots, the converse also holds by the Mostow-Prasad rigidity
theorem. Moreover, a suﬃcient condition for invertible knots to be strongly invertible was given by Boileau [6]. The ﬁniteness theorem of symmetries of 3-manifolds
proved by Kojima [31] by using the orbifold theorem [7, 8, 14, 18] implies that any
knot admits only ﬁnitely many strong inversions up to equivalence. Here two strong
inversions h and h′ of K are regarded to be equivalent if there is an orientationpreserving diﬀeomorphism φ of S 3 such that h′ = φhφ−1 . However, the number ofstrong inversions up to equivalence for a satellite knot can be arbitrary large [43,Lemma 5.4]. For torus knots and hyperbolic knots, the number is at most 2, as
described in Proposition 2.1 below. Recall that a knot K in S 3 is said to have cyclic
period 2 or free period 2, respectively, if there is an orientation-preserving smooth
involution f of S 3 that maps K to itself preserving orientation, such that Fix(f ) isS 1 or ∅.
Proposition 2.1. (1) The trivial knot admits a unique strong inversion up to equivalence.
(2) A nontrivial torus knot admits a unique strong inversion up to equivalence.
(3) An invertible hyperbolic knot admits exactly two or one strong inversions up
to equivalence according to whether it has (cyclic or free) period 2 or not.
The ﬁrst assertion is due to Marumoto [33, Proposition 2], and the remaining
assertions are proved in [42, Proposition 3.1] (cf. [2, Section 4]) by using the resultof Meeks-Scott [34] on ﬁnite group actions on Seifert ﬁbered spaces and the orbifold
theorem. Another key ingredient of the proof of the third assertion is the following
consequence of Riley’s observation [41, p.124] based on the positive solution of theSmith conjecture [36]. For a hyperbolic invertible knot K, the orientation-preserving
isometry group of the hyperbolic manifold S 3 \ K is the dihedral group D2n of order
2n for some n ≥ 1, i.e.,
Isom+ (S 3 \ K) ∼= ⟨f, h | f n = 1, h2 = 1, hf h−1 = f −1 ⟩.
4Here h extends to a strong inversion of K and f extends to a periodic map of
S 3 of period n which maps K to itself preserving orientation. If n is odd, then K
does not have cyclic nor free period 2, and any strong inversion of K is equivalent
to that obtained from h. If n = 2m is even, then f m extends to an involution of S 3
which gives cyclic or free period 2 of K, and any strong inversion of K is equivalent
to that obtained from exactly one of h and f h.
By [30, Proposition 1.2], the above Proposition 2.1(3) is reﬁned to the following
proposition concerning marked strongly invertible knots.
Proposition 2.2. Let K be an invertible hyperbolic knot.
(1) Suppose K does not have cyclic nor free period 2. Then K admits a uniquestrong inversion up to equivalence, and the two marked strongly invertible knots
associated with K are inequivalent. Thus there are precisely two marked strongly
invertible knots associated with K.
(2) Suppose K has cyclic period 2. Then K admits precisely two strong inversions
up to equivalence, and for each strong inversion, the associated two marked strongly
invertible knots are inequivalent. Thus there are precisely four marked strongly invertible knots associated with K.
(3) Suppose K has free period 2. Then K admits precisely two strong inversions
up to equivalence, and for each strong inversion, the associated two marked strongly
invertible knots are equivalent. Thus there are precisely two marked strongly invertible knots associated with K.
Since every 2-bridge knot admits cyclic period 2, we obtain the following corollary.
Corollary 2.3. Every hyperbolic 2-bridge knot has precisely four associated marked
strongly invertible knots up to equivalence.
The four marked strongly invertible knots associated with a hyperbolic 2-bridge
knot are (implicitly) presented in [42, Proposition 3.6] and [2, Section 4].
Remark 2.4. By Proposition 2.1(1) and (2), we can easily see that the trivial knot
has a unique associated marked strongly invertible knot and that every torus knot
has precisely two associated marked strongly invertible knots.
We note that Barbensi, Buck, Harrington and Lackenby [4] shed new light on
the strongly invertible knots in relation with the knotoids introduced by Turaev[48]. They prove that there is a 1 − 1 correspondence between unoriented knotoids,
up to “rotation”, and strongly invertible knots, up to “inversion” [4, Theorem 1.1].
Proposition 2.2 is a variant of their result [4, Theorem 1.3] concerning knotoids.
3. Invertible diagrams and invariant Seifert surfaces
In this section, we describe two proofs of the following basic proposition whichshows the existence of an invariant Seifert surface for every marked strongly invertible knot. One is due to Boyle and Issa [9], and the other is due to Hiura [26].
5Proposition 3.1. Every marked strongly invertible knot admits an invariant Seifert
surface. Namely, for every marked strongly invertible knot (K, h, δ), there is an hinvariant Seifert surface S for K such that Fix(h) ∩ S = δ.
Following [9, Deﬁnition 3.3] we say that a symmetric diagram representing astrongly invertible knot (K, h) is(1) intravergent if h acts as half-rotation around an axis perpendicular to theplane of the diagram (see Figure 3.1(a)), and(2) transvergent if h acts as half-rotation around an axis contained within theplane of the diagram (see Figure 3.5(a)).
3.1. Construction from an intravergent diagram and knotoid
As Boyle and Issa note in [9, Proposition 1], Seifert’s algorithm applied to an intravergent diagram produces an invariant Seifert surface. To be precise, for a marked
strongly invertible knot (K, h, δ), let Γ be an intravergent diagram of the strongly
invertible knot (K, h), such that δ is the crossing arc at the crossing through which
the axis passes. Then, by applying Seifert’s algorithm to Γ, we obtain an invariant
Seifert surface for (K, h, δ) (Figure 3.1(b)).
By cutting the over- or under-path which contains a ﬁxed point, we have arotationally symmetric kontoid diagram Γ′ (Figure 3.1(c)). By applying Seifert’ssmoothing to Γ′ , one obtains Seifert circles and an arc. Then replacing the arc by athin disk, we have an invariant Seifert surface for (K, h, δ) (see Figure 3.1(d)). Notethat theses surfaces in general have diﬀerent genera.

