Equivariant 4-genera of strongly invertible and periodic knots
EEQUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE ANDPERIODIC KNOTS
KEEGAN BOYLE AND AHMAD ISSA
Abstract.
We study the equivariant genera of strongly invertible and periodic knots. Ourtechniques include some new strongly invertible concordance group invariants, Donaldson’stheorem, and the g-signature. We find many new examples where the equivariant 4-genusis larger than the 4-genus. Introduction
Edmonds showed that periodic knots bound equivariant minimal genus Seifert surfaces[Edm84]. By contrast, the equivariant 4-genus of a periodic knot, or more generally any typeof symmetric knot, is rather subtle, and may vary between different symmetries on the sameknot. A symmetric knot ( K, ρ ) is a knot K ⊂ S together with a finite order diffeomorphism ρ : S → S such that ρ ( K ) = K . Given a symmetric knot ( K, ρ ), we define the equivariant -genus (cid:101) g ( K ) of K as the minimal genus of an orientable, smoothly properly embedded surface S ⊂ B with boundary K for which ρ extends to a diffeomorphism (cid:101) ρ : ( B , S ) → ( B , S )with the same order as ρ . If ρ is orientation preserving, the fixed point set of ρ is eitherempty or the unknot. In the latter case the fixed point set is either disjoint from K , in whichcase we call ( K, ρ ) periodic , or intersects K in exactly two points, in which we case we call( K, ρ ) strongly invertible . This paper is mainly concerned with the equivariant 4-genus ofperiodic and strongly invertible knots and the strongly invertible concordance group.As a motivating example, consider the knot 9 with strong inversions τ , τ and 2-periodicsymmetry ρ shown in Figure 1. The (non-equivariant) 4-genus of 9 is 1. In contrast,we show that (cid:101) g (9 , τ ) = 2 using Donaldson’s theorem and that (cid:101) g (9 , ρ ) = 3 using anapplication of the Riemann-Hurwitz formula. It is unknown whether (cid:101) g (9 , τ ) is 1 or 2. SeeAppendix B for more examples where we bound (cid:101) g ( K ). ρ τ τ Figure 1.
The knot 9 with strong inversions τ , τ and 2-periodic symmetry ρ . The strong inversion τ is rotation around an axis perpendicular to the planeof the diagram. Here (cid:101) g (9 , τ ) = 2, (cid:101) g (9 , ρ ) = 3, and (cid:101) g (9 , τ ) ∈ { , } . a r X i v : . [ m a t h . G T ] J a n KEEGAN BOYLE AND AHMAD ISSA
The above example contrasts with the existing literature which has focused primarily onobstructing slice knots from bounding equivariant slice disks. For a periodic knot K , suchobstructions have been found using a variety of techniques. In [Nai97], Naik showed that K must have linking number 1 with the axis of symmetry, and obtained further obstruc-tions using equivariant metabolizers and Casson-Gordon invariants. In [CK99], Cha and Koshowed that if K is equivariantly slice then certain twists of the quotient knot are slice in ahomology 4-ball, and then obtained obstructions by applying Casson-Gordon invariants. In[DN06], Davis and Naik used Reidemeister torsion to obtain restrictions on an equivariantversion of the Alexander polynomial when K is equivariantly slice.For strongly invertible knots, equivariant slice disk obstructions have been obtained bySakuma [Sak86] using Kojima and Yamasaki’s η -polynomial [KY79] and by Dai, Hedden,and Mallick [DHM20] as a consequence of their cork obstructions coming from HeegaardFloer homology.In this paper we more generally compare g ( K ) and (cid:101) g ( K ) for both periodic and stronglyinvertible knots. Our main results fall into three categories. First, we use Donaldson’s the-orem to give various examples where (cid:101) g ( K ) > g ( K ). Second, we use g -signatures to definean equivariant version of the signature giving lower bounds on equivariant 4-genera. Third,we define several strongly invertible concordance invariants based on some new topologicalconstructions.It has been known since shortly after Donaldson’s theorem was proved in the 1980s [Don87]that the theorem can often be used to obstruct the existence of a slice disk (see for example[Lis07]), and more generally can be used to obstruct g ( K ) = | σ ( K ) | / B lifts to the double branched cover of B over the surface. Donaldson’s theorem can oftenbe used to show that (cid:101) g ( K ) > | σ ( K ) | / B over a spanning surface with the Gordon-Litherland form, we prove the following theorem. Theorem 1.
Let K ⊂ S be a knot with a periodic or strongly invertible symmetry ρ : S → S . Suppose that F ⊂ S is a spanning surface for K for which the Gordon-Litherland pairing G F is positive definite and ρ ( F ) = F . If (cid:101) g ( K ) = − σ ( K ) / then there is an embedding oflattices ι : ( H ( F ) , G F ) → ( Z k , Id ) such that δ ◦ ι = ι ◦ ρ ∗ , where δ is an automorphism of ( Z k , Id ) with order ( δ ) = order ( ρ ) and k = − σ ( K ) / b ( F ) . In particular, the followingdiagram commutes. ( H ( F ) , G F ) ( Z k , Id)( H ( F ) , G F ) ( Z k , Id) ρ ∗ ι δι Theorem 1 can be applied, for example, when K is a periodic or strongly invertible knotwith an alternating diagram in which the symmetry is visible . In this case, F can bechosen as the positive definite checkerboard surface. Using a computer, we enumerated An alternating knot with a periodic symmetry of order > QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 3 all alternating symmetric diagrams through 11 crossings and checked the obstruction fromTheorem 1. In all examples we found with (cid:101) g ( K ) > | σ ( K ) | / g ( K ) = − σ ( K ) / (cid:101) g ( K ) > g ( K ). These periodic and strongly invertible examples are listedin Appendix B.The best lower bound we can obtain with Theorem 1 is (cid:101) g ( K ) ≥ g ( K ) + 1. However, weprove the following theorem which in particular shows that (cid:101) g ( K ) can be much larger than g ( K ). Theorem 2.
Let K be an n -periodic knot with quotient knot K . Then (cid:101) g ( K ) ≥ | n · σ ( K ) − σ ( K ) | n − . We prove this theorem using an equivariant concordance invariant (cid:101) σ ( K ) which we callthe g -signature of K . The g -signature of K is a natural equivariant generalization of theknot signature defined in terms of g -signatures of the double cover of B branched overan equivariant surface with boundary K . For periodic knots, we prove the lower bound (cid:101) g ( K ) ≥ | (cid:101) σ ( K ) | /
2. Moreover, if K is periodic with quotient K we are able to express (cid:101) σ ( K )purely in terms of σ ( K ) and σ ( K ) (see Theorem 10), from which we obtain Theorem 2.The following theorem gives a family of 2-periodic Montesinos knots K n for which Theorem2 shows that there is an arbitrarily large gap between (cid:101) g ( K n ) and g ( K n ). Theorem 3.
Let { K n } be the family of 2-periodic Montesinos knots with K n = M (1; − n, n + 2 , − n ) shown in Figure 2, where n is odd and positive. The difference (cid:101) g ( K n ) − g ( K n ) is unbounded.In fact, (cid:101) g ( K n ) = 2 n and g ( K n ) = 1 . n + 1- n - n Figure 2.
A family of 2-periodic knots K n (left) for n odd, with n = 3shown on the right. The boxes are twist regions with the labeled numberof half-twists. The period can be seen by performing a flype on the centralcrossing region (enclosed by a dotted loop in the right diagram), then rotatingthe entire diagram by π within the plane of the diagram.Theorem 2 shows that (cid:101) g ( K n ) ≥ n , but an argument involving the Riemann-Hurwitzformula shows that in fact (cid:101) g ( K n ) ≥ n . We also provide an example where Theorem 2 givesa stronger lower bound than the Riemann-Hurwitz formula argument (see Example 6.3). We follow the convention for Montesinos knots notation from [Iss18].
KEEGAN BOYLE AND AHMAD ISSA
For a strongly invertible knot K , one can attempt to define (cid:101) σ ( K ) analogously as the g -signature of the double cover of B branched over an equivariant surface S . Unfortunately,this turns out to depend on the choice of S . However, if we require that S is a butterflysurface , that is, the pointwise fixed arc on S is separating , then (cid:101) σ ( K ) no longer dependson the choice of S . In fact, we find that (cid:101) σ ( K ) is an equivariant concordance invariant. Thisnaturally leads us to define the butterfly 4-genus (cid:101) bg ( K ) of K as the minimal genus of abutterfly surface in B with boundary K . The g -signature gives the following lower boundon the butterfly 4-genus. Theorem 4. If K is a directed strongly invertible knot which bounds a butterfly surface in B , then (cid:101) bg ( K ) ≥ | (cid:101) σ ( K ) | / . In Example 6.6, we give a family of strongly invertible knots K n for which Theorem 4shows that (cid:101) bg ( K n ) − g ( K n ) is unbounded. However, we are unable to answer the analogousquestion about the equivariant 4-genus. Question 1.1.
Is there a family of strongly invertible knots K n for which (cid:101) g ( K n ) − g ( K n )is unbounded?Since an equivariant slice disc is always a butterfly surface, Theorem 4 can be used toobstruct (cid:101) g ( K ) = 0. We obtain further obstructions to (cid:101) g ( K ) = 0 by defining some new in-variants of the strongly invertible concordance group (cid:101) C . This group, first defined by Sakuma[Sak86], consists of equivariant concordance classes of directed strongly invertible knots, thatis, strongly invertible knots along with an orientation on the axis of symmetry and a choiceof half-axis. The group operation is given by equivariant connect sum. In [Sak86], Sakumadefined a strongly invertible concordance invariant as the Kojima-Yamasaki η -polynomial[KY79] of the two component link consisting of the axis of symmetry and the quotient oftwo parallel push-offs of a given strongly invertible knot. Similarly, we define new strongly in-vertible concordance invariants by associating two new 2-component links L b ( K ) and L qb ( K )with a directed strongly invertible knot K . Figure 3.
The butterfly link L b (3 +1 ) (center) and quotient butterfly link L qb (3 +1 ) (right) for the directed strong inversion 3 +1 (left) on the trefoil. Thesurgery band is shaded in gray.First, the butterfly link L b ( K ) is the 2-periodic link with linking number 0 constructed byperforming a band move on K along a band containing the chosen half-axis. Second, the The fixed arc plays the role of the butterfly’s thorax, separating the surface into symmetric wings. A direction is a choice of oriented half-axis which we use to define (cid:101) σ ( K ); see Definition 2.1 and Figure 4. QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 5 quotient butterfly link L qb ( K ) is the link consisting of the quotient of L b ( K ) by its periodicsymmetry, and the axis. See Figure 3 for an example of L b ( K ) and L qb ( K ). The terminology“butterfly link” refers to the fact that cutting a butterfly Seifert surface for K , if it exists,along its pointwise fixed arc gives a surface with boundary L b ( K ) (see Figure 6).In Section 4, we use L b and L qb to produce several group homomorphisms from the stronglyinvertible concordance group (cid:101) C . First, note that if K is equivariantly slice, then L b ( K )and L qb ( K ) are (strongly) slice links so that their η -polynomials vanish. Applying the η -polynomial to L b then gives a group homomorphism η ◦ L b : (cid:101) C → Z [ t, t − ]. Second, the linkingnumber of L qb ( K ) is an even integer 2 · (cid:101) lk ( K ), and this defines a group homomorphism (cid:101) lk : (cid:101) C → Z . Finally, we have the following surjective homomorphisms to the smooth concordancegroup C and the topological concordance group C top . Theorem 5.
The maps b : (cid:101) C → C and qb : (cid:101) C → C top defined as follows are surjective grouphomomorphisms.(i) b ( K ) is the smooth concordance class of one component of L b ( K ) .(ii) qb ( K ) is the topological concordance class of the non-axis component of L qb ( K ) . Unlike Sakuma’s polynomial, our concordance invariants are sensitive to the choice ofdirection on strong inversions. For a specific example, qb (3 − ) = m and qb (3 +1 ) = 0 , where3 +1 and 3 − are the same strong inversion with oppositely chosen half-axes (see Example 4.10).As another application, our concordance invariants show that neither strong inversion on 8 is equivariantly slice, even though their Sakuma polynomials vanish and 8 is slice. SeeAppendix A for the invariants of 8 and some other low-crossing examples.As a final note we show that strongly invertible knots do not always bound butterflysurfaces using the following theorem, which follows immediately from Proposition 5 andProposition 9. Theorem 6.
Let K be a strongly invertible knot.(i) If K bounds a butterfly Seifert surface in S , then the Arf invariant of K is .(ii) If K bounds a butterfly surface in B , then (cid:101) lk ( K ) = 0 . Organization.
