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2. The following result is also a direct consequence of Theorem 3.1:

Proposition 3.4 (Conjecture in Section 1.4 of [5]) The crossing number of a q- periodic alternating knot with q ≥ 3 is a multiple of q.

3. The Murasugi decomposition into atoms of 12a634 gives rise to an adjacency graph which is a tree of 2-vertices ([16], [17]). According to Visibility Theorem 3.1 and Lemma 3.2 (which is an application of Corollary 1 in [5]), we deduce that 12a634 is not 3-periodic. We thank C. Livingstone to point out the existence of a computer proof of this fact by S. Jabuka and S. Naik [11].

1.1 Organization of the paper

For the study of the visibility of the periodicity of the alternating knots on alternating projections, we will call upon the canonical decomposition of link projections recalled in §2, as it was done for the visibility of achirality of the alternating knots in [8]. The decomposition of a link projection Π is carried out by a family of canon- ical Conway circles which decomposes (S2, Π) into diagrams called jewels and twisted band diagrams; the arborescent part of Π is the union of the twisted band diagrams of Π. The decomposition of the diagram (S2, Π) by the canonical Con- way circles is a 2-dimensional version of the decomposition of Bonahon-Siebenman [1] of (S3,K) into an algebraic part (A, A∩K) and a non-algebraic part (N, N ∩K). Each component of ∂A = ∂N is a 2-sphere that cuts K in four points and is called a Conway sphere. We now assume that links and projections we consider are al- ternating. In our terminology, the 2-dimensional notion “arborescent” implies the 3-dimensional notion “algebraic” (see the definition for instance in [18]). The pro- jection of a Conway sphere on S2 is a Conway circle. The inverse is not true: there are “hidden” Conway spheres that do not project on Conway circles on alternating

2 projections ([18]). However since our point of view is strictly 2-dimensional and based only on alternating projections, this case does not affect us. The notion of flype in alternating projections (Fig. 7) is at the heart of our analysis and lies completely in their arborescent part. According to Menasco-Thistlethwaite’s Flyping theorem [14], two reduced alternating projections Π1 and Π2 of an isotopy class of an alternating link K are related by a finite sequence of flypes, up to homeo- morphisms of S2 onto itself. Starting from the canonical decomposition of a projection of Π(K) of K, we associate canonical and essential structure trees (as recalled in §2) that does not depend on the choice of an alternating projection. The canonical and essential structure trees are invariants of the isotopy class of alternating knots. For example, for rational links, their canonical structure tree is a linear tree with integer-weighted vertices and their essential structure tree is reduced to a vertex of rational weight.

In §3 we study how the q-periodicity acts on the essential Conway circles and on the diagrams of any alternating projection. With the help of Kerekjarto’s theorem [3] and Flyping Theorem of Menasco-Thistlethwaite we prove Theorem 3.1 and we obtain a q-periodic alternating projection by adjustments with flypes on any alternating prime q-periodic knot. With the help of the Murasugi decomposition into atoms for periodic alternating knots and a result in [5] which links the q-periodicity of an alternating knot to the q-periodicity of its atoms, we finally show that 12a634 is not 3-periodic.

2 Canonical Decomposition of a Projection

In this section we do not assume that link projections are alternating. A projection on S2 is the image of a link in S3 by a generic projection onto S2, hence a labeled graph with n 4-valent crossing-vertices labeled to reflect under and over crossings. In this paper the term “ diagram” will be used to refer to a different object (see below §2.1).

2.1 Diagrams

Let Σ be a compact connected planar surface embedded on the projection sphere S2. We denote by k + 1 the number of connected components of its boundary ∂Σ. Definition 2.1. The pair D = (Σ, Γ = Π ∩ Σ) where Π is a link projection is called

3 a diagram if for each connected component C of ∂Σ, C ∩ Π is composed exactly of 4 points. Remark 2.1. A (link) projection Π on S2 is a diagram (Σ, Π) where Σ= S2. Definition 2.2. (1) A trivial diagram is a diagram homeomorphic to T([∞]) (Fig. 1(a)). (2) A singleton is a diagram homeomorphic to Fig. 1(b).

Figure 1: (a) Tangle T([∞]) (b) A singleton Definition 2.3. A Haseman circle of a diagram D = (Σ, Γ) is a circle γ ⊂ Σ that intersects the projection Π exactly in 4 points away from crossing points. A Haseman circle is said to be compressible if γ bounds a disc ∆ in Σ such that (∆, Γ ∩ ∆) is either a trivial diagram or a singleton.

In what follows, Haseman circles are not compressible. We therefore only con- sider diagrams that are neither trivial diagrams nor singletons. Definition 2.4. A twisted band diagram (TBD) is a diagram homeomorphic to Fig. 3.

The signed weight of a crossing on a band is defined according to Fig. 2. It depends on the direction of the half-twist of the band supporting the crossing.

Figure 2: The signed weight of a crossing on a band

In Fig. 3 the boundary components of Σ are denoted γ1,...,γk+1 where k ≥ 0. The corresponding portion of the band diagram between the projection and the circles γi and γi+1 is called a twist region with |ai| crossing points. The sign of ai is the

4 Figure 3: A twisted band diagram

signed weight of the |ai| crossing points. The integer ai will be called an intermediate weight. If k + 1 = 1, the planar surface Σ is a disc and the twisted band diagram (Σ, Σ ∩ Π) is called a spire with |a1| ≥ 2 crossings. If k + 1 = 2, the twisted band diagram is a twisted annulus and we require that a1 + a2 =6 0.