Figure 3.1. A transvergent diagram and a symmetric knottoid
6

3.2. Construction from a transvergent diagram
In order to construct an invariant Seifert surface from a transvergent diagram of astrongly invertible knot (K, h), we consider the quotient θ-curve θ(K, h) deﬁned asfollows. Let π : S 3 → S 3 /h ∼= S 3 be the projection. Then O := π(Fix(h)) is a trivial
knot and k := p(K) is an arc such that O ∩ k = ∂k. Thus the union θ(K, h) := O ∪ kforms a θ-curve embedded in S 3 : we call it the quotient θ-curve of the strongly
invertible knot (K, h). For a marked strongly invertible knot (K, h, δ), set δ̌ := π(δ)
and Ǩ := k ∪ δ̌ (Figure 3.2).

Figure 3.2. The quotient θ-curve θ(K, h) = O ∪ k and the constituent knot Ǩ = k ∪ δ̌ = π(K ∪ δ). Note that θ(K, h) consists ofthree edges k = π(K), δ̌ = π(δ) and δ̌ c = π(cl(Fix(h)\δ)) = cl(O \ δ̌).
Observe that if S is an h-invariant Seifert surface for (K, h, δ), then its image
Š := π(S) in S 3 /h is a spanning surface for the knot Ǩ, i.e., a compact surface in
S 3 /h with boundary Ǩ, which is disjoint from the interior of the arc δ̌ c := cl(O \ δ̌).
Conversely, if Š is a spanning surface for Ǩ which is disjoint from the interior of thearc δ̌ c , then its inverse image π −1 (Š) is an h-invariant spanning surface for (K, h, δ),
namely an h-invariant spanning surface for K whose intersection with Fix(h) is equalto δ. However, π −1 (Š) is not necessarily orientable. The following proposition gives
a necessary and suﬃcient condition for π −1 (Š) to be orientable and so an invariant
Seifert surface for (K, h, δ).
Proposition 3.2. Under the above notation, the following hold for every marked
strongly invertible knot (K, h, δ).
(1) If S is an invariant Seifert surface for (K, h, δ), then its image Š = π(S)in S 3 /h is a spanning surface for Ǩ = k ∪ δ̌ disjoint from the interior of δ̌ c , andsatisﬁes the following condition.
(C) For any loop γ in Š, γ is orientation-preserving or -reversing in Š according
to whether the linking number lk(γ, O) is even or odd.
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(2) Conversely, if Š is a spanning surface for the knot Ǩ in S 3 /h which is disjoint
from the interior of δ̌ c and satisﬁes Condition (C), then its inverse image π −1 (Š)
in S 3 is an invariant Seifert surface for (K, h, δ)Proof. (1) Note that Š ∼= S/h is regarded as an orbifold which has δ̌ as reﬂector
line. We consider an “orbifold handle decomposition” Š = D ∪ (∪i Bi ), where (a) D
is an “orbifold 0-handle”, namely D is a disk such that ∂ Š ∩ D = δ̌, and (b) {Bi }are 1-handles attached to D, namely each Bi is a disk such that Bi ∩ D = ∂Bi ∩ ∂D
consists of two mutually disjoint arcs which are disjoint from the reﬂector line δ̌.
This handle decomposition of the orbifold Š lifts to the handle decomposition S =π −1 (D) ∪ (∪i π −1 (Bi )) of the surface S, where π −1 (D) is an h-invariant 0-handle andeach π −1 (Bi ) consists of a pair of 1-handles attached to the disk π −1 (D). For eachi, D ∪ Bi is an annulus or a Möbius band which is obtained as the quotient of theorientable surface π −1 (D ∪Bi ) ⊂ S by the (restriction of) the involution h. Let γi bea core loop of D ∪ Bi . Then we can see that the projection π : π −1 (D ∪ Bi ) → D ∪ Bi
is as shown in Figure 3.3(a), (b) according to whether lk(γi , O) is even or odd. Hence
D ∪ Bi is an annulus or a Möbius band accordingly. Since the homology classes of{γi } generate H1 (Š), this observation implies that Š satisﬁes Condition (C).
(2) This can be proved by traversing the above argument.
□\\

