If r Dehn surgery on a knot K in S³ yields a lens space, we call the pair (K, r) a lens space surgery, and we also say that K admits a lens space surgery, and that r is the coefficient of the lens space surgery. The task of classifying lens space surgeries, especially knots that admit lens space surgeries has been a focal point in low-dimensional topology. In 1990, Berge [9] pointed out a “mechanism” of known lens space surgeries, that is, doubly-primitive knots in the Heegaard surface of genus 2. Berge also gave a conjecturally complete list of such knots, described them by Osborne–Stevens’s “R-R diagrams” in [22], and classified such knots into three families, and into 12 types in detail:

• Knots in a solid torus (Berge–Gabai knots) : TypeI, II, ... and VI (Berge [10]).

  Dehn surgery along a knot in a solid torus whose resulting manifold is also a solid torus. TypeI consists of torus knots. TypeII consists of 2-cable of torus knots.

• Knots in a genus-one fiber surface : TypeVII and VIII (see Baker [6, 8] and also [24, 28]).

• Sporadic examples (a), (b), (c) and (d) : TypeIX, X, XI and XII, respectively.

Their surgery coefficients are also decided. They are called Berge’s lens space surgeries. The numbering VII–XII (after VI) are used by Baker [7, 8]. It is conjectured by Gordon [17, 18] that every knot of lens space surgery is a doubly-primitive knot. Recently, Greene [19] proved completeness of Berge’s list as a list of doubly-primitive knots.

In the present paper, we are concerned with the minor family (3). It is known that TypeIX and TypeXII (Berge’s (a) and (d)) are related, and that TypeX and TypeXI (Berge’s (b) and (c)) are related. Thus our targets are TypeIX and TypeX.

Notation 1.1

Throughout the paper, we let denote either TypeIX or TypeX, i.e., or X. Knots in are parametrized by an integer j with j ≠ 0, −1 ([9]), thus we call the knots k_IX(j) for TypeIX, and k_X(j) for TypeX, respectively. The knot is obtained from a torus knot by full-twists twice from a torus knot. For the precise construction of them, see Section 3.

Berge’s original classification (a)–(d) of sporadic examples in [9] was

• k_IX(j) with j > 0,

• k_X(j) with j > 0,

• k_X(j)! with j < −1,

• k_IX(j)! with j < −1,

respectively, see Deruelle–Miyaszaki–Motegi’s works [13, 14], where K! means the mirror image of a knot K.

The theory of A’Campo’s divide knots and links came from singularity theory of complex curves. A divide is originally a relative, generic immersion of a 1-manifold in the unit disk in ℝ², see Section 2. A’Campo [1, 2, 3, 4] formulated the way to associate to each divide P a link L(P) in S³. In the present paper, we regard a PL (piecewise linear) plane curve as a divide by smoothing the corners and controlling the size. Let X be the π/4-lattice defined by {(x, y)| cos πx = cosπy} in xy-plane. In this paper, we are interested in plane curves constructed as intersection of X and regions. When the region is L-shaped, we call such a curve L-shaped curve. See Figure 1, which is the starting examples of our project (Question 1.2). Two L-shaped curves present a same knot, the pretzel knot of type (−2, 3, 7) as a divide knot. Its 18-surgery and 19-surgery are lens spaces (by Fintushel–Stern [15]). Note that the areas of ℒ are equal to the coefficients of the lens space surgeries. They have different mechanism of lens space surgeries: 18-surgery is in TypeIII, 19-surgery is in TypeVII.

We are concerned with the following question:

Question 1.2

Is every knot of (Berge’s) lens space surgeries a divide knot?

Our main theorem is:

Theorem 1.3

Up to mirror image, every knot in TypeIX and TypeX in Berge’s list of lens space surgery is a divide knot.

In fact, we will construct plane curves that present the knots for and X, j ∈ ℤ with j ≠ 0, −1. They are a little complicated (generalized) L-shaped curves. In the next subsection, we roughly review the knots of Berge’s TypeIX and TypeX and explain how to construct . Theorem 1.5 is a more precise version of Theorem 13. We will also study the relation between the coefficients of lens space surgeries and the divide presentations.

We postpone the strict definition of (and P(j)) until Definition 3.1, after the definition of generalized L-shaped curves and the notion “type” of L-shaped curves in Section 2.

1.1. Presentation of the Sporadic Knots

First, we review the knots of Berge’s TypeIX and TypeX ( or X, j ∈ ℤ and j ≠ 0, −1). They are called knots of sporadic lens space surgeries. Table 1 is a list of some data on : the coefficients p of their lens space surgeries of as , the second parameter q of the resulting lens space L(p, q) and the genus of , which depends on the sign of j. Our convention about orientations of lens spaces is “p/q Dehn surgery along an unknot is –L(p, q)”.