In Section 2 we give some background on the strongly invertible con-cordance group. In Section 3 we define the equivariant genera of strongly invertible andperiodic knots, and prove Theorem 6(i). In Section 4 we define the butterfly link L b ( K ), thequotient butterfly link L qb ( K ), the axis-linking number (cid:101) lk ( K ), and we prove Theorem 5 andTheorem 6(ii). In Section 5 we discuss lifting group actions to the double branched cover,then prove Theorem 1. In Section 6 we discuss the g -signature (cid:101) σ ( K ), proving Theorem 2and Theorem 4. In Appendix A we provide a table of some low-crossing directed stronglyinvertible knots and their invariants. In Appendix B we provide a table of examples whereTheorem 1 shows that (cid:101) g ( K ) > g ( K ).1.2. Acknowledgments.
We would like to thank Liam Watson for his encouragement andinterest in this project, several helpful conversations, and his comments on an earlier draft.We would also like to thank Makoto Sakuma and Danny Ruberman for some helpful com-ments.
KEEGAN BOYLE AND AHMAD ISSA The strongly invertible concordance group
The strongly invertible concordance group (cid:101) C was first defined by Sakuma [Sak86]; werecall the basic definitions here. This group is an equivariant version of the usual smoothconcordance group C . We also recall the definitions of the forgetful homomorphism f : (cid:101) C → C and the doubling homomorphism r : C → (cid:101) C . Definition 2.1.
A strongly invertible knot separates the axis of symmetry into two compo-nents, each of which we call a half-axis . A direction on a strongly invertible knot is a choiceof half-axis, and a choice of orientation on the axis. We call a strongly invertible knot alongwith a choice of direction, a directed strongly invertible knot K . We do not require a choiceof orientation on the knot itself (since it is invertible). The axis-reverse of K is the samestrongly invertible knot with the opposite choice of orientation on the axis, the mirror mK of K is (as usual) given by reversing the orientation on the ambient S , and the antipode K − of K is given by choosing the other choice of half-axis with the same orientation.Note that the two fixed points of a directed strongly invertible knot have a natural orderinggiven by the choice of oriented half-axis. The point at the beginning of the half-axis is thefirst fixed point, and the point at the end of the half-axis is the second fixed point. Definition 2.2.
Two strongly invertible knots (
K, τ ) and ( K (cid:48) , τ (cid:48) ) are equivariantly isotopic or equivalent if there is an orientation-preserving homeomorphism φ : ( S , K ) → ( S , K (cid:48) )such that φ ◦ τ = τ (cid:48) ◦ φ . Furthermore, if K and K (cid:48) are directed and φ preserves the chosenoriented half-axis then we say that ( K, τ ) and ( K (cid:48) , τ (cid:48) ) are equivalent as directed stronglyinvertible knots . Definition 2.3.
Let K and K (cid:48) be directed strongly invertible knots. The equivariant connectsum , K (cid:101) K (cid:48) is the directed strongly invertible knot obtained by cutting K at its second fixedpoint, and K (cid:48) at its first fixed point, then gluing the two knots and axes in the way thatis compatible with the orientations on the axes and choosing the half-axis for K (cid:101) K (cid:48) as theunion of the half-axes for K and K (cid:48) (see Figure 4). Note that there is no twisting ambiguityalong the axis since the knots are strongly invertible. Figure 4.
A directed strongly invertible knot constructed by an equivariantconnect sum of a directed strong inversion on the figure eight knot and adirected strong inversion on the trefoil.
Definition 2.4.
Two directed strongly invertible knots K and K are (smoothly) equivari-antly concordant if there is a smooth proper embedding c : S × I (cid:44) → S × I equivariant withrespect to some smooth involution τ on S × I such that the following hold. QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 7 (i) τ (cid:12)(cid:12) S ×{ i } is the strong inversion on K i for i ∈ { , } .(ii) The orientations on the axes of K and K induce the same orientation on the fixed-point annulus F of τ , and the half-axes of K and K are contained in the samecomponent of F \ c ( S × I ).We denote by (cid:101) C the strongly invertible concordance group, that is, the group with el-ements given by equivalence classes of directed strongly invertible knots under equivariantconcordance, with group operation given by equivariant connect sum. See [Sak86] for a proofthat this group is well-defined. The inverse of a directed strongly invertible knot K ∈ (cid:101) C isthe axis-reverse of the mirror of K . We use the notation C for the usual smooth concordancegroup, and C top for the topological concordance group.There is an obvious group homomorphism f : (cid:101) C → C given by forgetting about the stronginversion on K , which we will refer to as the forgetful map . Note that K is isotopic to itsreverse (since it is strongly invertible), so we do not need to specify an orientation on K todefine f . There is also a group homomorphism r : C → (cid:101) C induced by K (cid:55)→ K rK , withthe strong inversion on K rK exchanging the two summands. Note that this map does notdepend on where the connect sum band is attached to K . The direction on r ( K ) = K rK isdefined as follows. Take the edge of the connect summing band contained in K and translateit along the band to coincide with the half-axis contained in the connect summing band (seeFigure 5). This gives an oriented choice of half-axis. See Figure 6 for an example of thestrong inversion on r ( K ) = K rK . K rK
Figure 5.
A schematic for the chosen directed strong inversion on r ( K ) = K rK . The shaded band is the connect-summing band for K rK .3. Equivariant Genera
In this section, we give a brief introduction to the equivariant 4-genus (cid:101) g ( K ) and definethe butterfly 4-genus (cid:101) bg ( K ) and butterfly 3-genus (cid:101) bg ( K ). In Proposition 4 we give anelementary lower bound on (cid:101) g ( K ) for periodic knots using the Riemann-Hurwitz formula.In Proposition 5 we prove that the Arf invariant obstructs strongly invertible knots frombounding a butterfly Seifert surface in S . Definition 3.1.
Let K be a periodic or strongly invertible knot in S with ρ : ( S , K ) → ( S , K ) the periodic symmetry or strong inversion. A surface F ⊂ B with ∂F = K ⊂ ∂B is an equivariant surface for ( K, ρ ) if F is connected, smoothly properly embedded in B , andthere exists a diffeomorphism ρ : ( B , F ) → ( B , F ) restricting to ρ on ∂B with order( ρ ) =order( ρ ). We call ρ an extension of ρ .In Definition 3.1, ρ can always be extended to B by taking the cone of ρ , although exoticextensions with a knotted fixed point disk are also possible (see for example [Gif66]). Thedifficulty is in finding an extension which respects a given surface. KEEGAN BOYLE AND AHMAD ISSA
Definition 3.2.
The equivariant 4-genus (cid:101) g ( K, ρ ) of a periodic or strongly invertible knot(
K, ρ ) is the minimal genus of an orientable equivariant surface for K . If ρ is clear fromcontext, we simply write (cid:101) g ( K ).In order to study the equivariant genus, it will be helpful to look at symmetric diagramswhich we define precisely here; see Appendix B for some examples. Definition 3.3.
Let (
K, ρ ) be a periodic or strongly invertible knot. A knot diagram for K is (i) transvergent if the order of ρ is 2, and ρ acts as rotation around an axis containedwithin the plane of the diagram, and(ii) intravergent if ρ acts as rotation around an axis perpendicular to the plane of thediagram.In either case, the diagram is called symmetric .Hiura proved that strongly invertible knots bound equivariant Seifert surfaces by con-structing such a surface starting with a transvergent diagram [Hiu17]. Another way to seethis is to first observe that every periodic or strongly invertible knot admits an intraver-gent diagram. Applying Seifert’s algorithm to such a diagram then produces an equivariantSeifert surface, which shows the following proposition. Proposition 1.
Every periodic or strongly invertible knot bounds an equivariant Seifertsurface.
This proposition shows that the equivariant 4-genus always exists by equivariantly pushingthe Seifert surface into B . Furthermore, for an alternating diagram, Seifert’s algorithmconstructs a minimal genus surface [Cro59, Mur58]. Thus for knots with an alternatingintravergent diagram, we obtain that (cid:101) g ( K ) ≤ g ( K ). Note however, Hiura gave examples ofstrongly invertible alternating knots for which there is no equivariant minimal-genus Seifertsurface [Hiu17]. The following proposition gives another useful way to obtain upper boundson (cid:101) g ( K ) from a symmetric diagram. Proposition 2.
Let ( K, ρ ) and ( K (cid:48) , ρ (cid:48) ) be strongly invertible or periodic knots. If there aresymmetric diagrams for K and K (cid:48) which are related by n equivariant crossing changes, then | (cid:101) g ( K ) − (cid:101) g ( K (cid:48) ) | ≤ n. In particular, if K (cid:48) is the unknot, then (cid:101) g ( K ) ≤ n .Proof. The n equivariant crossing changes give an equivariant genus n cobordism from K to K (cid:48) , obtained by attaching a pair of equivariant bands for each crossing. Gluing anequivariant surface for K (or K (cid:48) ) to this cobordism gives the stated inequality. Finally, theequivariant 4-genus of the unknot is always 0 (in the strongly invertible case, this followsfrom [Mar77, Proposition 2]). (cid:3) The equivariant 4-genera of a strongly invertible knot.
Given a strongly invert-ible knot bounding an orientable equivariant surface in B , there is a fixed arc connectingthe fixed points on the knot. If this fixed arc is separating, we call the surface a butterflysurface . (See Figure 6 for justification of the term butterfly surface.) To be precise, we givethe following definition. QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 9
Figure 6.
The bounded checkerboard surface for this diagram is a butterflysurface with boundary 8 r . The strong inversion is rotation around avertical axis. Definition 3.4. A butterfly surface is an orientable compact connected surface S along withan involution ρ on S such that the pointwise fixed set of ρ contains an arc disconnecting S .Let ( K, τ ) be a strongly invertible knot. An equivariant surface S ⊂ B with ∂S = K is a butterfly surface for K if ( S, τ (cid:12)(cid:12) S ) is a butterfly surface, where τ is an extension of τ to B .If K is directed, an equivariant Seifert surface S for K is a butterfly Seifert surface for K if ( S, τ (cid:12)(cid:12) S ) is a butterfly surface and the pointwise fixed arc on S coincides with the chosenhalf-axis.We have a few immediate comments about butterfly surfaces. First, butterfly surfacesnever contain pointwise fixed circles since the fixed arc separates the surface into two com-ponents which are exchanged by the involution. Second, the quotient of a butterfly surfaceis orientable. Third, the symmetry on a butterfly surface guarantees that its genus is alwayseven. Fourth, the genus determines the equivariant homeomorphism type of a butterfly sur-face (see for example [Dug19]). Finally, any disk with an orientation-reversing involution isnecessarily a butterfly surface since any fixed arc disconnects a disk. Definition 3.5.
The butterfly 4-genus (cid:101) bg ( K ) of a strongly invertible knot K is the minimalgenus of a butterfly surface in B for K .Note that (cid:101) g ( K ) ≤ (cid:101) bg ( K ) and that (cid:101) g ( K ) and (cid:101) bg ( K ) are both strongly invertible con-cordance invariants.3.2. The equivariant 4-genus of a periodic knot.
Unlike in the strongly invertible case,an equivariant surface in B with boundary a periodic knot has a 0-dimensional pointwisefixed set. Thus the quotient is a surface with boundary the quotient of the periodic knot.This leads to an inequality on the equivariant 4-genus coming from the Riemann-Hurwitzformula. To begin, we need the following well-known proposition. Proposition 3. If ρ : B → B is a finite order diffeomorphism with fixed-point set a diskthen the quotient of B by ρ is homeomorphic to B .Proof. By [Bre72, Corollary II.6.3 and Theorem III.5.4] the quotient of B is a simply con-nected homology 4-ball. Moreover, since the fixed-point set is a disk, the quotient is atopological manifold and hence homeomorphic to B by work of Freedman [Fre82]. (cid:3) By Edmonds’ theorem [Edm84], every periodic knot has a minimal genus Seifert surfacewhich is equivariant. However (cid:101) g ( K ) is not equal to g ( K ) in general. One way to see this is the following well-known consequence of Proposition 3 and the Riemann-Hurwitz formula;see the proof of [Nai97, Corollary 3.6]. For an example application, see the periodic symmetryon 11 in Appendix B where g (11 ) = 1 and (cid:101) g (11 ) = 3. Proposition 4.
Consider an n -periodic knot K with quotient K , and linking number λ between K and the axis of symmetry (arbitrarily oriented). Then (cid:101) g ( K ) ≥ n · g top ( K ) + ( n − | λ | − . Proof.