Figure 3.3. The band D ∪ Bi (bottom) and its inverse image
π −1 (D ∪ Bi ) (top). The linking number lk(γi , O) is even in (a) andodd in (b).
8Remark 3.3. (1) Condition (C) is equivalent to the condition that the homomorphism ι : H1 (Š; Z/2Z) → Z/2Z deﬁned by ι(γ) := lk(O, γ) (mod 2) is identical withthe orientation homomorphism.
(2) We have β1 (S) = 2β1 (Š), where β1 denotes the ﬁrst Betti number, because if
b is the number of the 1-handles {Bi } in the proof, then β1 (Š) = b and β1 (S) = 2b.
From a given transvergent diagram of a strongly invertible knot (K, h), we caneasily draw a diagram of the quotient θ-curve θ(K, h) (see Figure 3.5(b)). There arevarious ways of modifying the diagram into a “good” diagram from which we canconstruct a surface Š for θ(K, h) that satisﬁes the conditions in Proposition 3.2.
Hiura [26] gave an algorithm for obtaining such a diagram of θ(K, h) = O ∪ k as
follows (see Figure 3.4).
(a) Let k hook O in even times in the uniform way and intδ̌ ⊂ O be disjoint from k.
(b) Travel along k and enumerate the hooks from 1 to 2n. Then slide the hooks
so that they are paired and arranged from the top to the bottom along O. Notethat one travels between (2i − 1)st and (2i)th hooks in one of the four routine ways
according to the orientation of the hooks.
(c) Surgery k along the bands {Bi }ni=1 arising in the pairs of hooks to obtain a setof arcs k ′ . Arrangement in (b) allows us to reset the orientation of k ′ as depicted.
Apply Seifert algorithm to k ′ ∪ δ̌ to obtain a Seifert surface S ′ . Then attaching
the bands {Bi }ni=1 to S ′ yields a spanning surface Š for k ∪ δ̌ which satisﬁes theconditions in Proposition 3.2(2). Therefore, the inverse image S := π −1 (Š) is aninvariant Seifert surface for (K, h, δ).

Figure 3.4. Hiura’s algorithm to obtain an invariant Seifert surface
from an intravergent diagram
9

Example 3.4. Let (K, h, δ) be the marked strongly invertible knot with K = 83
as illustrated by the transvergent diagram in Figure 3.5(a). Then by applying thealgorithm to the diagram, we obtain a genus 2 invariant Seifert surface S for (K, h, δ)
as shown in Figure 3.5(c). This is a minimal genus Seifert surface for (K, h, δ)and so g(K, h, δ) = 2, as shown below. By Hatcher-Thurston [22, Theorem 1]
or by Kakimizu [28] succeeding the work of Kobayashi [29], 83 has precisely two
genus 1 Seifert surfaces up to equivalence. (They are obtained by applying Seifert’s
algorithm to the alternating diagram, where there are two diﬀerent ways of attachinga disk to the unique big Seifert circle.) Obviously they are interchanged by h, andhence not h-invariant.
Kakimizu [28] in fact showed that the two genus 1 Seifert surfaces are the only
incompressible Seifert surfaces for 83 . Thus 83 does not even admit an h-invariant
incompressible Seifert surface. So the result of Edmonds and Livingston [20, Corollary 2.2] that every periodic knot admits an invariant incompressible Seifert surface
does not hold for strongly invertible knots.