We ask the crossings on the same band to have the same signed weight. In other words, using flypes (Fig. 7) and Reidemeister move 2, we can reduce the number of crossing points of a twisted band diagram so that all non zero intermediate weights ai of a twisted band diagram have the same sign. Definition 2.5. (1) The crossings of a TBD (twisted band diagram) (Σ, Σ ∩ Π) are called the visible crossings of (Σ, Σ ∩ Π). (2) The sum a = Σai is called the total weight of the twisted band diagram (Σ, Σ∩Π). If k +1 ≥ 3 we may have a = 0. The absolute value of a is equal to the number of the visible crossings of (Σ, Σ ∩ Π).

Two Haseman circles are said to be parallel if they bound an annulus A ⊂ Σ such that the pair (A, A ∩ Γ) is diffeomorphic to Fig. 4.

Figure 4: Parallel Haseman circles

We define a Haseman circle γ to be boundary parallel if there exists an annulus A ⊂ Σ such that:

5 (1) the boundary ∂A of A is the disjoint union of γ and a boundary component of Σ; (2) (A, A ∩ Γ) is diffeomorphic to Fig. 4. Definition 2.6. A jewel is a diagram J such that: (1) it is not a twisted band diagram with k +1=2 and a = ±1 or with k +1=3 and a =0. (2) each Haseman circle of J is boundary parallel.

Fig. 5 depicts a jewel J = (Σ, Π ∩ Σ) where Σ is a planar surface with boundary ∂Σ= γ1 ∪ γ2 ∪ γ3 ∪ γ4.

Figure 5: A jewel

2.2 Families of Haseman circles for a projection

2.2.1 Canonical Conway circles

If not otherwise stated, the projections we consider are connected and prime. Definition 2.7. Let Π be a projection. A family of Haseman circles for Π is a set of Haseman circles satisfying the following conditions: (1) any two circles are disjoint and (2) no two circles are parallel.

Let H = {γ1, ..., γn} be a family of Haseman circles for Π. Let R be the closure of a 2 i=n connected component of S \ Si=1 γi. We call the pair (R, R ∩ Π) a diagram of Π determined by the family H. Definition 2.8. A family C of Haseman circles is an admissible family if each diagram determined by it is either a twisted band diagram or a jewel. An admissible family is minimal if removing a circle turns it into a family that is not admissible.

6 Theorem 2.1 is the main structure theorem about link projections proved in ([15], Theorem 1). It is essentially due to Bonahon and Siebenmann.

Figure 6: A non canonical Conway circle

Theorem 2.1. (Existence and uniqueness theorem of minimal admissible families) Let Π be a link projection in S2. Then: i) there exist minimal admissible families for Π; ii) any two minimal admissible families are isotopic by an isotopy respecting Π. Definition 2.9. An Haseman circle belonging to “the” minimal admissible family of Π noted Ccan is called a canonical Conway circle of the projection Π. Example 1. The Haseman circle C in Fig. 6 is not a canonical Conway circle.

The decomposition of Π into twisted band diagrams and jewels determined by Ccan will be called the canonical decomposition of Π. If there are no jewels in its canonical decomposition, the projection Π is said to be arborescent.

Figure 7: A projection with its canonical Conway circles

A canonical Conway circle can be of 3 types: (1) a circle that separates two jewels. (2) a circle that separates two twisted band diagrams. (3) a circle that separates a jewel and a twisted band diagram.

7 Example 2. Fig. 7 illustrates a projection Πe with its canonical Conway family: Ccan(Π)e = {γ1,γ1,1,γ2,γ3,γ3,1,γ3,2,γ4}.

Figure 8: 10∗∗∗ is a tangle sum of two 6∗

Remark 2.2. 1. As remarked in [8], our notion of jewel is more restrictive than the notion of John Conway polyhedron ([13] p. 139). We define a jewel graph GJ of a jewel J by collapsing each Haseman circle of J to a vertex. For John Conway, the graph of a basic polyhedron is a simple regular graph of valency 4. A basic polyhedron can therefore be a tangle sum of several jewel graphs. A jewel graph is simply a polyhedron in the sense of John Conway, indecomposable with regard to the tangle sum. The polyhedron 10∗∗∗ has a non-trivial Haseman circle (see Fig. 8). 2. The minimal projection of the torus link of type (2, m) can be considered as a twisted band diagram with k +1=0.

2.2.2 Essential Conway circles

Let Π be a projection on S2.

Definition 2.10. A (2-dimensional) tangle T of Π is a pair T = (∆, τ∆) where ∆ is 2 a disk in S , τ∆ is Π∩∆ and the boundary ∂∆ of ∆ intersects τ∆ exactly on 4 points. The boundary ∂T of T is the boundary ∂∆ of ∆.

Note that a (2-dimensional) tangle is the projection onto the equatorial disk of the 3-ball of a (3-dimensional) tangle which will be defined further in Definition 3.4. ′ ′ Definition 2.11. Two tangles T = (∆, τ∆) and T = (∆, τ∆) are isotopic if there exists a homeomorphism f : T → T ′ such that: (1) f is the identity on the boundary ∂∆ and ′ (2) f(τ∆)= τ∆.