Take a minimal genus orientable equivariant surface S in B with ∂S = K , equivariantwith respect to an extension ρ : B → B of the periodic symmetry. By Proposition 3, thequotient of B by ρ is homeomorphic to B . Furthermore, since the periodic symmetrypreserves the orientation on K , it preserves the orientation on S . Hence the fixed-point setof S is a finite set of points and the quotient is therefore a surface S ⊂ B with ∂S = K ⊂ S .Now g ( S ) ≥ g top ( K ), and the axis of symmetry intersects S in at least | λ | points so that S is a branched cover of S over at least | λ | points. The Riemann-Hurwitz formula gives that χ ( S ) = nχ ( S ) − ( n − b, where b is the number of branch points on S . Using that | λ | ≤ b and solving for g ( S ) = (cid:101) g ( K )gives the stated inequality. (cid:3) The equivariant 3-genera of a strongly invertible knot.
The main focus of thispaper is on the 4-genus, but in this section we take a brief detour to discuss some interestingproperties of the equivariant 3-genus and butterfly 3-genus of strongly invertible knots.
Definition 3.6.
The butterfly 3-genus (cid:101) bg ( K ) of a directed strongly invertible knot K is theminimal genus of a butterfly Seifert surface for K . If no such surface exists, then we write (cid:101) bg ( K ) = ∞ . See Figure 7 for some examples. Figure 7.
A butterfly Seifert surface for the directed strong inversion 5 b − (left) on 5 , showing that (cid:101) bg (5 b − ) ≤
2, and a butterfly Seifert surface for thedirected strong inversion 6 b − (right) on 6 , showing that (cid:101) bg (6 b − ) ≤
2. Thefixed arcs are indicated as dotted lines.Note that unlike the butterfly 4-genus, the definition of the butterfly 3-genus depends onthe direction on the strongly invertible knot. The following proposition shows that butterflySeifert surfaces do not always exist.
QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 11
Proposition 5.
Let K be a directed strongly invertible knot. If the Arf invariant a ( K ) = 1 ,then (cid:101) bg ( K ) = ∞ .Proof. Suppose that (cid:101) bg ( K ) < ∞ so that there exists a butterfly surface S ⊂ S with ∂S = K . By definition the fixed arc α separates S into two surfaces S and S . Let K = ∂S and K = ∂S . By thinking of each of S and S as a disk with bands attachedwe see that away from α , we can perform pass moves between the bands of S and S untilthey are unlinked (see [Kau85] for a definition of pass moves). Denote the isotoped unlinkedsurfaces S (cid:48) and S (cid:48) respectively. A boundary connect sum of S (cid:48) and S (cid:48) near α gives a surface S (cid:48) which is pass move equivalent to S . Hence a ( K ) = a ( ∂S ) = a ( ∂S (cid:48) ) since the Arf invariantis preserved by pass moves [Kau85]. However, the additivity of the Arf invariant underconnect sum implies that a ( ∂S (cid:48) ) = a ( K ) + a ( K ) = 0 where we use that K and K areisotopic since the strong inversion exchanges them. (cid:3) There are additional obstructions to the existence of a butterfly Seifert surface (see forexample Proposition 9). We also consider the equivariant 3-genus.
Definition 3.7.
The equivariant 3-genus (cid:101) g ( K ) is the minimal genus of an orientable surface S ⊂ S which has boundary K , and is equivariant with respect to the strong inversion.Edmonds’ theorem [Edm84] states that the equivariant 3-genus of a periodic knot is equalto its (non-equivariant) 3-genus. If an analogous result were to hold for strongly invertibleknots then the equivariant 3-genus would be additive under equivariant connect sum. How-ever, Hiura showed that for strongly invertible knots the equivariant 3-genus is sometimeslarger than the 3-genus [Hiu17]. Nonetheless we are able to show this additivity directly. Wecan also show the additivity of the butterfly 3-genus. Proposition 6.
Given directed strongly invertible knots K and K and an arbitrary knot K , the following hold.(i) (cid:101) bg ( K ) + (cid:101) bg ( K ) = (cid:101) bg ( K (cid:101) K ) .(ii) (cid:101) g ( K ) + (cid:101) g ( K ) = (cid:101) g ( K (cid:101) K ) .(iii) (cid:101) g ( K rK ) = (cid:101) bg ( K rK ) = 2 g ( K ) .Here we use the natural directed strong inversion on K rK ; see the end of Section 2. Recallthat (cid:101) refers to the equivariant connect sum; see Definition 2.3.Proof. First, given a pair of butterfly surfaces for K and K in S , the equivariant boundaryconnect sum of these surfaces is a butterfly surface for K (cid:101) K so that (cid:101) bg ( K ) + (cid:101) bg ( K ) ≥ (cid:101) bg ( K (cid:101) K ). Next, given a minimal genus butterfly surface (cid:101) S for K (cid:101) K , consider theequivariant sphere Σ ⊂ S decomposing K (cid:101) K as a connected sum. After performing anequivariant isotopy we may assume that Σ and (cid:101) S intersect transversely. Then the same typeof argument as in the non-equivariant setting (see for example [Lic97, Chapter 2]) decomposes (cid:101) S as the boundary sum of an equivariant Seifert surface S for K and an equivariant Seifertsurface S for K . Since the fixed arc in (cid:101) S is separating, so are the fixed arcs in S and S .Hence (cid:101) bg ( K ) + (cid:101) bg ( K ) ≤ (cid:101) bg ( K (cid:101) K ), proving (i). A similar argument proves (ii).Finally, a boundary sum of any minimal genus Seifert surface for K with itself gives abutterfly Seifert surface of genus 2 g ( K ) for K rK . (cid:3) Some new strongly invertible concordance invariants
In this section we associate to K a new 2-component link L b ( K ) with linking number 0which we call the butterfly link, and another 2-component link L qb ( K ) with even linkingnumber 2 · (cid:101) lk ( K ) which we call the quotient butterfly link. The integer (cid:101) lk ( K ) is a stronglyinvertible concordance invariant, and we apply Kojima and Yamasaki’s eta polynomial (seeDefinition 4.3) to each of L b ( K ) and (when (cid:101) lk ( K ) = 0) L qb ( K ) to get strongly invertible con-cordance invariants. Unlike Sakuma’s invariant [Sak86], our strongly invertible concordanceinvariants are able to distinguish some knots from their antipode and to obstruct the stronginversions on 8 from being equivariantly slice. We also define the group homomorphisms b and qb . Definition 4.1.
Let K be a directed strongly invertible knot. Consider a band whichattaches to K at the two fixed points of the strong inversion, and which runs parallel to thechosen half-axis. Performing a band move on K along this band produces a 2-componentlink, with linking number depending on the number of twists in the band. The butterfly link L b ( K ) of K is the 2-component 2-periodic link with linking number 0 obtained by this bandmove on K . Both components of L b ( K ) are oriented to agree with the orientation on theaxis on the strands of L b ( K ) parallel to the half-axis. See Figure 3 in the introduction foran example.The following proposition allows us to relate a butterfly surface for a knot K to an equi-variant surface for L b ( K ). Proposition 7.
Let S ⊂ B be a butterfly surface for a directed strongly invertible knot ( K, τ ) . Then the butterfly link L b ( K ) bounds a two component orientable surface S (cid:48) = S ∪ S in B with g ( S ) = g ( S ) = g ( S ) , and the components are exchanged by the extension of τ to B corresponding to S . Moreover, there exists an equivariant ( I × D ) ⊂ B such that ( ∂I ) × D ⊂ S (cid:48) , I × ( ∂D ) ⊂ ( S ∪ B ) and S (cid:48) = [ S \ ( I × ∂D )] ∪ [( ∂I ) × D ] , where B ⊂ S is a band with the corresponding band move on K giving L b ( K ) .Proof. Let τ : ( B , S ) → ( B , S ) be an extension of τ , with fixed set a disk F ⊂ B . Then S ∩ F is an arc α separating F into two components. Since ( K, τ ) is directed, there isa preferred choice of half-axis in S which is contained in a component D of F − α . Wecan further choose an equivariant tubular neighborhood N ( D ) of D and identify ∂N ( D )with the unit tangent bundle over D . Then ∂N ( D ) ∩ S is an S -subbundle of ∂N ( D ) | α .Consider an equivariant band B in S containing the chosen half-axis and intersecting K ina neighborhood of each fixed point. We can choose B so that ∂B − K is contained in ∂N ( D )and hence B ∪ S intersects ∂N ( D ) in an S -subbundle of ∂N ( D ) | ∂D . Note that there arefurthermore many choices of B corresponding to twists around the axis. Choose B so thatthe S -subbundle extends across all of D to an S -subbundle E of ∂N ( D ), which naturallyhas two sections D and D . We can now perform a band move on K along B to obtain alink L which bounds the (disconnected) surface[ S − N ( D )] ∪ D ∪ D . Then L is a 2-component link with linking number 0 since the components bound disjointsurfaces in B . Hence L = L b ( K ). Finally, we can extend E to an equivariant I -subbundleof N ( D ) to get the desired ( I × D ) ⊂ S with the properties stated in the proposition. (cid:3) QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 13
Remark 4.2.
In Proposition 7 we moreover have that if the 2-periodic link L b ( K ) boundsan equivariant 2-component surface in B (which is reconnected by the surgery band), thenreattaching the surgery band gives an equivariant surface for the strongly invertible knot. Inparticular, if L b ( K ) is an equivariantly slice link, then K is an equivariantly slice stronglyinvertible knot.Now that we have defined the butterfly link L b ( K ), we can obtain a strongly invertibleconcordance invariant by applying the Kojima-Yamasaki η -polynomial, which we now recall. Definition 4.3 ([KY79]) . Let L = K ∪ K be a 2-component link with lk( K , K ) = 0 andlet X = S \ K . Let (cid:101) X be the infinite cyclic cover of X so that H ( (cid:101) X ) is naturally a Z [ t, t − ]-module where t generates the group of covering transformations. Let (cid:96) be a homologicallongitude of K and let (cid:101) (cid:96) and (cid:101) K be nearby lifts of (cid:96) and K to (cid:101) X . Let f ( t ) ∈ Z [ t, t − ] sothat f ( t )[ (cid:101) (cid:96) ] = 0 ∈ H ( (cid:101) X ). Finally, let ζ be a 2-chain in (cid:101) X with ∂ζ = f ( t ) · (cid:101) (cid:96) and ζ transverseto t i (cid:101) K for all i . Then the Kojima-Yamasaki η -polynomial is η ( L ) = 1 f ( t ) ∞ (cid:88) i = −∞ Int( ζ, t i (cid:101) K ) t i ∈ Z [ t, t − ] , where Int( ζ, t i (cid:101) K ) is the signed count of intersection points between ζ and t i (cid:101) K in (cid:101) X . Remark 4.4.
The coefficient of t i in η ( L ) can be thought of as the linking number between t i (cid:101) K and (cid:101) K in (cid:101) X . In general, η ( L ) depends on the order of K and K .We will use the following properties of the η -polynomial. Theorem 7 ([KY79]) . Let L = K ∪ K be a 2-component link with lk ( K , K ) = 0 . Then η ( L ) has the following properties.(i) η ( L )( t ) = η ( L )( t − ) (ii) η ( L )(1) = 0 (iii) η ( L ) does not depend on the orientation of L .(iv) η ( L ) is a topological link concordance invariant. More precisely, if L ⊂ S × { } and L (cid:48) ⊂ S × { } bound a pair of disjoint topologically locally flatly embedded cylindersin S × I , then η ( L ) = η ( L (cid:48) ) . Moreover if L is the unlink then η ( L ) = 0 . Definition 4.5.
The butterfly polynomial η ( L b ( K )) of a strongly invertible knot K isKojima-Yamasaki’s eta polynomial applied to the butterfly link of K . Note that this doesnot depend on an ordering of the components of the link since L b ( K ) is symmetric.The following lemma shows that the η -polynomial is additive in certain circumstances. Lemma 1.
For i ∈ { , } , let L i = J i ∪ K i be a 2-component link in S . Let L = J ∪ K be a component-wise connect sum of L and L for which there is a connect-summing sphere Σ ⊂ S separating L and L which simultaneously decomposes J = J J and K = K K .If lk ( J i , K i ) = 0 for i ∈ { , } , then η ( L ) = η ( L ) + η ( L ) . Proof.