Figure 3.5. A transvergent diagram for 83 and an invariant Seifert surface10

4. Proof of Theorem 1.2
By Proposition 3.2 and Remark 3.3, we can characterize the equivariant genus
g(K, h, δ) in terms of the quotient θ-curve θ(K, h) = k ∪ δ̌ ∪ δ̌ c and the constituent
knot Ǩ = k ∪ δ̌ as follows.
Proposition 4.1. Let (K, h, δ) be a marked strongly invertible knot. Then g(K, h, δ)is equal to the minimum of β1 (Š) where Š runs over the spanning surfaces for theconstituent knot Ǩ = k ∪ δ̌ of the quotient θ-curve θ(K, h) that is disjoint from theinterior of the remaining edge δ̌ c and satisﬁes Condition (C).
By relaxing the deﬁnition of the crosscap number, γ(K), of a knot K introducedby Clark [13], we deﬁne the band number, b(K), to be the minimum of β1 (G) of all
spanning surfaces G for K. In another words, b(K) = min(2g(K), γ(K)). Then we
have the following corollary.
Corollary 4.2. g(K, h, δ) ≥ b(Ǩ).
For any marked strongly invertible knot (K, h, δ) with K a 2-bridge knot, theconstituent knot Ǩ is either the trivial knot or a 2-bridge knot (see [42, Proposition3.6]). In his master thesis [5] supervised by the last author, Bessho described amethod for determining the cross cap numbers of 2-bridge knots by using the resultof Hatcher and Thurston [22, Theorem 1(b)] that classiﬁes the incompressible andboundary incompressible surfaces in the 2-bridge knot exteriors. Hirasawa and Teragaito [25] promoted the method into a very eﬀective algorithm. For some classesof marked strongly invertible 2-bridge knots, we can apply Corollary 4.2 by using
Hirasawa-Teragaito method, as shown in the following example.
Example 4.3. For a positive integer n, let Kn be the the plat closure of a 4-string
braid (σ22 σ14 )n . Then Kn is the 2-bridge knot whose slope q/p has the continued
fraction expansion [2, 4, 2, 4, · · · , 2, 4] of length 2n. Here we employ the following
convention of continued fraction expansion, which is used in [25].
1[a1 , a2 , · · · , am ] :=

1

a1 −
a2 −

1
..

.−

1
am

Thus Kn is the boundary of a linear plumbing of unknotted annuli where the ithband has 2 or 4 right-handed half-twists according to whether i is odd or even. In
particular, g(Kn ) = n. Note that Kn is isotopic to the knot admitting the strong
inversion h in Figure 4.1(a). Let (Kn , h, δ) be the marked strongly invertible knot,
where δ is the long arc in Fix(h) bounded by Fix(h)∩Kn illustrated in Figure 4.1(a).
Then we have g(Kn , h, δ) = 2n = g(Kn ) + n, as explained below.
11Figure 4.1. A Seifert surface realizing the equivariant genus
Observe that (Kn , h, δ) is equivalent to the marked strongly invertible knot bounding the h-invariant Seifert surface S of genus 2g in Figure 4.1(b). Consider thequotient surface Š = S/h, which is a spanning surface for the knot Ǩ := k ∪ δ̌. We
show that the band number b(Ǩ) is equal to β1 (Š). Then it follows from Corollary 4.2 that S is a minimal genus Seifert surface for (Kn , h, δ). To this end, observe that Ǩ is the 2-bridge knot whose slope has the continued fraction expansion
C := [4, 2, 4, 2, · · · , 4, 2] of length 2n (Figure 4.1(c)). Since C does not contain an
entry 0, −1, 1 and since C does not contain a sub-sequence ±[· · · , 2, 3, · · · , 3, 2, · · · ]nor ±[· · · , 2, 2, · · · ], it follows from [25, Theorems 2 and 3], that the length 2n of C
is minimal among all continued fraction expansions of all fractions representing Ǩ.
Then it follows from [22, Theorem 1(b)] that b(Ǩ) = β1 (Š) = 2g as desired.
Theorem 1.2 follows from the above example and Proposition 1.5, which is provedat the end of Section 5. Though the above method is also applicable to a certain
family of marked strongly invertible 2-bridge knots, there are various cases where theabove simple method does not work. In fact, there is a case where the knot Ǩ = k∪ δ̌is trivial (see Figure 3.5). The second author [26] succeeded in treating such family
by studying the spanning surface Š that satisfy the condition in Proposition 4.1
through the hight function method by Hatcher-Thurston [22]. However, there still12remained cases where none of the methods described above work. In the sequelof this paper [24], we give a uniﬁed determination of the equivariant genera of all
marked strongly invertible 2-bridge knots by using Theorems 1.3 and 1.4, which isproved in the next section.
5. Proof of Theorems 1.3 and 1.4
For a knot K in S 3 , let E(K) := S 3 \ int N (K) be the exterior of K, where
N (K) is a regular neighborhood of K. If F is a Seifert surface for K, then after
an isotopy, F intersects N (K) in a collar neighborhood of ∂F , and F ∩ E(K) is asurface properly embedded in E(K) whose boundary is a preferred longitude, i.e., alongitude of the solid torus N (K) whose linking number with K is 0. Conversely,
any such surface in E(K) determines a Seifert surface for K. Thus we also call sucha surface in E(K) a Seifert surface for K.
Proof of Theorem 1.3. We give a proof imitating the arguments by Edmonds [19].
Let (K, h) be a strongly invertible knot, and let E := E(K) be an h-invariant
exterior of K. We continue to denote by h the restriction of h to E. Then Theorem
1.3 is equivalent to the existence of a minmal genus Seifert surface F ⊂ E for K
such that F ∩ h(F ) = ∅.
Fix an h-invariant Riemannian metric on E such that ∂E is convex. Choose apreferred longitude ℓ0 ⊂ ∂E of K such that ℓ0 and ℓ1 := h(ℓ0 ) are disjoint. Let F ∗be a smooth, compact, connected, orientable surface of genus g(K) with one hole,ψ0 : ∂F ∗ → ∂E an embedding such that ψ0 (∂F ∗ ) = ℓ0 , and set ψ1 := hψ0 . For
i = 0, 1, let Fi be the space of all piecewise smooth maps f : (F ∗ , ∂F ∗ ) → (E, ℓi )
properly homotopic to an embedding, such that f |∂F ∗ = ψi . Then we have thefollowing [19, Proposition 1].
Lemma 5.1. Each Fi contains an area minimizer, namely, there exists an elementfi ∈ Fi whose area is minimum among the areas of all elements of Fi . Moreover,any area minimizer in Fi is an embedding.
Since h is an isometric involution, it follows that fi is an area minimizer in Fi if
and only if hfi is an area minimizer in Fj , where {i, j} = {0, 1}. Thus Theorem 1.3
follows from the following analogue of [19, Theorem 2].
□
Theorem 5.2. For i = 0, 1, let fi be an area minimizer in Fi , and Fi ⊂ E theminimal genus Seifert surface for K obtained as the image of fi . Then F0 and F1
are disjoint.
Proof. The proof is the same as [19, Proof of Theorem 2], as explained below.
Suppose to the contrary that F0 ∩ F1 ̸= ∅. As is noted in [19, the 2nd paragraphof the proof of Theorem 2], by virtue of the Meeks-Yau trick introduced in [35] anddiscussed in [21], we can reduce to the case where F and h(F ) intersect transversely.
Hence we suppose that F and h(F ) intersect transversely.
13