Before a more precise version of the main theorem, we show some plane curves, see P(j) in Figure 2. Each curve P(j) is constructed as intersection of X and a region that consists of some rectangles sharing the left bottom corner. We call such a curve a generalized L-shaped curve, or an L-shaped curve for short. The precise definition and some notations will be given in Section 2. See Definition 3.1 for the strict definition of P(j) as L-shaped curves. For example, the curve P(3) will be defined as an L-shaped curve of type [(1, 7), (3, 5), (4, 3), (6, 2)]. We define plane curves from P(j) by adding a square twice in the sense [27, Lemma 4.2] as follows.

Definition 1.4.(Plane curves )

We let E_a (and E_b, respectively) denote the bottom edge (and the left edge) of the region of P(j), see Subsection 3.1 for the precise definition. For an integer j with j ≠ 0, −1, depending on whether or X, we construct a region of as follows, see Figure 3.

[TypeIX] We add a square along the bottom edge E_a first, and add another square along the lengthened left edge Eb¯.

[TypeX] We add a square along the left edge E_b first, and add another square along the lengthened bottom edge Ea¯.

We remark that, by the first square addition along an edge E_a (or E_b, respectively), the other edge E_b (or E_a) is lengthened as Eb¯ (or Ea¯). The second square is added along the lengthened one. For an edge E, we let l(E) denote the length of E. We have l(E_a) = |2j|, l(E_b) = |2j + 1| and l(Eb¯)=l(Ea¯)=l(Ea)+l(Eb).

The concrete version of the main theorem (Theorem 1.3) is:

We review theory of A’Campo’s divide knots and links briefly. We are interested in plane curves constructed as intersection of the π/4-lattice X and a region. We generalize L-shaped plane curve and decide the parametrizing notation.

2.1. Torus Knots

We start with a presentation of a (positive) torus knot as a divide knot. Let (a, b) be a pair of positive integers and B(a, b) a curve defined as an intersection of the π/4-lattice X and an a × b rectangle ℛ(a, b) whose every vertex is placed at a lattice point (∈ ℤ²), see Figure 4. If (a, b) is coprime, B(a, b) is a billiard curve in ℛ(a, b) with slope ±1.

2.2. Basic Facts on Divide Knots

The theory of A’Campo’s divide knots and links comes from the singularity theory of complex curves. A divide P is (originally) a relative, generic immersion of a 1-manifold in the unit disk D in ℝ². A’Campo [1, 2, 3, 4] formulated the way to associate to each divide P a link L(P) in S³. We regard S³ as

and the original construction is

where T_uP is the subset consisting of vectors tangent to P in the tangent space T_uD of D at u. In the present paper, we regard a PL (piecewise linear) plane curve as a divide by smoothing corners and controlling the size.

Some characterizations of (general) divide knots and links are known, and some topological invariants of L(P) can be gotten from the divide P directly. Here, we list some of them.

2.3. Curves defined by Regions

In xy-plane, a lattice point or an integer vector (k, l) ∈ ℤ² is called even or odd, if k + l is even or odd, respectively. Double points of the π/4-lattice X = {(x, y)| cos πx = cosπy} are at even points. We are interested in curves constructed as intersection of X and a region, as a generalization of Lemma 2.1.

The lattice X has some symmetries: We let r_x denote the reflection along y-axis, r_x(x, y) = (−x, y), R_π/2 the π/2-rotation (along the origin), and +v⃗_ev a parallel translation by an even vector. Symmetry of the lattice X is generated by r_x, R_π/2 and some v⃗_ev. A curve constructed as intersection of X and a region ℛ does not change by the action (on ℛ) of the symmetry of X. We also use a parallel translation +v⃗_od by an odd vector to place a region ℛ well.

2.4. L-shaped Curves

We generalize L-shaped curves from the old version. It is an extension of “L-shaped curves” in [27, Section 3.2], but the notation (parametrization) is changed.

2.5. Deformation of Curves, Couture Move

We show a lemma on a deformation of divide presentations of torus knots of type (n, 2n − 1) geometrically. We call the method Couture move for short. The move consists of some Δ-moves in Lemma 2.2(7). It was pointed out in the private communication of the author and Olivier Couture in the opportunity of a conference “Singularities, knots, and mapping class groups in memory of Bernard Perron” held in Sept. 2010. By the move, we can deform the divide presentations of T(n, 2n−1) from the curve B(n, 2n−1) in Lemma 2.1 to a generalized L-shaped curve of special type. An example is shown in Figure 9.

We give a precise definition of the plane curves P(j), verify the formula in Lemma 1.8 and prove Theorem 1.5 and the lemmas. We start the proof with Baker’s description in [6, 8] of sporadic knots, and we use Couture moves on divides in the second half of the proof.

3.1. Precise Definition of Curves

We define divides P(j) by the method introduced in the last section.

3.2. Proof of the Main Theorem

In the first half of the proof, we study the knots in the usual diagram and Dehn surgery description. In the second half, we will use divide presentation.