8 Definition 2.12. A rational tangle is a tangle such that all its canonical Con- way circles are concentric and delimit twisted annuli, with the exception of the innermost circle which is the boundary of a spire, as shown in Fig. 9. A maximal rational tangle of a link projection Π is a rational tangle that is not strictly included in a larger rational tangle of Π.

Let T be a rational tangle. We now consider T under the cardan form T[a0,...,am] (or equivalently under the standard form described in [12]) illustrated in Fig. 9 such i that the twisted band diagrams have weights bi =(−1) ai with i =0,...,m and such that the first weight band b0 is horizontal.

To T[a0,...,am] where a0 ∈ Z and a1,...,am ∈ Z −{0}, we assign the continued fraction 1 [a , a , ··· a ]= a + 0 2 m 0 1 a + 1 . 1 .. + am

r If T is not the trivial tangle T([∞]) (Fig. 1(a)), the rational number s = [a0, a2, ··· am] with (r, s) = 1 and r > 0 is called the fraction F(T ). By convention, the fraction of the trivial tangle T([∞]) is: F(T([∞]) := ∞.

The fraction is an isotopy invariant of the tangle T . It means that with the expan- r sion of s in another continuous fraction [d0,...,dk], we get another cardan tangle T[d ,...,d ] isotopic to T[a ,...,a ]. We will use T r to denote a rational tangle 0 k 0 m s r with fraction s .

r r Remark 2.3. Let s be a rational number with r > 0 and (r, s)=1. Then s has an expansion [a0,...,am] where the ai’s are all positive or all negative; it is called a homogeneous continued fraction. If furthermore, a0 and am are not equal to ±1, the continued fraction [a0,...,am] is said to be strictly homogeneous. If [a0,...,am] is a homogeneous continued fraction, the cardan tangle T[a0,...,am] is an alternating tangle.

Definition 2.13. An essential Conway circle of an alternating projection Π is a canonical Conway circle that is not properly contained in a maximal rational tangle.

In a rational link projection, there are no essential Conway circles.

9 Figure 9: T[a0,...,am]

Let Π be a non-rational link projection. By removing from the minimal admissible family Ccan of Π all concentric Conway circles of each maximal rational tangle T of (S2, Π) except its boundary circle ∂T , we obtain the essential Conway family of Π denoted Cess(Π). Remark 2.4. The set of essential Conway circles is empty for an alternating projec- tion if only if the projection is one of the three following cases: a) a standard torus knot projection of type (2,s) (in this case it can be considered as a twisted band diagram with empty boundary (Remark 2.2.2)), b) a jewel without boundary c) a minimal projection of a rational knot.

Figure 10: The essential Conway family of the projection Πe illustrated in Fig. 7

10 ′ ′′ Figure 11: A projection with its set Cess = {C, C , C }

Example 3. 1) Fig. 10 illustrates the essential Conway family Cess(Π)e of the projec- tion Πe of Fig. 7: Cess(Π)e = {γ1,γ2,γ3,γ3,1,γ3,2,γ4}.

′ ′′ 2) In the projection depicted in Fig. 11, C , C and C = ∂T[a0,...,am] are essential Conway circles while dotted circles are just canonical Conway circles.

2.3 Canonical and Essential Structure Trees

We now focus the canonical decomposition of alternating link projections.

2.3.1 Flypes and Flyping Theorem

Let Π be a n-crossing regular projection of a link L ֒→ S3 on the projection plane 2 S . As in [14], consider n disjoint small “crossing ball” neighbourhoods B1,...,Bn of the crossing points c1,...cn of Π. Then assume that L coincides with Π, except that inside each Bi the two arcs forming α(ci) = Π ∩ Bi are perturbed vertically to form semicircular overcrossing and undercrossing arcs, which lie on the boundary of Bi. This relationship between the link L and its projection Π is expressed as L = λ(Π)

11 ([14]). Note that there is a homeomorphism of pairs (S3, L) → (S3,λ(Π)). We call λ(Π) a realized projection (or a realized diagram in the terms of [2]) for L.

We can consider the ambient space S3 to be R3 ∪{∞}, and we shall take the 2-sphere S2 on which the regular projection Π lies to be S2 = {x ∈ R3 : kxk =1}. We assume 3 1 3 that the knot λ(Π) lies within the neighborhood N = {x ∈ R : 2 ≤kxk ≤ 2 }. 3 3 Definition 2.14. Let g : (S ,λ(Π1)) → (S ,λ(Π2)) be a homeomorphism of pairs. The homeomorphism of pairs g is flat if g is pairwise isotopic to a homeomorphism of pairs h with the condition that h maps N onto itself and h| = h ×id 1 3 for some N 0 [ 2 , 2 ] 2 2 orientation-preserving homeomorphism h0 : S → S . We call h0 the principal part of the flat homeomorphism g.

By asking the crossing balls to be sent on the crossing balls, a flat homeomorphism of pairs is an isomorphism of realized projections as defined in [2].

Definition 2.15. An isomorphism of realized projections h : λ(Π) → λ(Π∗) is 3 ˜3 a homeomorphism of pairs h :(S , L) → (S , Le) such that: (1) h(S2)= S2, (2) h(Bi)= Bei and (3) h(α(ci)) = α(˜ci).

We recall the definition of a flype as described in [14].