By the arguments in [19, the 3rd to the 6th paragraphs of the proof of Theorem
2] using the area minimality and the incompressibility of Fi (i = 0, 1) and theasphericity of E, it follows that every component of F0 ∩ F1 is essential in both F0
and F1 .
By the arguments in [19, the 7th to the ﬁnal paragraphs of the proof of Theorem2], we see that there is a submanifold W in E satisfying the following conditions.
(a) ∂W = A ∪ B where A = W ∩ F0 and B = W ∩ F1 .
(b) Both (F0 \ A) ∪ B and (F1 \ B) ∪ A are minimal genus Seifert surfaces.
Since (F0 \ A) ∪ B and (F1 \ B) ∪ A have corners, smoothing them reduces area andyields two minimal genus Seifert surfaces, at least one of which has less area thanf0 or f1 , a contradiction.
□
Proof of Theorem 1.4. Throughout the proof we use the following terminology: fora topological space X and its subspace Y , a closed-up component of X \Y means theclosure in X of a component of X \ Y . Let (K, h, δ) be a marked strongly invertible
knot and let E and h be as in the proof of Theorem 1.3. Then Fix(h) ⊂ E is the
disjoint union of two arcs δ ⊔ δ c , where δ denotes the intersection of the original δwith E and δ c = Fix(h) \ δ. Then, by the assumption of Theorem 1.4, there is aminimal genus Seifert surface F ⊂ E such that F ∩ h(F ) = ∅. Let Eδ and Eδc be theclosed-up components of E(K) \ (F ∪ h(F )) containing δ and δ c , respectively. Then
there is a minimal genus Seifert surface S ⊂ E for (K, h, δ) such that ∂S is containedin the interior of the annulus ∂E ∩ ∂Eδ . We will construct from S a minimal genus
Seifert surface for (K, h, δ) which is properly embedded in Eδ \ (F ∪ h(F )).
Claim 1. We can choose S so that S intersects F ∪ h(F ) transversely and so thatevery component of S ∩ (F ∪ h(F )) is essential in both S and F ∪ h(F ).
Proof. Since F ∩ h(F ) = ∅, we can h-equivariantly isotope S so that it intersects
F ∪ h(F ) transversely. Then S ∩ (F ∪ h(F )) consists of simple loops, because theboundaries of S and F ∪ h(F ) are disjoint.
Suppose ﬁrst that S ∩ (F ∪ h(F )) contains a component that is inessential inF ∪ h(F ). Let α be one such component which is innermost in F ∪ h(F ), and let ∆be the disk in F ∪ h(F ) bounded by α. Then ∆ ∩ S = ∂∆, h(∆) ∩ S = ∂(h(∆)),
and ∆ ∩ h(∆) = ∅. Let S ′ be the h-invariant surface obtained from S by surgery
along ∆ ∪ h(∆), i.e., S ′ is obtained from S by removing an h-invariant open regularneighborhood of ∂(∆ ∪ h(∆)) in S and capping oﬀ the resulting four boundary
circles with nearby parallel h-equivariant copies of ∆ and h(∆). Let Sb′ be thecomponent of S ′ containing ∂S. Then Sb′ is a Seifert surface for (K, h, δ) such thatg(Sb′ ) ≤ g(S) = g(K, h, δ). Hence Sb′ is also a minimal genus Seifert surface for(K, h, δ). Moreover |Sb′ ∩ (F ∪ h(F ))| ≤ |S ∩ (F ∪ h(F ))| − 2, where | · | denotes thenumber of the connected components.
Suppose next that S ∩ (F ∪ h(F )) contains a component that is inessential in S.
Let α be one such component which is innermost in S. Then α is also inessential
14