We start with Baker’s Dehn surgery description of the knots in Figure 12 from [6]. Throughout the paper, we fix

It is easy to see that the framed sublink of thick seven components presents S³, by usual weighted tree diagram in the right half of Figure 12. We call the sublink the non-trivial diagram of S³. In the resulting S³, the thin component is the sporadic knot as a knot.

[Case j > 0] First, we assume that j > 0 until the final paragraph. To get a usual diagram of the knot , we have to chase the thin component (with framing) during the deformation from the non-trivial diagram to the empty diagram of S³. It is not easy but straight forward. In the middle of the process, we reach a diagram in Figure 13, where boxed +1 in the diagram means a right handed full-twist. The thin component k(j) is a torus knot T(j, j + 1) with framing j(j + 1), which is a coefficient of reducible Dehn surgery. We name the four component link L(j) = k(j) ∪ u₋₁ ∪ u_α ∪ u_β, where u_x the x-framed unknotted component for x = −1, α, β.

Second, we decompose the full-twist at the boxed +1 as two half-twists, denoted by +12 in triangles, and deform the diagram by isotopy as in Figure 14. Note that the half-twist in the right is of j + 1 strings. We can see that L(j) is strongly invertible with respect to the horizontal axis, see Lemma 2.2(5). As a quotient of the involution, ignoring the crossing data (over or under), we have a plane curve properly immersed in the half plane. The curve can be modified as in Figure 15. This is a divide presentation of L(j). It can be checked by Couture–Perron’s method in Figure 6. This process is related to the original construction of divide knots: For a link of singularity of a complex plane curve, the divide is a real part of a “good” perturbation (called real Morsification) of the equation of the singularity.

In the divide presentation of L(j) in Figure 15, a line segment ℓ is placed slightly different whether j is even or odd. We name the plane curve C(j) = c(j)∪ℓ∪a∪b, where c(j) presents k(j) = T(j, j+1) by Lemma 2.1, since c(j) is an L-shaped curve of type [(j, j + 1), (j + 1, 1)] and is isotopic to B(j, j + 1). The line segments ℓ, a, b present u₋₁, u_α, u_β, respectively, as a divide presentation.

The linking matrix of L(j) with a suitable orientation of the link

is equal to the matrix of the number of intersection points of the components of the divide by Lemma 2.2(3).

From now on, we go into the second half of the proof, and study the knots and links by divide presentation. We use Δ-moves on divides freely, see Lemma 2.2(7). We have two (or three) steps: (i) Blow-down along ℓ (take a full-twist along u₋₁, (ii) (Only if j is odd) Modify the curve by some Δ-moves, and (iii) Deform the curve by Couture moves. Our goal is the divide P(j) with edges ā, b̄, where ā (and b̄) is a small parallel push-off of the bottom edge E_a (the left edge E_b) of the region of P(j) into the interior.

(Step (i)) We take a right handed full-twist of k(j) ∪ u_α ∪ u_β along the unknot u₋₁. We name the resulting link

This full-twist is done as a blow-down along the line ℓ, i.e., by adding a square by Lemma 2.3. In the case where j is even, we slide ℓ and b by Δ-moves as the first picture in Figure 16 and add a square. Otherwise, we add a square along the bottom edge as in Figure 17.

(Step (ii)) If j is odd, the curve has a terminal point at the right bottom corner. We move the terminal point (and its segment) up along the right edge, as the second deformation of in Figure 17. We also slide the other added part at the bottom to the right by some Δ-moves.

(Step (iii)) The resulting curve is near L-shaped, but the lines are not in the required position. Here we use Couture move, see the second halves of Figure 16 in the case j is even, and Figure 17 in the case j is odd. By some obvious Δ-moves, we have the required curve P(j) ∪ ā ∪ b̄, which presents L(j)¯ as a divide link.

The sublink uα¯∪uβ¯ is a Hopf link and the framings are (α+1, β+1) = (−1, −2) for TypeIX, or = (−2, −1) for TypeX. By the construction of in Definition 1.4, and by the correspondence between adding a square to a divide and a full-twist of the divide link in Lemma 2.3, we have the required divide presentation of with j > 0.

ww [Case j < −1] Finally, we study the case j < −1. The method is similar. We define an integer j′ by j = −(j′+1) for figures. Starting with Baker’s description in Figure 12, we have a link L(j) in Figure 18.

Divide description of L(j) is in Figure 19, contrast to Figure 15. Especially, the non-trivial component c(j) and c(j′) (ex. c(−6) and c(5)) are the same curves presenting T(j, j + 1) = T(j′ + 1, j′) but the other segment components are placed differently.

In Figure 20, we show some deformations. Figures 18, 19, 20 (Case j < −1) are contrast to Figures 13, 15, 16 and 17 (Case j > 0), respectively.