Definition 2.16. Let Π1 be a projection with the pattern described in Fig. 12(a). A 3 3 standard flype of (S ,λ(Π1)) is any homeomorphism f which maps (S ,λ(Π1)) to 3 a pair (S ,λ(Π2)) where Π2 is the pattern described in Fig. 12(b), in such a way that: (1) f sends the 3-ball BA into itself by a rigid rotation about an axis in the projection plane, (2) f fixes pointwise the 3-ball BB, (3) f moves the crossing visible on the left of Fig. 12(a) to the crossing visible on the right of Fig. 12(b).

Definition 2.17. Let Π1 be any projection. Then a flype is any homeomorphism 3 3 ′ ′ f :(S ,λ(Π1)) → (S ,λ(Π1)) of the form f = g1 ◦ f ◦ g2 where f is a standard flype and g1 and g2 are flat homeomorphisms.

If the tangle A of Fig. 12 contains no crossing, then the standard flype determined by that figure is a flat homeomorphism; therefore according to the above definition, any flat homeomorphism is a flype. We call it a trivial flype.

12 A 2-dimensional description of a standard flype, as in Fig. 12, is sufficient since it corresponds to a unique standard flype up to isomorphism of realized diagrams. The crossing that moves during the (2-dimensional) flype is an active crossing point of the flype.

Terminology. If there is no possible confusion, we will also designate the (2- dimensional) flype by flype.

Figure 12: A flype

We can now precisely locate where flypes can be performed with respect to the canon- ical Conway decomposition of a prime alternating reduced link. Theorem 2.2. [15] (Position of flypes) Let Π be a prime alternating reduced link projection in S2 and suppose that a flype can be done in Π. Then, its active crossing point belongs to a diagram determined by Ccan(Π). The flype moves the active crossing point either within the twist region to which it belongs or to another twist region of the same twisted band diagram.

Figure 13: An efficient flype Remark 2.5. (1) We are only interested in efficient flypes that move the active crossing point from one twist region to another in the same twist band diagram.

(2) Let T be a TBD of Π and C1,C2,...,Ck,C1 be its cyclic sequence of Ccan|T . A flype on T does not modify the order of occurence of the Conway circles Ccan|T in the cyclic sequence.

13 Definition 2.18. (1) The set of the twist regions of a given twisted band diagram is called a flype orbit (Fig. 14). (2) The cardinal of Ccan|T is the valency k +1 of the TBD T .

Figure 14: A flype orbit Corollary 2.1. [15] (1) A flype moves an active crossing point inside the flype orbit to which it belongs. (2) Two distinct flype orbits are disjoint.

This implies that an active crossing point belongs to one and only one TBD. Since two TBD have at most one canonical Conway circle, Corollary 2.1 can be interpreted as a loose kind of commutativity of flypes.

2.3.2 Canonical Structure Tree A(L).

Fundamental to our purposes is the following Menasco-Thistlethwaite Flyping Theo- rem [14]:

Let Π1 and Π2 be two reduced, prime, oriented, alternating projections of links. If 3 3 f : (S ,λ(Π1)) → (S ,λ(Π2)) is a homeomorphism of pairs, then f is a composition of flypes and flat homeomorphisms.

Since two realizations of alternating projections in S2 of the same isotopy class of an oriented prime alternating link in S3 are related by flypes, their canonical and essential structure trees constructed as described below, are isomorphic.

Construction of the canonical structure tree A(L).

Let L be a prime alternating link and let Π be an alternating projection of L. Let Ccan be the canonical Conway family for Π. We construct the canonical structure tree A(L) as follows: its vertices are in bijection with the diagrams determined by Ccan and

14 its edges are in bijection with the canonical circles; the vertices of each edge represent the diagrams which both have in their boundary, the canonical circle γ corresponding to the edge. Since S2 has genus zero, the constructed graph is a tree. We label the vertices of A(L) as follows: if a vertex represents a twisted band diagram, we label it by its total weight a and if it represents a jewel, we label it with the letter J.

In the case of a tangle T whose boundary is a canonical Conway circle γ, the canonical structure tree A(T ) of T is a graph such that all its edges have two vertices at the extremities except for an“open” edge (with a single vertex) which represents the circle γ. For an example, see Fig. 16.

Proposition 2.1. The canonical structure tree A(L) is independent of the alternating projection chosen to represent L.

Proof. Let Π be an alternating link projection in S2. By [15], Theorem 1: i) there exist minimal Conway families for Π and ii) any two minimal Conway families are isotopic, by an isotopy which respects Π.

A flype or a flat homeomorphism does not modify the canonical structure tree. By Flyping Theorem, we conclude that the canonical structure tree is independent of the chosen alternating projection Π and we can speak of the canonical structure tree of L (and not only of Π).

Definition 2.19. The alternating knot K is arborescent if each vertex of A(K) has an integer weight. Example 4. The link K0 which has the projection Πe represented by Fig. 7, has its canonical structure tree A(K0) given by Fig. 15(a).

Remark 2.6. If the projection Π is arborescent, we can encode Π with a weighted planar tree `ala Bonahon-Siebenman (§5 in [15]) which is a canonical structure tree with more complete information.

2.3.3 Essential Structure Tree A˜(L).

Construction of the essential structure tree A˜(L).