in F ∪ h(F ) by the incompressibility of F ∪ h(F ). So, by the argument in thepreceding paragraph, we obtain a minimal genus Seifert surface Sb′ for (K, h, δ) suchthat |Sb′ ∩ (F ∪ h(F ))| ≤ |S ∩ (F ∪ h(F ))| − 2.
By repeating the above arguments, we can ﬁnd a desired minimal genus Seifert
surface for (K, h, δ).
□Let p : Ẽ → E be the inﬁnite cyclic covering, and let Ej (j ∈ Z) be the closed-up
components of Ẽ \ p−1 (F ∪ h(F )) satisfying the following conditions.
(1) Ẽ = ∪j∈Z Ej .
(2) Ej projects homeomorphically onto Eδ or Eδc according to whether j is evenor odd.
(3) Fj := Ej−1 ∩ Ej projects homeomorphically onto F or h(F ) according towhether j is even or odd.
For each j ∈ Z, let hj be the involution on Ẽ which is a lift of h such that hj (Ej ) =Ej . Note that hj (Ei ) = E2j−i and that Fix(hj ) is a properly embedded arc in Ej
which projects to δ or δ c according to whether j is even or odd. The compositionτ := hj+1 hj is independent of j, and gives a generator of the covering transformationgroup of Ẽ, such that τ (Ei ) = Ei+2 and τ (Fi ) = Fi+2 for every i.
Note that p−1 (S) = ⊔i∈Z S2i , where S2i is the lift of S preserved by h2i . Letr := max{j ∈ Z | S0 ∩ Ej ̸= ∅}, and assume that the number r is minimized among
all minimal genus Seifert surfaces S for (K, h, δ) that satisfy the conclusion of Claim
1. Then Theorem 1.3 is equivalent to the assertion that r = 0.
Suppose to the contrary that r > 0. Let S0− be the closed-up component of Ẽ \ S0containing S−2 = τ −1 (S0 ). Set W̃ := S0− ∩ Er and W := p(W̃ ). Since W̃ ⊂ Er ,p|W̃ : W̃ → W is a homeomorphism. (See Figure 5.1.)
Claim 2. W is a (possibly disconnected) compact 3-manifold contained in int E suchthat ∂W = A ∪ B, where A = S ∩ W , B = p(Fr ) ∩ W , and A ∩ B = p(S0 ∩ Fr ) ⊂S ∩ p(Fr ). Here p(Fr ) = F or h(F ) according to whether r is even or odd.
Proof. Note that S0− ∩ ∂ Ẽ is the half-inﬁnite annulus in ∂ Ẽ which forms the closedup component of ∂ Ẽ \ ∂S0 disjoint from ∂F1 . On the other hand, Er ∩ ∂ Ẽ is theannulus in ∂ Ẽ bounded by ∂Fr and ∂Fr+1 . Since r > 0 by the assumption, these
imply that S0− ∩ ∂ Ẽ and Er ∩ ∂ Ẽ are disjoint. Hence W̃ is disjoint from ∂ Ẽ and soW̃ ⊂ int Ẽ. Note that fr S0− = S0 and fr Er = Fr ⊔Fr+1 intersect transversely (Claim1), where S0− ∩ Fr+1 = ∅. Hence W̃ is a compact 3-manifold contained in int Ẽ, suchthat ∂ W̃ = Ã ∪ B̃, where Ã = (fr S0− ) ∩ W̃ = S0 ∩ W̃ , B̃ = (fr Er ) ∩ W̃ = Fr ∩ W̃ , andÃ ∩ B̃ = (S0 ∩ W̃ ) ∩ (Fr ∩ W̃ ) = S0 ∩ Fr . Since p|W̃ : W̃ → W is a homeomorphism,
these imply the claim.
□
Claim 3. W ∩ h(W ) = ∅.
15