Let (cid:101) X J be the infinite cyclic cover of S \ J . In this infinite cyclic cover, Σ \ J liftsto an infinite strip (cid:101) Σ ∼ = R × (0 , (cid:101) X J along (cid:101) Σ splits (cid:101) X J into two components: (cid:101) X J and (cid:101) X J , which are the infinite cyclic covers of S \ J and S \ J respectively. Let (cid:101) K, (cid:101) (cid:96) K ⊂ (cid:101) X J be nearby lifts of K and the homological longitude of K respectively. Then (cid:101) K = (cid:101) K (cid:101) K and (cid:101) (cid:96) K = (cid:101) (cid:96) K (cid:101) (cid:96) K , where (cid:102) K i and (cid:101) (cid:96) K i are nearby lifts of K i and a homologicallongitude of K i for i ∈ { , } . Let f ( t ) ∈ Z [ t, t − ] such that f ( t )[ (cid:101) (cid:96) K ] = 0 ∈ H ( (cid:101) X J )and f ( t )[ (cid:101) (cid:96) K ] = 0 ∈ H ( (cid:101) X J ). Then for i ∈ { , } , let ζ i be a 2-chain in (cid:101) X J i ⊂ (cid:101) X J with ∂ζ i = f ( t ) · (cid:101) (cid:96) K i transverse to all translates of (cid:101) K i . Let ζ be the 2-chain in (cid:101) X J with ∂ζ = f ( t ) · (cid:101) K which is obtained by gluing ζ and ζ together along bands realizing translates of the connectsum (cid:101) (cid:96) K = (cid:101) (cid:96) K (cid:101) (cid:96) K . Then the points of intersection between ζ and t i (cid:101) K are precisely thepoints of intersection between ζ and t i (cid:101) K and between ζ and t i (cid:101) K . Thus for all i we havethat Int( ζ, t i (cid:101) K ) = Int( ζ , t i (cid:101) K ) + Int( ζ , t i (cid:101) K ) . Hence η ( L ) = η ( L ) + η ( L ). (cid:3) As a consequence of this lemma, we obtain the following proposition.
Proposition 8.
The butterfly polynomial η ( L b ( − )) : (cid:101) C → Z [ t, t − ] is a group homomorphismfrom the strongly invertible concordance group to the additive group of (symmetric) Laurentpolynomials.Proof. Let K and K (cid:48) be directed strongly invertible knots. It is enough to check that if K is equivariantly slice, then η ( L b ( K )) = 0, and that η ( L b ( K (cid:101) K (cid:48) )) = η ( L b ( K )) + η ( L b ( K (cid:48) )) . If K is slice, then the 2-component link L b ( K ) is also slice by Proposition 7. Hence byTheorem 7, η ( L b ( K )) = 0. Next, the equivariant connect-summing sphere for K (cid:101) K (cid:48) realizes L b ( K (cid:101) K (cid:48) ) as a component-wise connect sum of L b ( K ) and L b ( K (cid:48) ) as in Lemma 1. Hence η ( L b ( K (cid:101) K (cid:48) )) = η ( L b ( K )) + η ( L b ( K (cid:48) )) . (cid:3) Using the butterfly link, we define some other interesting invariants.
Definition 4.6.
The axis-linking number (cid:101) lk ( K ) of K is the linking number between onecomponent of L b ( K ) and the axis of symmetry. Definition 4.7.
The quotient butterfly link L qb ( K ) of K is the 2-component link consistingof the quotient of L b ( K ) by the 2-periodic action and the axis in the quotient.Note that the linking number between the components of L qb ( K ) is 2 · (cid:101) lk ( K ). The followingproposition shows that (cid:101) lk ( K ) provides an obstruction to the existence of a butterfly surfacein B with boundary K . Proposition 9.
Let K be a directed strongly invertible knot. If K bounds a butterfly surfaceof genus g in B , then L qb ( K ) bounds an orientable two component surface topologicallylocally flatly embedded in B consisting of a disk with boundary the axis and a genus g/ surface bounding the other component of L qb ( K ) . In particular, (cid:101) lk ( K ) = 0 .Proof. Suppose K bounds a genus g butterfly surface in B , invariant with respect to anextension τ : B → B of the strong inversion and let F ⊂ B be the disk of fixed points.Then by Proposition 7, the 2-component link L b ( K ) bounds an orientable two componentsurface S = S ∪ S , with the components exchanged by τ . This surface is disjoint from QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 15 F . Taking the quotient of ( B , S, F ) by τ gives a pair of orientable surfaces with boundary L qb ( K ) in a topological 4-ball by Proposition 3. The quotient of F is a disk, and the quotientof S is a genus g/ L qb ( K ) have linking number 0 so that (cid:101) lk ( K ) = 0. (cid:3) Proposition 10.
The axis-linking number (cid:101) lk ( − ) : (cid:101) C → Z and the eta polynomial of thequotient butterfly link η ( L qb ( − )) : ker ( (cid:101) lk ) → Z [ t, t − ] are group homomorphisms.Proof. Let K and K (cid:48) be directed strongly invertible knots. It is enough to check the following.(1) If K is equivariantly slice then (cid:101) lk ( K ) = 0 and η ( L qb ( K )) = 0.(2) (cid:101) lk ( K (cid:101) K (cid:48) ) = (cid:101) lk ( K ) + (cid:101) lk ( K (cid:48) ).(3) If (cid:101) lk ( K ) = (cid:101) lk ( K (cid:48) ) = 0, then η ( L qb ( K (cid:101) K (cid:48) )) = η ( L qb ( K )) + η ( L qb ( K (cid:48) )).To show (1), suppose K is equivariantly slice. By Proposition 9, (cid:101) lk ( K ) = 0 and L qb ( K )bounds a pair of disjoint disks in B . Hence by Theorem 7, η ( L qb ( K )) = 0. To show (2) and(3), consider the quotient of the equivariant connect-summing sphere for K (cid:101) K (cid:48) by τ . Thisquotient sphere decomposes L qb ( K (cid:101) K (cid:48) ) as a component-wise connect sum of L qb ( K ) and L qb ( K (cid:48) ) as in Lemma 1. This decomposition implies (cid:101) lk ( K ) + (cid:101) lk ( K (cid:48) ) = (cid:101) lk ( K (cid:101) K (cid:48) ), and byLemma 1 that η ( L qb ( K (cid:101) K (cid:48) )) = η ( L qb ( K )) + η ( L qb ( K (cid:48) )) provided (cid:101) lk ( K ) = (cid:101) lk ( K (cid:48) ) = 0. (cid:3) Example 4.8.
Consider the knot K = K13n1496 with the strong inversion shown in Figure8. A straightforward computation shows that K has trivial Alexander polynomial, whichby work of Freedman [Fre82, Theorem 1.13] implies that K is topologically slice. On theother hand, Rasmussen’s s -invariant is 2 so that K is not smoothly slice. With respect tothe strong inversion, η ( L qb ( K )) = 2 t − − t (cid:54) = 0. Hence by a similar argument asin Proposition 9, a topological slice disk can not be made equivariant with respect to anydiffeomorphism B → B extending the strong inversion on K . In particular, Freedman’sconstruction of a topological slice disk can not be made equivariantly with respect to adiffeomorphism extending this strong inversion. Figure 8.
A directed strong inversion K13n1496 + . Theorem 5.
The maps b : (cid:101) C → C and qb : (cid:101) C → C top defined as follows are surjective grouphomomorphisms.(i) b ( K ) is the smooth concordance class of one component of L b ( K ) . (ii) qb ( K ) is the topological concordance class of the non-axis component of L qb ( K ) .Proof. It suffices to show that if K is equivariantly smoothly slice then b ( K ) is smoothly sliceand qb ( K ) is topologically slice, that b ( K (cid:101) K ) is isotopic (and hence smoothly concordant)to b ( K ) b ( K ), and that qb ( K (cid:101) K ) is isotopic (and hence topologically concordant) to qb ( K ) qb ( K ).Suppose K is equivariantly slice. Then as we have seen in the proof of Proposition 8,each component of L b ( K ) bounds a smooth disk in B . In particular b ( K ) is smoothly slice.Furthermore, the two components bound a disjoint equivariant pair of disks, so that theimage of one disk in the quotient has no self-intersection points. Then since the quotient isa topological B by Proposition 3, qb ( K ) is topologically slice.Now consider K (cid:101) K and an equivariant sphere S separating K and K . Then the same S realizes b ( K (cid:101) K ) as b ( K ) b ( K ). Furthermore, since S is equivariant the quotientrealizes qb ( K (cid:101) K ) as qb ( K ) qb ( K ). (cid:3) Remark 4.9.
Note that b ( K ) can be easily computed directly as the union of the chosen half-axis and (either) half of the strongly invertible knot. For example in Figure 7, b (5 b − ) = 3 and b (6 b − ) = 4 are both immediately visible from the diagrams for the butterfly surfaces.Recall the group homomorphism r : C → (cid:101) C given by r ( K ) = K rK defined at the end ofSection 2. Note that b ( r ( K )) = K and qb ( r ( K )) = K . This implies that r is injective, and b and qb may be thought of as retracts from (cid:101) C → C and (cid:101)
C → C top respectively.The invariants η ( L b ( K )) , η ( L qb ( K )) , and (cid:101) lk ( K ) behave nicely under taking the axis-reverse(reversing the orientation on the axis) or mirror of K (see Proposition 11), but seem to havesubtle behavior when taking the antipode (taking the other choice of half-axis). This is anadvantage over Sakuma’s polynomial however, since these invariants sometimes distinguishantipodes in the strongly invertible concordance group as seen in the following example. Example 4.10.
Consider the two elements of the strongly invertible concordance group 3 +1 and 3 − given by the two choices of half-axis for the unique strong inversion on the right-handed trefoil (see the table in Appendix A). We have that qb (3 +1 ) is the unknot, but qb (3 − )is the left-handed trefoil. Since the unknot and trefoil are not concordant in the topologicalconcordance group, 3 +1 and 3 − are not strongly invertibly concordant by Theorem 5. Inparticular 3 +1 (cid:101) − ), where inv(3 − ) is the inverse of 3 − in (cid:101) C , is slice but not equivariantlyslice. Proposition 11.
Let K be a directed strongly invertible knot with mirror mK and axis-reverse aK . (The axis-reverse is K with the orientation on the axis reversed.) Then(i) η ( L b ( K )) = η ( L b ( aK )) = − η ( L b ( mK )) ,(ii) η ( L qb ( K )) = η ( L qb ( aK )) = − η ( L qb ( mK )) provided η ( L qb ( K )) is defined, and(iii) (cid:101) lk ( K ) = (cid:101) lk ( aK ) = − (cid:101) lk ( mK ) .Proof. Theorem 7 implies that η ( L b ( K )) = η ( L b ( aK )) and η ( L qb ( K )) = η ( L qb ( aK )). Sincereversing the orientation of both components of the link does not change the linking number,we also have that (cid:101) lk ( K ) = (cid:101) lk ( aK ). Now since η ◦ L b is additive by Proposition 8, we knowthat η ( L b ( K )) = − η ( L b ( maK )) = − η ( L b ( mK )) (since maK is the inverse of K in (cid:101) C ), andsimilarly for (cid:101) lk and η ◦ L qb by Proposition 10. (cid:3) QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 17 Equivariant Donaldson’s theorem obstruction
In this section we prove Theorem 1 and use it to give examples of strongly invertibleand periodic knots where the equivariant 4-genus is different from the (non-equivariant) 4-genus (see Appendix B for a table of examples). To do so we consider the question of whena 3-manifold with a symmetry bounds an equivariant positive-definite 4-manifold and useDonaldson’s theorem to obtain an obstruction in some cases. We start by showing thatcertain symmetries of B can be lifted to Σ( B , F ), the double branched cover of B over anequivariant surface F . Proposition 12.