15 Figure 15: Canonical structure tree and essential structure tree of K0

On the same lines of the construction of the canonical structure tree A(L), we con- essential structure tree struct the Ae(L). The vertices of Ae are in bijection with the diagrams determined by the set Cess(Π) and the edges are in bijection with the circles of Cess(Π). The extremities of an edge corresponding to an essential Conway circle γ are two vertices associated to the two diagrams having γ in their boundary.

As in the case with the canonical structure tree, Flyping Theorem implies that: Proposition 2.2. The essential structure tree Ae(L) is independent of the minimal projection chosen to represent K.

The essential structure tree of a tangle: To a tangle T with an essential Conway circle γ as boundary, we associate an essential structure tree denoted Ae(T ) which has only one “open edge” with one vertex-end. The unique edge of Ae(T ) corresponds to γ.

Remark 2.7. 1. If T p is a maximal rational tangle, A(T p ) is a linear graph com- q q e p posed only with an “open” edge and one vertex labelled by q .

Q Z 2. A vertex in Ae(K) with weight ∈ \ is monovalent and its union with its single edge corresponds to a maximal rational tangle in Π.

3. Only monovalent vertices of an essential structure tree of a link can have weights that are ∈ Q \ Z

r The essential structure tree A(T r ) is reduced to a vertex of weight . e s s

16 Figure 16: (a) Canonical structure tree of A(T[−1, −2]) and (b) Essential structure tree of Ae(T[−1, −2])

Example 5. A(T[−1, −2]) and Ae(T[−1, −2]) are described in Fig. 16. Remark 2.8. The essential structure tree Ae(K) is reduced to a single vertex if and only if K is a rational link T r or is a link described by a jewel without boundary. s

3 On Visibility Theorem 3.1

This section is about the proof of the Visibility Theorem 3.1 for q-periodic alternating prime knots: Theorem 3.1. Let K be an oriented prime alternating knot that is q-periodic with q ≥ 3. Then there exists a q-periodic alternating projection Πe for K.

We first recall the definition of a q-periodic knot in S3. Definition 3.1. A knot K is q-periodic if there is a (auto)-homeomorphism Φ of pairs (S3,K) of period q which satisfies the following conditions: 2π 3 (1) Φ is a q -rotation about a “line” (circle) α in S and (2) α ∩ K = ∅. Φ is called a q-homeomorphism of (S3,K). Remark 3.1. Let K be a q-periodic knot with Φ a q-homeomorphism of (S3,K). For r q each divisor p of q, K is p-periodic with its p-homeomorphism Ψr = Φ where r = p .

For our ends, we introduce the following notion:

17 Figure 17: A non-strictly 4-periodic projection

Definition 3.2. 1) If a knot K is q-periodic but not strictly q-periodic then K is rq-periodic for some r ≥ 2. 2) If a projection Π is q-periodic but not strictly q-periodic then Π is rq-periodic for some r ≥ 2.

Example 6. Let Π be an alternating projection described in Fig.17 where Y is an alternating tangle with boundary an essential Conway circle. The big red circles do not belong to Cess(Π) and hence do not appear on A˜(K). The projection Π is not strictly 4-periodic since as shown by Fig. 17, Π is an 8-periodic projection. Note that the projection Π depicted in Fig.17 is not a knot projection, regardless the tangle Y (see Proposition 3.2 below).

From now on, by

q-periodic projections, we mean strictly q-periodic projections.

Our objective being the study of periodicity, we reformulate the Flyping theorem in the following form: Theorem 3.2. Let Φ:(S3,K) → (S3,K) be an orientation preserving homeomor- phism of pairs where K is a prime alternating knot. Let λ(Π) be a realized projection of a reduced alternating projection Π of K and h : (S3,K) → (S3,λ(Π)) be a home- omorphism of pairs. Then the isomorphism of realized projections ΦΠ = h ◦ Φ ◦ −1 3 3 h : (S ,λ(Π)) → (S ,λ(Π)) can be expressed as ΦΠ = φ ◦ F where φ is a flat homeomorphism and F is a composition of standard flypes on λ(Π) unless F is the identity. Remark 3.2. 1. For our purposes, we separate the flat homeomorphisms from the flypes. Therefore, the standard flypes involved in Flyping Theorem 3.2 above are not trivial.

18 2. Standard flypes and flat homeomorphisms “essentially” commute. By “essen- tially” commute, we mean that if f is a standard flype and h is a flat homeo- morphism, then f ◦ h = h ◦ f ′ where f ′ = h−1 ◦ f ◦ h is also a standard flype. Then it is possible to express ΦΠ = φ ◦ F .

Let K ⊂ S3 be a prime (strictly) q-periodic alternating knot with Φ : (S3,K) → (S3,K) its corresponding rotation about an axis of order q. Let Π be a reduced alternating projection of K and λ(Π) its realized diagram. By Flyping Theorem, Φ is conjugate through maps of pairs to an isomorphism ΦΠ on λ(Π) (onto itself) which is a composition of a flat homeomorphism with standard flypes. Definition 3.3. An essential Conway sphere of Π is a 2-dimensional sphere ly- 3 1 3 ing in the interior of N = {x ∈ R : 2 ≤ kxk ≤ 2 } such that its projection on the projection plane is an essential Conway circle.

Consider the set Cess(Π) of essential Conway circles of Π and its corresponding set Sess(Π) of essential Conway spheres.