Figure 5.1. Schematic picture of p : Ẽ → E, where r = 2. The
picture does not reﬂect the assumption that the boundaries of F ,h(F ) and S are disjoint.
Proof. Since the restriction p|Er is a homeomorphism onto its image p(Er ) (which isequal to Eδ or Eδc according to whether r is even or odd) and since the involution
h|p(Er ) is pulled back to the involution hr |Er by p|Er , it follows that p|Er restricts toa homeomorphism from W̃ ∩ hr (W̃ ) onto W ∩ h(W ). On the other hand, we haveW̃ ∩ hr (W̃ ) = (S0− ∩ Er ) ∩ hr (S0− ∩ Er ) = (S0− ∩ hr (S0− )) ∩ Er = ∅ ∩ Er = ∅,where the third identity is veriﬁed as follows. Recall that S0− is the closed-up
component of Ẽ \ S0 containing S−2 . Thus hr (S0− ) is the closed-up componentof Ẽ \ hr (S0 ) = Ẽ \ S2r containing hr (S−2 ) = S2r+2 . Since r > 0, this implies
S0− ∩ hr (S0− ) = ∅. Hence W ∩ h(W ) = p(W̃ ∩ hr (W̃ )) = ∅ as desired.
□
We perform an h-equivariant cut and paste operation on S ∪ F ∪ h(F ) alongW ∪h(W ), and produce surfaces S ′ = Sb′ ⊔Sc′ , F ′ = Fb′ ⊔Fc′ and h(F ′ ) = h(Fb′ )⊔h(Fc′ )as follows.
16

(1) S ′ := (S \ (A ∪ h(A)) ∪ (B ∪ h(B)), Sb′ is the component of S ′ containing ∂S,
and Sc′ := S ′ \ Sb′ .
(2) F ′ := (F \B)∪A, Fb′ is the component of F ′ containing ∂F , and Fc′ := F ′ \Fb′ .
Then Sb′ is a Seifert surface for (K, h, δ), and both Fb′ and h(Fb′ ) are Seifert surfaces
for K. Let Σ be the disjoint union of copies of Sc′ , Fc′ and h(Fc′ ). Then Σ is a possibly
empty, closed, orientable surface, such that χ(Σ) = χ(Sc′ ∪ Fc′ ∪ h(Fc′ )). (Note thatthe intersection among Sc′ , Fc′ and h(Fc′ ) consists of disjoint loops.) Since S andF ∪ h(F ) intersects in essential loops by Claim 1, none of the components of Σ is a
2-sphere and therefore χ(Σ) ≤ 0. Hence
χ(Sb′ )+χ(Fb′ )+χ(h(Fb′ )) ≥ χ(Sb′ )+χ(Fb′ )+χ(h(Fb′ ))+χ(Σ) = χ(S)+χ(F )+χ(h(F )).
Since g(F ) = g(K) ≤ g(Fb′ ), this implies
χ(Sb′ ) ≥ χ(S) + (χ(F ) − χ(Fb′ )) + (χ(h(F )) − χ(h(Fb′ ))) ≥ χ(S).
Hence g(Sb′ ) ≤ g(S) = g(K, h, δ), and therefore Sb′ is also a minimal genus Seifert
surface for (K, h, δ).
Note that the h0 -invariant lift of Sb′ is obtained from S0 ⊂ Ẽ by cut and paste
operation along W̃ ∪ h0 (W̃ ) and so it is contained in the region of Ẽ bounded by
F1−r and Fr . After a small h-equivariant isotopy of Sb′ , the lift is contained in theinterior of that region, and hence the number r for Sb′ is strictly smaller than theoriginal r. This contradicts the minimality of r. Hence we have r = 0 as desired. □The following example illustrates Theorems 1.3, 1.4. and Corollary 1.6.

Figure 5.2. The invariant Seifert surface S and its compressing loop
α on the positive side
Example 5.3. Let S be the invariant Seifert surface for K = 83 constructed in
Example 3.4. Then the loop α in Figure 5.2 is a compressing loop of S on thepositive side, and h(α) is a compressing loop of S on the negative side. Notethat α and h(α) intersect nontrivially, and therefore {α, h(α)} does not yield an17