Let F ⊂ S be a closed connected (not necessarily orientable) surface andlet ρ : S → S be an orientation-preserving finite order diffeomorphism such that ρ ( F ) = F .Suppose that ρ has a fixed point in S \ F . Let q : Σ( S , F ) → S be the double branchedcovering projection. Then ρ lifts to a smooth map (cid:101) ρ : Σ( S , F ) → Σ( S , F ) , that is, q ◦ (cid:101) ρ = ρ ◦ q and order ( (cid:101) ρ ) = order ( ρ ) . Furthermore there are exactly two such lifts, and if A is the fixed-point set of ρ and C is a component of A \ F , then exactly one of these two lifts fixes q − ( C ) pointwise.Proof. By the Equivariant Tubular Neighborhood theorem there is a ρ -equivariant tubularneighborhood of F which we denote by N ( F ) ⊂ S . Let X = S \ int( N ( F )) be the exteriorof F in S . Denote by (cid:101) X the two-fold cyclic cover of X corresponding to the kernel of π ( X ) → H ( X ; Z / Z ) ∼ = Z / Z . Let s ∈ X be a fixed point under ρ and let (cid:101) s ∈ q − ( s ) ⊂ (cid:101) X be a lift of s to (cid:101) X . We implicitly use s (resp. (cid:101) s ) as the basepoint of π ( X ) (resp. π ( (cid:101) X )).Let G ≤ π ( X ) denote the image of π ( q ) : π ( (cid:101) X ) → π ( X ). Since G is the uniqueindex 2 (and hence normal) subgroup of π ( X ), it is a characteristic subgroup. Hence, theimage of π ( ρ ◦ q ) : π ( (cid:101) X ) → π ( X ) is also G . By the covering space lifting property, sinceIm π ( ρ ◦ q ) ⊆ Im π ( q ), there exists a unique map (cid:101) ρ : ( (cid:101) X, (cid:101) s ) → ( (cid:101) X, (cid:101) s ) such that q ◦ (cid:101) ρ = ρ ◦ q .The tubular neighborhood N ( F ) ⊂ S has the structure of a D -bundle over F . Thebranched cover decomposes as Σ( S , F ) = (cid:101) X ∪ ∂ (cid:101) X W , where W = q − ( N ( F )) is naturally a D -bundle over F ∼ = q − ( F ) by lifting the fibers of N ( F ) to Σ( S , F ). It suffices to showthat (cid:101) ρ (cid:12)(cid:12) ∂ (cid:101) X extends to a map on W such that q ◦ (cid:101) ρ = ρ ◦ q holds on all of Σ( S , F ). By abuseof notation we also denote the zero section of W by F . Denote by W (cid:12)(cid:12) x the D fiber of W over x ∈ F ⊂ W . By the equivariant tubular neighborhood theorem, the action of ρ on N ( F ) ⊂ S is fiber preserving. Hence, the map (cid:101) ρ restricts to ∂W (cid:12)(cid:12) x → ∂W (cid:12)(cid:12) ρ ( x ) and this canbe extended over the disk W (cid:12)(cid:12) x to a map W (cid:12)(cid:12) x → W (cid:12)(cid:12) ρ ( x ) . This can be done smoothly over all x ∈ F ⊂ W so that q ◦ (cid:101) ρ = ρ ◦ q .To see that order( (cid:101) ρ ) = order( ρ ), note by induction that q ◦ (cid:101) ρ i = ρ i ◦ q for i ≥
1. Inparticular this implies that (cid:101) ρ i (cid:54) = Id for 1 ≤ i < order( ρ ) and q ◦ (cid:101) ρ n = q for n = order( ρ ).Hence (cid:101) ρ n is a branched covering transformation fixing the point (cid:101) s which is disjoint from thebranch set q − ( F ). Thus (cid:101) ρ n = Id so order( (cid:101) ρ ) = n = order( ρ ).The above construction produces the unique lift (cid:101) ρ fixing (cid:101) s . Composing with the (branched)deck transformation gives another lift exchanging the two points in q − ( s ), and these are theonly two lifts of ρ . Let C be a component of A \ F where A is the fixed-point set of ρ , andlet (cid:101) C = q − ( C ). We argue that there is a unique lift (cid:101) ρ : Σ( S , F ) → Σ( S , F ) of ρ whichfixes (cid:101) C pointwise. To see the existence of such a lift, first choose s ∈ C and let (cid:101) s be a liftof s to (cid:101) C in the construction above. Note that (cid:101) ρ | (cid:101) C is a deck transformation of the two-fold cover q | (cid:101) C . Since (cid:101) ρ has a fixed point (cid:101) s and C is connected, (cid:101) ρ must restrict to the identityon (cid:101) C . Thus the lift (cid:101) ρ | (cid:101) X is independent of the choice of (cid:101) s . The uniqueness of (cid:101) ρ follows fromthe uniqueness of the above construction by noting that (cid:101) ρ is independent of the particularchoice of (cid:101) s ∈ (cid:101) C since all of (cid:101) C is fixed pointwise. (cid:3) As a corollary we have the following version of Proposition 12 for surfaces in B . Corollary 1.
Let K ⊂ S be a knot and F ⊂ B a properly embedded, connected (notnecessarily orientable) surface with ∂F = K . Let ρ : B → B be an orientation-preservingfinite order diffeomorphism such that ρ ( F ) = F . Suppose that ρ has a fixed point in B \ F .Let q : Σ( B , F ) → B be the double branched covering projection. Then ρ lifts to a smoothmap (cid:101) ρ : Σ( B , F ) → Σ( B , F ) , that is, q ◦ (cid:101) ρ = ρ ◦ q and order ( (cid:101) ρ ) = order ( ρ ) . Furthermorethere are exactly two such lifts, and if A is the fixed-point set of ρ and C is a component of A \ F , then exactly one of these two lifts fixes q − ( C ) pointwise.Proof. Double ( B , F ) to obtain a closed connected surface F in S . Apply Proposition 12to get a lift (cid:101) ρ : Σ( S , F ) → Σ( S , F ). Restricting (cid:101) ρ to Σ( B , F ) gives the desired lift. (cid:3) In order to have well-defined invariants, we now pin down one of the two lifts from Corollary1.
Definition 5.1.
Let K ⊂ S be a periodic knot bounding an equivariant surface ( F, ρ ) in B . Let A be the fixed point set of ρ and let C be the component of A \ F containing theaxis in S . The distinguished lift of ρ is the unique lift (cid:101) ρ : Σ( B , F ) → Σ( B , F ) pointwisefixing q − ( C ), where q : Σ( B , F ) → B is the branched covering projection. Definition 5.2.
Let K ⊂ S be a directed strongly invertible knot bounding an equivariantsurface ( F, ρ ) in B . The axis of symmetry in S is made up of two half-axes: the half-axisdistinguished by the direction on K , and the other half-axis η . Let C be the componentof A \ F containing η . The distinguished lift of ρ is the unique lift (cid:101) ρ : Σ( B , F ) → Σ( B , F )pointwise fixing q − ( C ), where q : Σ( B , F ) → B is the branched covering projection. Remark 5.3.
In the periodic case the distinguished lift in fact pointwise fixes all of q − ( A ),however we will not make use of this. In the strongly invertible case the non-distinguishedlift fixes q − ( C (cid:48) ) where C (cid:48) is the component of A \ F containing the half-axis distinguishedby the direction on K .Before proving Theorem 1, we show that for an equivariant spanning surface the action ofthe distinguished lift on the intersection form can be simply described in terms of the Gordon-Litherland form [GL78]. To begin, we briefly recall the Gordon-Litherland form. Let K be aknot in S and let F be a compact, connected (not necessarily orientable) surface embeddedin S with ∂F = K . Gordon and Litherland define a bilinear form G F : H ( F ) × H ( F ) → Z as follows. Suppose α, β ∈ H ( F ) are represented by embedded oriented multicurves a , b in F . We can push off 2 b into S \ F , obtaining b . (Locally, one copy of b is pushed offeach side of F .) Define G F ( α, β ) := Lk( a, b ) to be the linking number of a and b . Gordonand Litherland show that the form G F is a well-defined symmetric bilinear form. If F isorientable, G F is the symmetrized Seifert form.Let (cid:98) F be the surface obtained from F by pushing int( F ) into int( B ). Let Σ( B , (cid:98) F ) denotethe double branched cover of B over (cid:98) F . Gordon and Litherland showed that ( H ( F ) , G F ) ∼ =( H (Σ( B , (cid:98) F )) , Q ), where Q is the intersection pairing on H (Σ( B , (cid:98) F )). QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 19
Proposition 13.
Let K be a knot in S and suppose it has a strongly invertible or periodicsymmetry ρ : ( S , K ) → ( S , K ) . Let F be a compact, connected embedded surface in S with ∂F = K and ρ ( F ) = F . Let (cid:101) ρ : Σ( B , (cid:98) F ) → Σ( B , (cid:98) F ) be a lift of ρ (as in Corollary 1).Then under the identification of the intersection form ( H (Σ( B , (cid:98) F )) , Q ) ∼ = ( H ( F ) , G F ) ,the induced map of lattices (cid:101) ρ ∗ : ( H (Σ( B , (cid:98) F )) , Q ) → ( H (Σ( B , (cid:98) F )) , Q ) is equivalent to ± ( ρ (cid:12)(cid:12) F ) ∗ : ( H ( F ) , G F ) → ( H ( F ) , G F ) . Furthermore:(i) If K is periodic and (cid:101) ρ is the distinguished lift of ρ (as in Definition 5.1), then theinduced map on lattices is equivalent to +( ρ | F ) ∗ .(ii) Suppose K is strongly invertible and directed, and (cid:101) ρ is the distinguished lift of ρ (asin Definition 5.2). The direction distinguishes a half-axis η + ; call the other half-axis η − . The induced map on lattices is equivalent to +( ρ | F ) ∗ if F contains η + , and isequivalent to − ( ρ | F ) ∗ if F contains η − .Proof. The symmetry ρ : S → S extends to a symmetry ρ : B → B given by the cone of ρ . Following [GL78, proof of Theorem 3], Σ( B , (cid:98) F ) can be constructed as follows. Let D denote the manifold obtained by cutting open B along the trace of an equivariant isotopywhich pushes int( F ) into int( B ). The manifold D is homeomorphic to B and the partexposed by the cut is given by an equivariant tubular neighborhood N of F in S . Let ι : N → N be the involution given by reflecting each fiber. Let D be another copy of D .Then Σ( B , (cid:98) F ) = ( D ∪ D ) / ( x ∈ N ⊂ D ∼ ι ( x ) ∈ N ⊂ D ) . The isomorphism φ : ( H ( F ) , G F ) → ( H (Σ( B , (cid:98) F )) , Q ) is given as follows. Let a be a 1-cyclein F , then φ ([ a ]) = [(cone on a in D ) − (cone on a in D )] . Let (cid:101) ρ : Σ( B , (cid:98) F ) → Σ( B , (cid:98) F ) be a lift of ρ : B → B as in Corollary 1. Note that there aretwo cases: (cid:101) ρ maps the interior of D to the interior of D , or (cid:101) ρ maps the interior of D tothe interior of D . We first assume that it is mapped into the interior of D . The map (cid:101) ρ restricted to the branched set F is equal to ρ (cid:12)(cid:12) F . Thus, (cid:101) ρ ( a ) = ρ ( a ). For i ∈ { , } , the coneof a in D i is mapped to a disk in D i with boundary ρ ( a ). Hence, (cid:101) ρ ∗ ( φ ([ a ])) = [(cone on ρ ( a ) in D ) − (cone on ρ ( a ) in D )]= φ ( ρ ([ a ])) . If instead (cid:101) ρ maps the interior of D into the interior of D , we similarly get (cid:101) ρ ∗ ( φ ([ a ])) = [(cone on ρ ( a ) in D ) − (cone on ρ ( a ) in D )]= − φ ( ρ ([ a ])) . Now let q : Σ( B , (cid:98) F ) → B be the branched covering map. If K is periodic with axis γ ,and (cid:101) ρ is the distinguished lift (see Definition 5.1), then q − ( γ ) ⊂ Σ( B , (cid:98) F ) is pointwise fixedby (cid:101) ρ . Since ∂D \ N contains points of q − ( γ ), (cid:101) ρ (int( D )) = int( D ). Hence the induced mapof lattices is given by +( ρ | F ) ∗ .Alternatively, suppose K is strongly invertible and directed. The fixed-point axis in S consists of a half-axis η + distinguished by the direction on K , and another half-axis η − .Suppose furthermore that η + ⊂ F and that (cid:101) ρ is the distinguished lift (see Definition 5.2).Then (cid:101) ρ fixes q − ( η − ) pointwise and q − ( η − ) (cid:54)⊂ N . Hence (cid:101) ρ has fixed points in ∂D \ N andso (cid:101) ρ (int( D )) = int( D ). Thus the induced map of lattices is given by +( ρ | F ) ∗ . If (cid:101) ρ pointwise fixes q − ( η − ) then (cid:101) ρ pointwise fixes q − ( η + ) ∪ q − ( η − ), which is topologicallya graph with two vertices and four parallel edges. This is impossible since the fixed-point setof (cid:101) ρ | ∂ Σ( B , (cid:98) F ) must be a 1-manifold. Hence (cid:101) ρ interchanges the two components of q − (Int( η + )).If F contains η − then q − (Int( η + )) has a component in D \ N and a component in D \ N .Hence (cid:101) ρ must be the lift interchanging int( D ) and int( D ). Thus the induced map of latticesis given by − ( ρ | F ) ∗ . (cid:3) We now have the tools needed to prove the main theorem of this section, which cansometimes be used to show that (cid:101) g ( K ) > | σ ( K ) | / Theorem 1.
Let K ⊂ S be a knot with a periodic or strongly invertible symmetry ρ : S → S . Suppose that F ⊂ S is a spanning surface for K for which the Gordon-Litherland pairing G F is positive definite and ρ ( F ) = F . If (cid:101) g ( K ) = − σ ( K ) / then there is an embedding oflattices ι : ( H ( F ) , G F ) → ( Z k , Id ) such that δ ◦ ι = ι ◦ ρ ∗ , where δ is an automorphism of ( Z k , Id ) with order ( δ ) = order ( ρ ) and k = − σ ( K ) / b ( F ) . In particular, the followingdiagram commutes. ( H ( F ) , G F ) ( Z k , Id)( H ( F ) , G F ) ( Z k , Id) ρ ∗ ι δι Proof.