A 2-dimensional tangle in Definition 2.10 is the projection on the equatorial disk of the 3-ball B of a 3-dimensional tangle defined as follows: Definition 3.4. A 3-dimensional tangle is a pair (B, t), where B is a 3-ball and t is a proper 1-submanifold of B meeting ∂B in four points. We say that two tangles are equivalent if they are homeomorphic as pairs. We say that (B, t) is trivial if it is equivalent to the pair (B0, t0), where B0 = 3 2 2 2 {(x1, x2, x3) ∈ R : x1 + x2 + x3 ≤ 1} and t0 consists of the points of B0 for which x1 ∈{1/2, −1/2} and x3 =0.

From such a tangle diagram T , we can create a 3-dimensional tangle λ(T ) by means of a suitably small vertical perturbation near each crossing of the diagram; the ambient space of λ(T ) is considered as a 3-ball for which the disk region of T is an equatorial slice.

Corollary 3.1. ΦΠ induces a permutation σΦΠ on the set of essential Conway spheres q Sess(Π) such that σΦΠ is the identity permutation.

Proof. The corollary is deduced from the following facts: 1) ΦΠ sends crossing balls to crossing balls and therefore induces a non trivial permu- tation σ(ΦΠ) on the set of crossing balls of λ(Π) and likewise a non trivial permutation

19 q of the set of crossings of Π such that σ(ΦΠ) is the identity permutation. 2) the existence and unicity of the family of essential Conway circles Cess(Π) of Π (Theorem 2.1) imply the existence and unicity (up isotopy) of the family of essential Conway spheres.

Denote by Φ˜ the automorphism induced by ΦΠ on the tree A˜(K).

Corollary 3.2. ˜ ˜q The automorphism Φ on Ae(K) satisfies Φ = Identity.

q Proof. By Corollary 3.1, the permutation σ(ΦΠ) on Cess(Π) is the identity permuta- tion on Cess(Π). Hence it induces the identity on the essential structure tree Ae(K).

3.1 Visibility of the q-periodicity of alternating knots on S2.

We now define the notion of visibility of a q-periodicity of an alternating knot.

Definition 3.5. Let K be an alternating q-periodic knot. The q-periodicity of K 2Π is visible if K displays the q-periodicity of K as a q -rotation on an alternating projection called a q-visible projection.

In §3.2 and §3.3, we will describe how the q-periodicity of an alternating knot K acts on the structure trees by studying how it is reflected on the set of essential Conway circles as well as on the diagrams of Π(K).

According to Flyping Theorem, we have two cases:

q (1) Suppose no flypes are needed. Hence ΦΠ = φ is flat and φ = Id. By Ker´ekj´arto’s theorem ([3]), the principal part of φ which is a homeomorphism of (S2, Π) is topologi- cally conjugate to a rotation of S2 of order q without fixed points on Π. Consequently, the q-periodicity of K is visible on an alternating projection Π of K.

Remark 3.3. 1. If (S2, Π) is a q-periodic jewel without boundary, its q-periodicity is visible on Π because the jewels have no TBD and then no flypes are necessary to realize ΦΠ. 2. A torus knot of type (2, q) (q must therefore be odd) displays the q-periodicity on a standard alternating projection.

20 (2) In what follows, we will deal with the case where flypes may be involved.

3.1.1 Action of ΦΠ on Structure Trees.

Let Π be a reduced alternating projection of K and Cess(Π) be its set of essential Conway circles. Suppose further that Cess(Π) is not empty.

Since the essential structure trees Ae(K) and Ae(ΦΠ(K)) of a prime oriented alternating q-periodic knot K are isomorphic graphs, we can interpret this isomorphism as an automorphism Φe of the essential structure tree Ae(K).

Since the graph Ae(K) is a tree, the fixed point set F ix(Φe) is a non-empty subtree. So we have two possibilities:

1. Case where F ix(Φe) contains an edge E, corresponding to a Conway circle which is ΦΠ invariant.

2. Case where F ix(Φe) is reduced to a vertex V0. Remark 3.4. If Ae(K) is reduced to a single vertex V0 then K is either a rational link or a link corresponding to a jewel without boundary and the automorphism Φe is obviously the identity map.

3.2 and the essential decomposition of 2 Fix(Φe) (S , Π)

Figure 18: Orientation of the boundary points

§ In order to describe the two cases of F ix(Φe) stated in 3.1.1 in terms of the essential decomposition of (S2, Π), let us first describe how the boundary points of a tangle are oriented.

21 Let T be a tangle of a projection Π. The intersection points of ∂T ∩ Π are called the boundary points of T . By the orientation and the connectivity of Π, the four boundary points of T are oriented such that two are entry points and both others are exit points (see Fig. 18). Up to a change in the global orientation of the strands and π up to a rotation of angle 2 , we have the two possible configurations described in Fig. 18.

Lemma 3.1. Let γ be a canonical or essential Conway circle of Π. If γ is ΦΠ- invariant then q =2.

Proof. Note that each orbit of the action of ΦΠ has q elements except the two orbits composed by the fixed points.

Let ∆1 and ∆2 be the two disks in the projection sphere such that γ is ∂∆1 = ∂∆2.

The two disks ∆1 and ∆2 are either permuted or invariant by ΦΠ.