h-equivariant compression of S. Consider a parallel copy S+ of S on the positiveside, and let F+ be the minimal genus Seifert surface obtained by compressing S+
along a parallel copy of α on S+ . Then F− := h(F+ ) is obtained from the parallel
copy h(S+ ) of S on the negative side through compression along a parallel copy ofh(α) ⊂ h(S+ ). The minimal genus Seifert surfaces F+ and F− = h(F+ ) are disjoint,
and S lies in a region between them.
As noted in Example 3.4, K = 83 has precisely two minimal genus Seifert surfaces
up to equivalence, and the Kakimizu complex M S(K) consists of a single edge andtwo vertices. The automorphism h∗ of M S(K) induced by h is the reﬂection in thecenter of the 1-simplex.
Finally, we prove Proposition 1.5 for ﬁbered knots.
Proof of Proposition 1.5. Let p : E(K) → S 1 be the ﬁbering whose ﬁbers are minimal genus Seifert surfaces for K. By the result of Tollefson [45, Theorem 2], we may
assume that the involution h on E(K) preserves the ﬁbering and that the involutionȟ on S 1 induced from h is given by ȟ(z) = z̄, where we identify S 1 with the unit circle on the complex plane. Then the inverse image p−1 (±1) gives a pair of h-invariant
Seifert surfaces for K with genus g(K). Since these give Seifert surfaces for the twomarked strongly invertible knots associated with (K, h), we have g(K, h, δ) = g(K)as desired.
□
6. Equivariant 4-genus
In this section, we review old and new studies of equivariant 4-genera of symmetricknots. For a periodic knot or a strongly invertible knot K, one can deﬁne theequivariant 4-genus g̃4 (K) to be the minimum of the genera of smooth surfaces in B 4
bounded by K that are invariant by a periodic diﬀeomorphism of B 4 extending theperiodic diﬀeomorphism realizing the symmetry of K. (Here, the symbol expressing
the symmetry is suppressed in the symbol g̃4 (K).) Of course, g̃4 (K) is bounded
below by the 4-genus g4 (K), and it is invariant by equivariant cobordism.
The equivarinat cobordism of periodic knots was studied by Naik [38], where
she gave criteria for a given periodic knot to be equivariantly slice, in terms of thelinking number and the homology of the double branched covering. By using thecriteria, she presented examples of slice periodic knots which are not equivariantly
slice. (See Davis-Naik [17] and Cha-Ko [12] for further development.)
The equivariant cobordism of strongly invertible knots was studied by the third
author [42], where the notion of directed strongly invertible knots was introduced
so that the connected sum is well-deﬁned, and it was observed that the set C˜ of thedirected strongly invertible knots modulo equivariant cobordism form a (possibly
noncommutative) group with respect to the connected sum. He then introduced apolynomial invariant η : C˜ → Z⟨t⟩ which is a group homomorphism, and presented
a slice strongly invertible knot that is not equivariantly slice.
18

These two old works show that there is a gap between the (usual) 4-genus g4 (K)and the equivariant 4-genus g̃4 (K) for both periodic knots and strongly invertible
knots.
In a recent series of works [9, 10, 3, 11], Boyle and his coworkers launched to a
project of systematic study of equivariant 4-genera of symmetric knots. It should
be noted that they treat not only periodic/strongly-invertible knots, but also freely
periodic knots and strongly negative amphicheiral knots. In [9, Theorems 2 and 3],
Boyle and Issa gave a lower bound of g̃4 (K) of a periodic knot K in terms of thesignatures of K and its quotient knot, and showed that the gap between g4 (K) andg̃4 (K) can be arbitrary large. They also introduced the notion of the butterﬂy 4e (K) of a directed (or marked) strongly invertible knot K, gave an estimate
genus bg4
e (K) in terms of the g-signature, and showed that the gap between g4 (K) andof bg
4e (K) can be arbitrary large [9, Theorem 4]. In [16], Dai, Mallick and Stoﬀregenbg4
introduced equivariant concordance invariants of strongly invertible knots using knot
Floer homology, and showed that the gap between the equivariant 4-genus g̃4 (K)and the genus g(K) for marked strongly invertible knots can be arbitrary large,
proving the conjecture [9, Question 1.1] by Boyle and Issa. In fact, they proved
that, for the knot Kn deﬁned as the connected sum Kn := (T2n,2n+1 #T2n,2n+1 )# −(T2n,2n+1 #T2n,2n+1 ), the equivariant 4-genus g̃4 (Kn ), with respect to some strong
inversion, is at least 2n − 2 [16, Theorem 1.4], whereas g4 (Kn ) = 0. Here T2n,2n+1
is the torus knot of type (2n, 2n + 1), and “−” denoted the reversed mirror image.
On the other hand, since Kn is ﬁbered, the (3-dimensional) equivariant generaof the marked strongly invertible knots associated with Kn are equal to the genus
g(Kn ) = 4n(2n − 1) (see Proposition 1.5), which is bigger than the estimate 2n − 2of g̃4 (Kn ) from below. As far as the authors know, there is no known algebraic
invariant that gives an eﬀective estimate of the (3-dimensional) equivariant generaof marked strongly invertible knots, though the (3-dimensional) genus of any knot
is estimated by the the Alexander polynomial and moreover it is determined by the
Heegaard Floer homology (Ozsvath-Szabo [39, Theorem 1.2]). As is noted in [16],there has been a renewed interest in strongly invertible knots from the view pointof more modern invariants (see Watson [49] and Lobb-Watson [32]). Thus we would
like to pose the following question.
Question 6.1. Is there an algebraic invariant of a marked strongly invertible knotthat gives an eﬀective estimate of the equivariant genus, or more strongly, determinesit?

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21

Department of Mathematics, Nagoya Institute of Technology, Showa-ku, Nagoya
city, Aichi, 466-8555, Japan
Email address: hirasawa.mikami@nitech.ac.jp
Inuyama-Minami High School, Hasuike 2-21, Inuyama City, Aichi, 484-0835, Japan
Email address: ryhiura@gmail.com
Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University,
3-3-138, Sugimoto, Sumiyoshi, Osaka City 558-8585, Japan
Email address: sakuma@hiroshima-u.ac.jp22