Let ρ F : B → B be the cone on ρ : S → S and let ˆ F ⊂ B be a properlyembedded surface equivariantly isotopic to F . Let S ⊂ B be an orientable surface whichis equivariant with respect to some extension ρ S : B → B of ρ such that ∂S = K and g ( S ) = (cid:101) g ( K ) = − σ ( K ) /
2. If K is strongly invertible, we choose a direction on K so thatthe distinguished half-axis is contained in F . Now let (cid:101) ρ ˆ F : Σ( B , ˆ F ) → Σ( B , ˆ F ) be thedistinguished lift of ρ ˆ F and (cid:101) ρ S : Σ( B , S ) → Σ( B , S ) be the distinguished lift of ρ S as inDefinition 5.1 or 5.2. Gluing these together, we obtain( X, (cid:101) ρ ) = (Σ( B , ˆ F ) , (cid:101) ρ ˆ F ) ∪ ( − Σ( B , S ) , (cid:101) ρ S ) . Since rank( H (Σ( B , S ))) = rank(2 g ( S )) = − σ ( K ) (see for example [GL11, Lemma 1]) and σ ( − Σ( B , S )) = − σ ( K ) (see [KT76]), we have that − Σ( B , S ) is positive definite. Theintersection form on Σ( B , ˆ F ) is isomorphic to G F by [GL78, Theorem 3]. Hence Σ( B , ˆ F )is positive definite by hypothesis and therefore X is positive definite (see for example [IM20,Proposition 7]).Since X is smooth and positive definite, Donaldson’s theorem [Don87] implies that theintersection form ( H ( X ) , Q X ) is isomorphic to ( Z k , Id), where k = rank H ( X ) = rank H (Σ( B , S )) + rank H (Σ( B , ˆ F )) = − σ ( K ) + b ( F ) . Let ι : Σ( B , ˆ F ) → X be the inclusion map. We then have the commutative diagram on theleft of Figure 9 where all maps preserve intersection pairings. The commutative diagram onthe right of Figure 9 is then obtained by identifying ( H (Σ( B , ˆ F )) , Q Σ( B , ˆ F ) ) ∼ = ( H ( F ) , G F )and ( H ( X ) , Q X ) ∼ = ( Z k , Id), and noting that (cid:16)(cid:101) ρ (cid:12)(cid:12) Σ( B ,F ) (cid:17) ∗ = (cid:0) ρ (cid:12)(cid:12) F (cid:1) ∗ by Proposition 13.Finally, (cid:101) ρ ∗ : Z k → Z k is an automorphism since (cid:101) ρ is a diffeomorphism, and order( (cid:101) ρ ∗ ) =order( (cid:101) ρ ) = order( ρ ). (cid:3) QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 21 H (Σ( B , ˆ F )) H ( X ) ( H ( F ) , G F ) ( Z k , Id ) H (Σ( B , ˆ F )) H ( X ) ( H ( F ) , G F ) ( Z k , Id ) (cid:101) ρ ∗ ι ∗ ι ∗ (cid:16)(cid:101) ρ (cid:12)(cid:12) Σ( B ,F ) (cid:17) ∗ (cid:101) ρ ∗ ι ∗ ι ∗ (cid:0) ρ (cid:12)(cid:12) F (cid:1) ∗ = Figure 9.
A commutative diagram for the equivariant lattice embedding usedin the proof of Theorem 1.To apply this theorem, it will be convenient to recall that given a knot diagram K ,the Gordon-Litherland form of a checkerboard surface is determined by the correspondingcheckerboard graph which we briefly describe (see [GL78] for details). Choosing a black andwhite checkerboard coloring of the knot diagram determines a black checkerboard surface F spanning K . We denote by G ( F ) the planar multigraph with vertices corresponding to whiteregions of the checkerboard coloring and edges corresponding to crossings between pairs ofdistinct white regions. Additionally, we weight the edges of G ( F ) with ± − G ( F ) with negative the sum of the weightsof incident edges and denote by w ( e ) and w ( v ) the weight of an edge e and a vertex v respectively. See Figure 10 for an example. We call G ( F ) the checkerboard graph of F .Now H ( F ) ∼ = Z (cid:104) v , v , . . . , v n (cid:105) / ( v + v + · · · + v n ) where v , v , . . . , v n are the verticesof G ( F ). This isomorphism is given by mapping v i to the class of the loop in F goingcounterclockwise once around the white region corresponding to v i . Under this isomorphism,the Gordon-Litherland form G F is determined by (cid:104) v i , v j (cid:105) = (cid:40) w ( v i ) , i = j (cid:80) e ∈ E ( v i ,v j ) w ( e ) , i (cid:54) = j, where E ( v i , v j ) is the set of edges between v i and v j .Suppose ( K, ρ ) is a periodic or strongly invertible knot which has an alternating in-travergent or transvergent diagram (see Definition 3.3). This diagram then has an equi-variant checkerboard surface F with positive definite Gordon-Litherland form and the sym-metry of the diagram induces a planar symmetry of the checkerboard graph. The map ρ ∗ : H ( F ) → H ( F ) can then be determined directly from the planar symmetry of thecheckerboard graph.Theorem 1 may be used to potentially obstruct (cid:101) g ( K ) = − σ ( K ) / ι : ( H ( F ) , G F ) → ( Z k , Id) making the diagram in Theorem 1commute. More precisely, first enumerate all lattice embeddings ( H ( F ) , G F ) → ( Z k , Id)(there are finitely many) which could potentially serve as such an ι . For each embedding,then show that there does not exist a map δ : ( Z (cid:104) e , . . . , e k (cid:105) , Id) → ( Z (cid:104) e , . . . , e k (cid:105) , Id) makingthe diagram commute. Note that δ maps each e i to ± e j for some j and hence there are onlyfinitely many possibilities for δ . In some cases this δ may exist so that Theorem 1 does notprovide an obstruction.In the case that g ( K ) = − σ ( K ) /
2, this strategy can be used to produce examples with (cid:101) g ( K ) > g ( K ); see Example 5.4 and more generally Appendix B. Finally, we note that thesignature is easy to compute in this setting. Choosing an arbitrary orientation on K , the Gordon-Litherland formula for the signature gives σ ( K ) = σ ( G F ) − n , where n is the numberof positive crossings in the alternating diagram. 44 343 Figure 10.
The knot 9 with a strong inversion given by rotation aroundan axis perpendicular to the page (left) and the checkerboard graph (right) forthe shaded checkerboard surface. Every edge has weight −
1, and the vertexweights are labeled.
Example 5.4.
Consider the knot K = 9 with shaded checkerboard surface F as shown inFigure 10. We see that K has an alternating diagram in which a strong inversion is givenby rotation around an axis perpendicular to the plane of the diagram and F is equivariantwith respect to this strong inversion. The Gordon-Litherland pairing is described by thecorresponding checkerboard graph as shown in the figure. One can check that σ ( K ) = − g ( K ) = 1 ([MS84]; see also [LM20]). By Theorem 1, if (cid:101) g ( K ) = − σ ( K ) / ι : ( H ( F ) , G F ) → ( Z , Id). We used a computerprogram to check that there are exactly two distinct lattice embeddings ι , ι : ( H ( F ) , G F ) → ( Z , Id) up to automorphisms of ( Z , Id), and it so happens that ρ ∗ ◦ ι i = ι j for { i, j } = { , } (see Figure 11). If there were an automorphism δ : ( Z , Id) → ( Z , Id) such that ι i ◦ ρ ∗ = δ ◦ ι i ,then ι j = δ ◦ ι i , contradicting that ι i and ι j are distinct up to automorphisms of ( Z , Id) for { i, j } = { , } . Thus (cid:101) g ( K ) > g ( K ). In fact, one can show that (cid:101) g ( K ) = 2. Indeed,performing an equivariant pair of crossing changes on a pair of opposite crossings closest tothe axis in Figure 10 gives the unknot, and thus by Proposition 2, (cid:101) g ( K ) ≤ e – e – e + e e + e + e – e + e – e – e – e + e + e – e – e + e – e – e + e + e – e + e – e – e e – e – e + e e + e + e – e – e + e – e Figure 11.
The unique pair of lattice embeddings (up to automorphism) of H of a checkerboard surface for the knot 9 into Z (cid:104) e , e , e , e , e , e (cid:105) . QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 23
Remark 5.5.
In Example 5.4, Theorem 1 was used to show that the smooth equivariant4-genus of K is 2. In fact, we can show that the topological equivariant 4-genus of K is also2 by essentially the same argument. In Theorem 1 the smoothness is only needed in orderto apply Donaldson’s theorem to conclude that a certain closed positive definite 4-manifold X has diagonalizable intersection form. However, in this example b ( X ) = 6 and in factfor algebraic reasons there is only one positive definite unimodular lattice of each rank ≤ The g -signature of knots In this section, we use the g -signature for 4-manifolds (see for example [Gor86]) to gen-eralize the knot signature to the equivariant setting. We use this generalization to provea lower bound on the equivariant 4-genus of a periodic knot and on the butterfly 4-genusof a strongly invertible knot. We begin by recalling the definition of the g -signature for4-manifolds. Definition 6.1. [Gor86, Section 1] Let X be a compact oriented 4-manifold with a finiteorder orientation-preserving diffeomorphism ρ : X → X . The intersection form on H ( X ; Z )induces a hermitian form ϕ : H × H → C where H = H ( X ; C ) = H ( X ; Z ) ⊗ C , by ϕ ( x ⊗ a, y ⊗ b ) = ab ( x · y ) . Then ϕ ( x ⊗ a, y ⊗ b ) = ϕ ( ρ ∗ x ⊗ a, ρ ∗ y ⊗ b ) for all ( x ⊗ a ) , ( y ⊗ b ) ∈ H . We may choose a ρ -invariant direct sum decomposition H = H + ⊕ H − ⊕ H which is orthogonal with respect to ϕ and where ϕ is positive definite, negative definite, and zero on H + , H − , and H respectively.Then the g -signature is (cid:101) σ ( X, ρ ) = trace( ρ ∗ | H + ) − trace( ρ ∗ | H − ) . It is usually easier to compute the g -signature using the following well-known proposition. Proposition 14.
Let X be a compact oriented 4-manifold with a finite order n orientation-preserving diffeomorphism ρ : X → X , and let ϕ : H × H → C be the intersection pairingwhere H = H ( X ; C ) . Then (cid:101) σ ( X, ρ ) = n − (cid:88) j =0 ω j σ ( ϕ (cid:12)(cid:12) H ( ω j ) ) , where ω = e πi/n is an n th root of unity and H ( ω j ) is the ω j -eigenspace of ρ ∗ : H ( X ; C ) → H ( X ; C ) .Proof. Let λ , . . . , λ k be the eigenvalues of ρ ∗ (cid:12)(cid:12) H + and λ k +1 , . . . , λ l be the eigenvalues of ρ ∗ (cid:12)(cid:12) H − .Since ρ n = Id, the λ j are n th roots of unity. Then (cid:101) σ ( X, ρ ) = trace( ρ ∗ | H + ) − trace( ρ ∗ | H − )= k (cid:88) j =1 λ j − l (cid:88) j = k +1 λ j = n − (cid:88) j =0 ω j ( p j − n j )= n − (cid:88) j =0 ω j σ ( ϕ (cid:12)(cid:12) H ( ω j ) ) , where p j = |{ i ≤ k : λ i = ω j }| and n j = |{ i > k : λ i = ω j }| . (cid:3) With this in hand, we define the g -signature for certain group actions on knots.6.1. Periodic knots and the g-signature.Definition 6.2.
Let K be an n -periodic knot in S with ρ : ( S , K ) → ( S , K ) a generatorof the periodic symmetry. Let F ⊂ B be an orientable equivariant surface for ( K, ρ ), and ρ : ( B , F ) → ( B , F ) be an extension of ρ . Let (cid:101) ρ : Σ( B , F ) → Σ( B , F ) be the distinguishedlift of ρ to the branched double cover (see Definition 5.1). The g-signature (cid:101) σ ( K, ρ ) of K isthe average of the g -signatures (cid:101) σ (Σ( B , F ) , (cid:101) ρ i ), that is, (cid:101) σ ( K, ρ ) = 1 n − n − (cid:88) i =1 (cid:101) σ (Σ( B , F ) , (cid:101) ρ i ) . When the periodic symmetry is clear we write (cid:101) σ ( K ) for (cid:101) σ ( K, ρ ).Note that in Definition 6.2, since we take the average of the g -signatures, (cid:101) σ ( K ) is inde-pendent of the choice of generator ρ for the n -periodic symmetry. Additionally, the lift (cid:101) ρ isindependent of the orientation on F and hence (cid:101) σ ( K ) does not depend on the orientation of K . We now show that the g -signature is well-defined. Theorem 8.