Assume ΦΠ permutes the two disks ∆1 and ∆2. Since ΦΠ preserves γ ∩ K and there are no fixed points by ΦΠ on K, we have that the set γ ∩ K is either an orbit of ΦΠ with q = 4 points, or two orbits with each q = 2 points. Assume q = 4. Let Sγ be the Conway sphere corresponding to γ which is invariant

by ΦΠ. The homeomorphism ΦΠ|Sγ is of order two and preserves the orientation of Sγ. Therefore, by Ker´ekj´arto theorem, it is topologically conjugate to a rotation of

order 2 of Sγ . Thus ΦΠ|Sγ has two fixed points, which must also be the fixed points

of ΦΠ. This implies that ΦΠ|Sγ preserves the orientation. Then the two connected 3 components of S − Sγ are invariant by ΦΠ, but this contradicts the hypothesis that ΦΠ(∆1) = ∆2. Therefore the two disks ∆1 and ∆2 are invariant by ΦΠ.

⋆ Assume ΦΠ|∆1 preserves the orientation. Then by Ker´ekj´arto’s theorem, ΦΠ|∆1 is topologically equivalent to a rotation of ∆1. As in the above case, since ΦΠ preserves γ ∩ Π and there are no fixed points on Π, the set γ ∩ Π is either an orbit of ΦΠ with q = 4 points or two orbits each with q = 2. The case q = 4 is excluded: since ΦΠ(∆i) = ∆i and γ ∩ Π is an orbit, the points in γ ∩ Π would be all entry points or all exit points, but this is impossible (for the orientation of the boundary points of a tangle of a projection as described above).

2 ⋆ ⋆ Assume ΦΠ|∆1 reverses the orientation and q > 2. Then ΦΠ|∆1 preserves the 2 orientation of ∆1 and has period q/2 > 1. Then ΦΠ|∆1 is conjugate to a rotation

such that its fixed points are also the fixed points of ΦΠ|∆1 , but this contradicts the

hypothesis that ΦΠ|∆1 reverses the orientation.

22 Proposition 3.1. If q ≥ 3, there are no edges in F ix(Φe).

Proof. Assume there is an edge E of Ae(K) fixed by Φe. The edge E corresponds to a Conway circle γE invariant by ΦΠ. Then the proposition follows from Lemma 3.1.

Corollary 3.3. If q ≥ 3, F ix(Φe)= V0 where V0 is a vertex of Ae(K).

3.3 Proof of Visibility Theorem 3.1 and Applications

3.3.1 Proof of Visibility Theorem 3.1

According to Remark 2.8, if the q-periodic knot K is a jewel without boundary or a torus knot of type (2, q), we are done. Since the non-torus rational knots are only 2-periodic (see for instance Theorem 3.1 in [10]), the hypothesis q ≥ 3 excludes the case of rational knots. All that remains is the case of a projection Π whose Cess(Π) is not empty. According to Corollary 3.3, the set F ix(Φ˜) is reduced to a vertex V0 representing a jewel or a TBD.

Proof.

1. Case where F ix(Φ˜) = V0 where V0 corresponds to a jewel J0 with non-empty boundary. Let γ1,...,γk be the boundary components of J0. Each essential Conway circle 2 γi bounds on S a disk ∆i which does not meet the interior of J0. Consider

the tangles Ti = (∆i, τ∆i ) where i =1,...,k. Hence the k underlying discs are distinct. Since J0 is a jewel, no flypes can occur in J0. Since ΦΠ does not leave the edges invariant, there are no invariant boundary circles and by Ker´ekj´arto’s

theorem applied to ΦΠ|J0 is topologically conjugate to a rotation and has two fixed points in the interior of J0. By using a flat homeomorphism, we can modify

Π such that ΦΠ|J0 is a rotation and acts freely on the k boundary components

23 of J0. After this modification, we continue to denote the new projection and homeomorphism respectively by Π and ΦΠ. Each circle γi has q images in its orbit. Thus k = nq and we have k distinct tangles Ti = (∆i, Π ∩ ∆i) with underlying disks ∆i where i =1,...k. Note that the k boundary components of J0 correspond to the k adjacent vertices to V0 q−1 Consider the disk ∆1 and its images φ(∆1),...,φ (∆1) denoted by

∆1, ∆2,..., ∆q−1

and the corresponding tangles

(∆1, τ∆1 ), (∆2, τ∆2 ),..., (∆q−1, τ∆q−1 )

i−1 where τ∆i = Π ∩ ∆i = ΦΠ (τ∆1 ) for i =1,...,q. j−1 j−1 Consider ∆j = φ (∆1). Then φ (τ∆1 ) ⊂ ∆j. Since φ(τ∆1 ) is equivalent

up to standard flypes to τ∆2 and since the disks ∆1 and ∆2 are distinct, we

can independently modify the projection τ∆2 = Π ∩ ∆2 by flypes such that ′ τ∆2 is replaced by φ(τ∆1 ). With these modifications, we have a new Π and ′ ′ ′ ′ ′ ′ ′ ΦΠ such that ΦΠ |∆1 does not need flypes. Hence ΦΠ |∆1 = φ |∆1 is flat and q ′ q φ|∆1 = ΦΠ|∆1 = Id. By repeating this process when necessary in all the orbits, we get a new alternating projection Π of K admitting the symmetry Φb which b bΠ is a q-rotation whose the two fixed points are inside J0.

2. Case where F ix(Φ˜) = V0 where V0 corresponds to a TBD T0 = (Σ, Σ ∩ Π). Since T0 is invariant by ΦΠ, the number of the visible crossings of Σ ∩ Π is mq. Since there are no edges invariant by ΦeΠ, the boundary components of Σ are distributed in ΦΠ-orbits of q elements and the total number of these components is sq for some integer s ≥ 1. By Ker´ekj´arto’s theorem, φ is equivalent to a rotation of order q with the two fixed points in Σ. We first modify the projection Π and Σ such that: - Σ is contained in a 2-sphere invariant by the q-rotation r,

- the visible crossings of T0 are in q twist regions, such that each twist region has s crossings and the q twist regions are symmetric with respect to the rotation r and

- the boundary components of T0 are sq circles distributed in s orbits of the action of r. ′ ′ ′ ′ This results in a new projection Π with the such modified TBD T0 = (Σ , Σ ∩ Π′). Then by a process similar to that described above, we will perform flypes if

24 necessary inside the sq tangles whose boundaries are the boundary components of T0, to obtain a projection Πb displaying the symmetry of order q.

Question: Are there any restrictions on the values of q in Visibility Theorem 3.1?:

Proposition 3.2. For prime alternating knots where F ix(Φ˜) = V0 with V0 corre- sponding to a TBD, only the periods q ≡ 1 mod2 are possible.

For each tangle of a projection Π, there are four boundary points located in North- West (NW), North-East (NE), South-East (SE) and South-West (SW). The projec- tion Π connects these four points in pairs with three possible connection paths (Fig. 19). If Π connects, (1) NW to NE and SW to SE, we have the H-connection path, (2 NW to SW and NE to SE, we have the V-connection path, (3) NW to SE and NE to SW, we have the H-connection path

Figure 19: Connection paths of a tangle

Proof. The proof is straightforward by an examination of the possible connection paths (Fig. 19) on the tangles of the TBD. In the case of knots, q cannot be even. It is interesting to compare this proposition to the result of [2].

Conclusion: Hence for an alternating periodic knot K with period q ≥ 3, there always exists a q-periodic alternating projection of K. The only possible obstruction case for a q-periodic alternating projection is when q = 2. Theorem 3.1 is the equiva- lent of the Order 4 Theorem 7.1 ([7]) in the study of the visibility of the +- achirality of alternating knots.

25 3.3.2 Applications

(1) Seifert’s algorithm applied to a q-periodic alternating projection of a knot K gives rise to a Seifert surface having the genus of K (see for instance [9]):

Proposition 3.3. There exists a q-equivariant orientable surface of K with minimal genus for q ≥ 3.

(2) From Visibility Theorem 3.1, we have:

Proposition 3.4. The crossing number of a prime alternating knot that is q-periodic with q ≥ 3 is a multiple of q.

(3) We now use Visibility Theorem 3.1 with the Murasugi decomposition of alternating links ([16],[17]) to study the 3-periodicity of the knot 12a634. We have

Proposition 3.5. The knot 12a634 is not 3-periodic.

Figure 20: 12a634 =31 ∗ 910

Figure 21: Adjacency graph G(12a634)

26 Proof. Let us consider the Murasugi decomposition of the knot 12a634 (Fig. 20) and its adjacency graph G(12a634). With the notations of [17], we have the Murasugi decomposition of 12a634 as: 12a634 =31 ∗ 910

where the knot 910 is the mirror image of 910. Thus, the adjacency graph G(12a634) is a tree with 2 vertices corresponding to the trefoil knot 31 and the knot 910 (Fig. 21).

The following lemma is useful for our analysis: Lemma 3.2. Suppose that a prime non-splittable oriented link L has a q-periodic alternating diagram and that its Murasugi adjacency graph is a tree with 2 vertices. Then the two constituent atoms of L are q-periodic.

Proof. According to Corollary 1 (in [5]), since the adjacent graph G(L) is a tree, its periodic automorphism has a fixed point which corresponds to a q-periodic atom. Moreover in the case where G(L) has only two vertices, the periodic automorphism is reduced to the identity and the two atoms are therefore both q-periodic.

By Theorem 3.1, if the knot 12a634 were 3-periodic, it would admit a 3-periodic alternating projection and we would be able to apply Lemma 3.2. However since the knot 910, one of the two constituent atoms of 12a634 is a non-torus rational knot, hence it is not 3-periodic; it is only 2-periodic (Theorem 6.1 in [10]). Hence by Lemma 3.2, we can conclude that 12a634 is not 3-periodic.

With this result, like S. Jabuka and S. Naik [11], we thus complete the tabulation of the q-periodic prime alternating twelve-crossing knots where q is an odd prime but our proof is not supported on computer calculations.

Remark 3.5. The Murasugi decomposition with its adjacency graph enables to deduce that the knot 12a634 is not q-periodic for any q ≥ 3, chiral and non-invertible ([16], [17]).

4 Addendum

There are overlapping results between the paper of Keegan Boyle [2] and this one. Both these papers use flypes as a main tool, but differ in their techniques. Since

27 non-trivial flypes lie completely in the arborescent part of alternating projections and since the canonical structure tree inherits the q-periodicity of a q-periodic alternating knot, we can adjust by flypes to derive a q-periodic alternating projection. Therefore our proof is somewhat constructive and also deals with the case of q-periodicity with q even.

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