The g -signature of an n -periodic knot ( K, ρ ) is independent of the choice oforientable equivariant surface and extension of the periodic symmetry to B .Proof. For i ∈ { , } , let F i be an orientable equivariant surface for K in B and let ρ i : ( B , F i ) → ( B , F i ) be an extension of ρ to B . Then let ( S , F ) = ( B , F ) ∪ ( − B , − F )with ρ : ( S , F ) → ( S , F ) the symmetry which restricts to ρ i on ( B , F i ) for i ∈ { , } . Bythe same argument as in Proposition 4.1 of [Nai97], the surface F bounds an equivariant3-manifold M ⊂ S .Thinking of S = ∂B , we can extend ρ to an n -periodic symmetry (which we againrefer to as ρ ) of B by taking the cone of ρ . Under this extension, we can perform anequivariant isotopy pushing M into the interior of B fixing ∂M to get a 3-manifold (cid:99) M properly embedded in B . The symmetry ρ on B lifts to a symmetry (cid:101) ρ on the doublebranched cover Σ( B , (cid:99) M ) of B over (cid:99) M by a similar argument to that of Proposition 12.This lift can be chosen to restrict to the lift of ρ i on each Σ( B , F i ) ⊂ Σ( S , F ) used inthe definition of (cid:101) σ ( K ). For 0 < j < n , since ∂ (Σ( B , (cid:99) M ) , (cid:101) ρ j ) = (Σ( S , F ) , (cid:101) ρ j ), we havethat (cid:101) σ (Σ( S , F ) , (cid:101) ρ j ) = 0 by [Gor86, Section 1]. Hence by Novikov additivity (again, see[Gor86, Section 1]),0 = (cid:101) σ (Σ( S , F ) , (cid:101) ρ j ) = (cid:101) σ (Σ( B , F ) , (cid:101) ρ j ) − (cid:101) σ (Σ( B , F ) , (cid:101) ρ j ) , for 0 < j < n . Averaging these gives the desired result. (cid:3) Proposition 15.
The g -signature of a periodic knot is an equivariant concordance invariant.Proof. Let K and K (cid:48) be equivariantly concordant. Then the union of an equivariant surface S in B with boundary K and the equivariant concordance cylinder is an equivariant surface S (cid:48) in B with boundary K (cid:48) . The map H (Σ( B , S ); C ) → H (Σ( B , S (cid:48) ); C ) induced byinclusion is an equivariant isomorphism. Hence (cid:101) σ ( K ) = (cid:101) σ ( K (cid:48) ). (cid:3) Theorem 9. If K is a periodic knot, then | (cid:101) σ ( K ) | ≤ (cid:101) g ( K ) . QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 25
Proof.
Let S ⊂ B be a minimal genus orientable equivariant surface with ∂S = K . Then itis well known (see for example [GL11, Lemma 1]) thatrank( H (Σ( B , S ); C )) = rank( H ( S ; C )) = 2 g ( S ) = 2 (cid:101) g ( K ) . Fix i ∈ { , , . . . , n − } . Applying Proposition 14 to (Σ( B , S ) , (cid:101) ρ i ), where (cid:101) ρ is the lift of ρ from the definition of (cid:101) σ ( K ), we have | (cid:101) σ (Σ( B , S ) , (cid:101) ρ i ) | = (cid:12)(cid:12)(cid:12)(cid:12) n − (cid:88) j =0 ω j σ ( ϕ (cid:12)(cid:12) H ( ω j ) ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n − (cid:88) j =0 | σ ( ϕ (cid:12)(cid:12) H ( ω j ) ) | ≤ rank( H (Σ( B , S ); C )) = 2 (cid:101) g ( K ) . Hence by the triangle inequality, | (cid:101) σ ( K ) | ≤ (cid:101) g ( K ). (cid:3) The following theorem allows us to express the g -signature purely in terms of the signatureof K and the signature of the quotient K . Theorem 10.
Let K be n-periodic with quotient knot K . Then (cid:101) σ ( K ) = n · σ ( K ) − σ ( K ) n − . Proof.
Let S ⊂ B be an orientable equivariant surface with ∂S = K , equivariant withrespect to ρ : B → B extending the periodic action on K . Let (cid:101) ρ : Σ( B , S ) → Σ( B , S )be the lift of ρ as in Definition 6.2. The quotient of ( B , S ) by ρ is ( B, S ) where B is atopological 4-ball by Proposition 3 and S is an orientable surface with boundary K .Let G be a finite group of order n acting on a compact connected oriented 4-manifold X .Then by [Gor86, Section 6], n · σ ( X/G ) − σ ( X ) = (cid:88) g ∈ G \{ Id } (cid:101) σ ( X, g ) . Taking X = Σ( B , S ) and G = (cid:104) ρ (cid:105) , we get that X/G = Σ(
B, S ) and hence n · σ ( K ) − σ ( K ) = n − (cid:88) i =1 (cid:101) σ (Σ( B , S ) , (cid:101) ρ i ) , where we use that σ ( X ) = σ ( K ) and σ ( X/G ) = σ ( K ) by [KT76]. Dividing both sides by n − (cid:3) As a corollary of Theorem 9 and Theorem 10 we obtain the following key theorem.
Theorem 2.
Let K be an n -periodic knot with quotient knot K . Then (cid:101) g ( K ) ≥ | n · σ ( K ) − σ ( K ) | n − . The following two examples use Theorem 2 to obtain a difference of more than 1 betweenthe 4-genus and the equivariant 4-genus. The first example, which we state as a theorem,gives a family of 2-periodic knots K n for which | (cid:101) g ( K n ) − g ( K n ) | is unbounded. Theorem 3.
Let { K n } be the family of 2-periodic Montesinos knots with K n = M (1; − n, n + 2 , − n ) We follow the convention for Montesinos knots notation from [Iss18]. shown in Figure 12, where n is odd and positive. The difference (cid:101) g ( K n ) − g ( K n ) is unbounded.In fact, (cid:101) g ( K n ) = 2 n and g ( K n ) = 1 . n + 1- n - n Figure 12.
A family of 2-periodic knots K n (left) for n odd. The boxes aretwist regions with the labeled number of half-twists. When n = 3, we have K = K14a19410 (right). The period can be seen by performing a flype on thecentral crossing region (enclosed by a dotted loop in the right diagram), thenrotating the entire diagram by π within the plane of the diagram. Performingthe green and blue band moves shown gives the unknot so that g ( K n ) = 1. Proof.
For all positive odd n , K n has signature −
2, and the quotient knot is the left-handedtorus knot T (2 , n ) which has signature n −
1. Thus by Theorem 2, we have that (cid:101) g ( K n ) ≥ n .On the other hand, performing the pair of band moves shown in Figure 12 gives the unknotand hence g ( K n ) ≤
1. We also have g ( K n ) ≥
1, since σ ( K ) = − g ( K n ) = 1.For an upper bound on the equivariant 4-genus, we note that performing Seifert’s algorithmon the diagram in Figure 12 gives a genus 2 n surface for K n . Hence by Edmonds’ theorem, (cid:101) g ( K n ) ≤ n . We note that since the linking number with the axis is ± (2 n + 3), Proposition4 gives the stronger lower bound (cid:101) g ( K n ) ≥ n so that in fact (cid:101) g ( K ) = 2 n . (cid:3) As in the previous example, Proposition 4 sometimes gives a better lower bound thanTheorem 2. However, we provide the following example where Theorem 2 gives a strongerbound than Proposition 4.
Example 6.3.
Consider the periodic knot K shown in Figure 13 which has σ ( K ) = −
4, andobserve that the quotient knot is the left-handed trefoil which has signature 2. By Theorem2, we have that (cid:101) g ( K ) ≥
4. On the other hand, the linking number between K and theaxis of symmetry is 1 and the genus of the trefoil is 1, so that Proposition 4 only gives that (cid:101) g ( K ) ≥
2. Furthermore, changing the starred crossings gives a genus 1 cobordism (see forexample [LM19, Lemma 5(ii)]) to 12n466 which has 4-genus 2 (see for example [LM20]) sothat g ( K ) ≤ Strongly invertible knots and the g-signature.Definition 6.4.
Let (
K, τ ) be a directed strongly invertible knot which bounds a butterflysurface F ⊂ B , and let (cid:101) τ : Σ( B , F ) → Σ( B , F ) be the distinguished lift from Definition5.2. The g -signature (cid:101) σ ( K, τ ) of K is the g -signature (cid:101) σ (Σ( B , F ) , (cid:101) τ ). If the directed stronginversion is clear we write (cid:101) σ ( K ) for (cid:101) σ ( K, τ ), and if (
K, τ ) does not bound a butterfly surface,we define (cid:101) σ ( K, τ ) = ∞ . QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 27
Figure 13.
A 22-crossing periodic knot. The period can be seen by perform-ing a flype on the tangle enclosed by the dotted loop, then rotating the entirediagram by π within the plane of the diagram.To prove that the g -signature of a directed strongly invertible knot is well-defined as stated,we need the following lemma. Lemma 2.
Let ( S , F ) = ( B , F ) ∪ ( − B , − F ) where F and F are butterfly surfaces fora directed strongly invertible knot ( K, τ ) . Then F ⊂ S = ∂B bounds a properly embedded -manifold M ⊂ B which is equivariant with respect to an involution on B extending theinvolution of ( S , F ) .Proof. For i ∈ { , } , applying Proposition 7 to F i gives that the butterfly link L b ( K ) isa 2-periodic link bounding an orientable equivariant surface F (cid:48) i . Then F (cid:48) = F (cid:48) ∪ F (cid:48) is aclosed surface in S . Since F and F are butterfly surfaces, the quotient of F (cid:48) by τ | F (cid:48) isan orientable surface. Hence the argument proving Proposition 4.1 of [Nai97] applies, givingthat the surface F (cid:48) bounds an equivariant 3-manifold M (cid:48) ⊂ S .Thinking of S = ∂B , we can extend τ to an involution (which we again refer to as τ ) of B by taking the cone of τ : S → S . Under this extension, we can perform anequivariant isotopy pushing M (cid:48) into the interior of B fixing ∂M (cid:48) to get a 3-manifold (cid:99) M (cid:48) properly embedded in B . Now for i ∈ { , } , the last part of Proposition 7 applied to F i gives an ( I × D ) ⊂ B . Gluing these together along their shared boundary in S , we obtainan equivariant 3-manifold N = ( I × D ) ⊂ S such that F (cid:48) = [ F \ ( I × ∂D )] ∪ [( ∂I ) × D ].Then performing an equivariant isotopy to ( (cid:99) M (cid:48) ∪ N ) ⊂ B so that it is properly embeddedin B gives the desired 3-manifold M . (cid:3) The following theorem shows that the g -signature is well-defined. Theorem 11.
The g -signature of a directed strongly invertible knot ( K, τ ) is independent ofthe choice of butterfly surface and extension of the strong inversion to B .Proof. For i ∈ { , } , let F i be a butterfly surface for K in B and let τ i : ( B , F i ) → ( B , F i )be an extension of τ to B . Then let ( S , F ) = ( B , F ) ∪ ( − B , − F ) with τ : ( S , F ) → ( S , F ) the symmetry which restricts to τ i on ( B , F i ) for i ∈ { , } . By Lemma 2, thesurface F bounds a properly embedded 3-manifold M ⊂ B which is equivariant under someextension of τ to B . The symmetry τ on B lifts to a symmetry (cid:101) τ on the double branched cover Σ( B , (cid:99) M )of B over (cid:99) M by a similar argument to that of Proposition 12. This lift can be chosento restrict to the lift of τ i on each Σ( B , F i ) ⊂ Σ( S , F ) used in the definition of (cid:101) σ ( K ).Since ∂ (Σ( B , (cid:99) M ) , (cid:101) τ ) = (Σ( S , F ) , (cid:101) τ ), we have that (cid:101) σ (Σ( S , F ) , (cid:101) τ ) = 0 by [Gor86, Section1]. Hence by Novikov additivity (again, see [Gor86, Section 1]),0 = (cid:101) σ (Σ( S , F ) , (cid:101) τ ) = (cid:101) σ (Σ( B , F ) , (cid:101) τ ) − (cid:101) σ (Σ( B , F ) , (cid:101) τ ) . (cid:3) Remark 6.5.
In general, the g -signature depends on the direction of K . In fact by Propo-sition 13, if K has a butterfly Seifert surface then (cid:101) σ ( K ) = − (cid:101) σ ( K − ).The same proof as for Proposition 15, but applied to butterfly surfaces, gives the followingproposition. Proposition 16.
The g -signature of a directed strongly invertible knot is an equivariantconcordance invariant. As for the usual knot signature, we have the following additivity property.
Proposition 17.
The g -signature (for strongly invertible knots) is additive under equivariantconnect sum.Proof. Let K and K be directed strongly invertible knots which bound butterfly surfaces S and S in B respectively. Then K = K (cid:101) K bounds a butterfly surface S in B where( B , S ) is the boundary connected sum of ( B , S ) with ( B , S ). Passing to the doublebranched covers, H (Σ( B , S )) is isomorphic to H (Σ( B , S )) ⊕ H (Σ( B , S )) and thisisomorphism respects the corresponding involutions from the definition of the g -signature.Hence (cid:101) σ ( K ) = (cid:101) σ ( K ) + (cid:101) σ ( K ). (cid:3) We now have the main theorem of Section 6.2, giving a lower bound on the butterfly genus.
Theorem 4. If K is a directed strongly invertible knot which bounds a butterfly surface in B , then (cid:101) bg ( K ) ≥ | (cid:101) σ ( K ) | / .Proof. The proof is identical to Theorem 9. (cid:3)
Note that (cid:101) bg ( K ) does not depend on the direction of K so that this theorem gives a lowerbound on the butterfly 4-genus for each choice of direction.As the following example shows, the g -signature can be used to prove that the butterfly4-genus may be arbitrarily larger than the 4-genus. Example 6.6.
Consider the butterfly surface S for the directed strongly invertible knot( K, τ ) = 9 a − shown in Figure 14. As in Definition 6.4, (cid:101) σ ( K ) = (cid:101) σ (Σ( B , S ) , (cid:101) τ ), andby Proposition 13 we identify (cid:101) τ ∗ : H (Σ( B , S )) → H (Σ( B , S )) with τ ∗ : H ( S ) → H ( S )(using C coefficients) where the intersection form is identified with the Gordon-Litherlandpairing G S .Take the basis { a, b, c, d } for H ( S ) shown in Figure 14. The strong inversion exchanges a with − b and c with − d . We will now compute the g -signature using Proposition 14. The(+1)-eigenspace H (+1) of τ ∗ has basis { a − b, c − d } , and the ( − H ( −
1) of τ ∗ has basis { a + b, c + d } . We then have G S (cid:12)(cid:12) H (+1) = (cid:20) − − − − (cid:21) and G S (cid:12)(cid:12) H ( − = (cid:20) − − (cid:21) , QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 29 a bc d
Figure 14.
The directed strong inversion 9 a − (left). The chosen half-axisis shown as a dotted line and the shaded surface is a butterfly surface S . Onthe right is a basis for H ( S ).where G S is the Gordon-Litherland pairing. In particular, σ ( G S (cid:12)(cid:12) H (+1) ) = − σ ( G S (cid:12)(cid:12) H ( − ) =2, so that (cid:101) σ (9 a − ) = − K n be the equivariant connect sum of K with itself n times. Since (cid:101) σ is additiveunder equivariant connect sum by Proposition 17, we see that (cid:101) σ ( K n ) = − n . By Theorem9, (cid:101) bg ( K ) ≥ n , and since S is a genus 2 butterfly surface for K , (cid:101) bg ( K ) ≤ n . Hence (cid:101) bg ( K ) = 2 n . However K is (non-equivariantly) slice, so K n is as well. Thus the differencebetween the butterfly 4-genus and the non-equivariant 4-genus can be arbitrarily large. Appendix A. Table of genera and invariants for strongly invertible knots
In this appendix we provide a table of strongly invertible concordance invariants for low-crossing directed strongly invertible knots. First, we summarize the notation used in thetable. • L b ( K ) is the butterfly link; see Definition 4.1. • L qb ( K ) is the quotient butterfly link; see Definition 4.7. • f is the underlying knot, forgetting the strong inversion. • b is one component of the butterfly link L b ( K ); see Theorem 5. • qb is the non-axis component of the quotient butterfly link L qb ( K ); see Theorem 5. • (cid:101) lk ( K ) is the linking number between one component of L b ( K ) and the axis of sym-metry; see Definition 4.6. • η is Sakuma’s η -polynomial [Sak86]. • η ( L b ( K )) is the Kojima-Yamasaki η -polynomial of the butterfly link; see Definition4.3. • η ( L qb ( K )) is the Kojima-Yamasaki η -polynomial of the quotient butterfly link; seeDefinition 4.3. • g is the (non-equivariant) smooth 4-genus. • (cid:101) g is the equivariant 4-genus; see Definition 3.2. • (cid:101) bg ( K ) is the butterfly 4-genus; see Definition 3.5. • (cid:101) σ is the g -signature; see Definition 6.4.The η -polynomial (cid:80) a i t i is denoted [ a , a , a , . . . ] (recall that a i = a − i ). A dash “-” indicatesthat the invariant is not defined. Specifically, η ( L qb ( K )) is undefined when (cid:101) lk ( K ) (cid:54) = 0, and (cid:101) σ ( K ) is undefined when (cid:101) lk ( K ) (cid:54) = 0 or (cid:101) lk ( K − ) (cid:54) = 0. We compute η ( L b ( K )) only in the caseswhere one component of L b ( K ) is the unknot, in which case the polynomial can be computedusing the method in [Sak86]. The strong inversions in the table are given by rotation arounda vertical axis, and the half-axes distinguished by the directions are indicated with a dottedline segment. The superscript + and − indicate opposite choices of half-axes (antipodes).Mirrors are not included since the invariants are essentially the same (see for exampleProposition 11). Similarly, the orientation on the axis is omitted since none of the numericalinvariants depend on it.Name Diagram Invariants3 +1 f : 3 g : 1 b : 0 η : [ − , , (cid:101) g : 1 qb : 0 η ( L b ) : [0] (cid:101) bg : ∞ (cid:101) lk : 2 η ( L qb ) : - (cid:101) σ : - QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 31 − f : 3 g : 1 b : 0 η : [ − , , (cid:101) g : 1 qb : m η ( L b ) : [ − , , (cid:101) bg : ∞ (cid:101) lk : 2 η ( L qb ) : - (cid:101) σ : -4 +1 f : 4 g : 1 b : 0 η : [ − , , , − (cid:101) g : 1 qb : 0 η ( L b ) : [0] (cid:101) bg : ∞ (cid:101) lk : − η ( L qb ) : - (cid:101) σ : -4 − f : 4 g : 1 b : 3 η : [ − , , , − (cid:101) g : 1 qb : 5 η ( L b ) : not computed (cid:101) bg : ∞ (cid:101) lk : 2 η ( L qb ) : - (cid:101) σ : -5 +1 f : 5 g : 2 b : 0 η : [ − , , (cid:101) g : 2 qb : 0 η ( L b ) : [0] (cid:101) bg : ∞ (cid:101) lk : 2 η ( L qb ) : - (cid:101) σ : -5 − f : 5 g : 2 b : 0 η : [ − , , (cid:101) g : 2 qb : 4 η ( L b ) : [0] (cid:101) bg : ∞ (cid:101) lk : 2 η ( L qb ) : - (cid:101) σ : - a + f : 5 g : 1 b : 0 η : [ − , , , − (cid:101) g : 1 qb : 0 η ( L b ) : [ − , , (cid:101) bg : ∞ (cid:101) lk : 4 η ( L qb ) : - (cid:101) σ : -5 a − f : 5 g : 1 b : m η : [ − , , , − (cid:101) g : 1 qb : m η ( L b ) : not computed (cid:101) bg : ∞ (cid:101) lk : 3 η ( L qb ) : - (cid:101) σ : -5 b + f : 5 g : 1 b : 0 η : [ − , , , − , (cid:101) g : 1 qb : 0 η ( L b ) : [0] (cid:101) bg : 2 (cid:101) lk : 0 η ( L qb ) : [ − , , (cid:101) σ : 2 b − f : 5 g : 1 b : 3 η : [ − , , , − , (cid:101) g : 1 qb : 5 η ( L b ) : not computed (cid:101) bg : 2 (cid:101) lk : 0 η ( L qb ) : [ − , , (cid:101) σ : -2 The butterfly 4-genus (cid:101) bg does not depend on the choice of half-axis. Hence (cid:101) bg (5 b + ) = (cid:101) bg (5 b − ) = 2. This follows from the computation of (cid:101) σ (5 b − ), see Remark 6.5. The butterfly Seifert surface in Figure 7 (left) can be pushed equivariantly into B . To see that thesymmetry on 5 shown in Figure 7 depicts 5 b − , it is sufficient to calculate b . QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 33 a + f : 6 g : 0 b : 0 η : [ − , , , , , − (cid:101) g : 1 qb : 0 η ( L b ) : [ − , , (cid:101) bg : ∞ (cid:101) lk : 4 η ( L qb ) : − (cid:101) σ : -6 a − f : 6 g : 0 b : m η : [ − , , , , , − (cid:101) g : 1 qb : 9 η ( L b ) : not computed (cid:101) bg : ∞ (cid:101) lk : 4 η ( L qb ) : − (cid:101) σ : -6 b + f : 6 g : 0 b : 0 η : [4 , − , − , (cid:101) g : 1 qb : 0 η ( L b ) : [0] (cid:101) bg : 2 (cid:101) lk : 0 η ( L qb ) : [ − , , (cid:101) σ : 0 b − f : 6 g : 0 b : 4 η : [4 , − , − , (cid:101) g : 1 qb : m η ( L b ) : not computed (cid:101) bg : 2 (cid:101) lk : 0 η ( L qb ) : [ − , , (cid:101) σ : 08 +9 f : 8 g : 0 b : 0 η : [0] (cid:101) g : 1 qb : 0 η ( L b ) : [2 , , − (cid:101) bg : ∞ (cid:101) lk : 0 η ( L qb ) : [0] (cid:101) σ : - This follows from the computation of (cid:101) σ (6 b − ), see Remark 6.5. The butterfly Seifert surface in Figure 7 (right) can be pushed equivariantly into B . To see that thesymmetry on 6 shown in Figure 7 depicts 6 b − , it is sufficient to calculate b . − f : 8 g : 0 b : 0 η : [0] (cid:101) g : 1 qb : m η ( L b ) : not computed (cid:101) bg : ∞ (cid:101) lk : − η ( L qb ) : − (cid:101) σ : - QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 35
Appendix B. Table of examples with non-trivial Donaldson’s theoremobstruction
In this appendix we give a table of all periodic and strongly invertible knots with a sym-metric alternating diagram having at most 11 crossings for which Theorem 1 provides anobstruction to the equivariant 4-genus (cid:101) g ( K ) being equal to | σ ( K ) | /
2. In all cases, we infact have g ( K ) = | σ ( K ) | / (cid:101) g ( K ) > g ( K ). The notation (cid:63) → K (cid:48) indicates that K (cid:48) is the knot obtained by equivariantly changing the crossings marked with a (cid:63) . We thencompute upper bounds on (cid:101) g using Proposition 2 and the computations of (cid:101) g in AppendixA. The transvergent symmetries are shown as rotation around a vertical dotted line, and theintravergent symmetries are shown as rotation around a dot in the center of the diagram.Unlike in Appendix A, the strongly invertible knots here are not directed.9 : transvergent, periodic 9 : intravergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot unknot9 : intravergent, strongly invertible 9 : intravergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot unknot : transvergent, periodic 9 : transvergent, strongly invertible σ : − σ : 0 g : 1 g : 0 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → note. : transvergent, strongly invertible 10 : transvergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot10 : transvergent, strongly invertible 10 : transvergent, strongly invertible σ : 0 σ : − g : 0 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → This was also shown in [Sak86, Appendix II] and [DHM20, Theorem 1.11]. The lower bound (cid:101) g ≥ (cid:101) g ≤ g = 3 is obtained byEdmonds’ theorem [Edm84]. This was also shown in [DHM20, Theorem 1.11].
QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 37 : transvergent, strongly invertible 11 : intravergent, strongly invertible σ : 0 σ : − g : 0 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot unknot11 : transvergent, strongly invertible 11 : intravergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot11 : transvergent, strongly invertible 11 : intravergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot : transvergent, periodic 11 : intravergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → note. : transvergent, strongly invertible 11 : transvergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot unknot11 : transvergent, strongly invertible 11 : transvergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot The lower bound (cid:101) g ≥ (cid:101) g ≤ g = 3 is obtained byEdmonds’ theorem [Edm84]. QUIVARIANT 4-GENERA OF STRONGLY INVERTIBLE AND PERIODIC KNOTS 39 : transvergent, strongly invertible 11 : transvergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot11 : transvergent, strongly invertible 11 : transvergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot unknot11 : intravergent, strongly invertible 11 : intravergent, strongly invertible σ : − σ : − g : 1 g : 1 (cid:101) g ≥ (cid:101) g ≥ (cid:101) g ≤ (cid:101) g ≤ (cid:63) → (cid:63) → unknot References [Boy19] Keegan Boyle. Odd order group actions on alternating knots. 2019. Available at https://arxiv.org/abs/1906.04308 .[Boy20] Keegan Boyle. Involutions of alternating links. 2020. Available at https://arxiv.org/abs/2002.08191 .[Bre72] Glen E. Bredon.
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Department of Mathematics, University of British Columbia, Canada
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