Article
Kyungpook Mathematical Journal 2020; 60(2): 255277
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Divide Knot Presentation of Knots of Berge's Sporadic Lens Space Surgery
Yuichi Yamada
Dept. of Mathematics, The University of ElectroCommunications, 151, Chofu gaoka, Chofu, Tokyo, 1828585, Japan
email : yyyamada@eone.uec.ac.jp
Received: March 28, 2018; Revised: June 7, 2019; Accepted: June 10, 2019
Abstract
Divide knots and links, defined by A’Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a sequel of the author’s previous result that every knot in the major subfamilies of Berge’s lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an Lshaped curve as a divide knot. In the present paper, Lshaped curves are generalized and it is shown that every knot in the minor subfamilies, called sporadic examples of Berge’s lens space surgery, is presented by a generalized Lshaped curve as a divide knot. A formula on the surgery coefficients and the presentation is also considered.
Keywords: Dehn surgery, lens space, plane curve
1. Introduction
If

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 2cable of torus knots.

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

k _{IX} (j ) withj > 0, 
k _{X} (j ) withj > 0, 
k _{X} (j )! withj < −1, 
k _{IX} (j )! withj < −1,
respectively, see Deruelle–Miyaszaki–Motegi’s works [13, 14], where
The theory of A’Campo’s
We are concerned with the following question:
Question 1.2
Our main theorem is:
Theorem 1.3
In fact, we will construct plane curves that present the knots for and X,
We postpone the strict definition of (and
1.1. Presentation of the Sporadic Knots
First, we review the knots of Berge’s TypeIX and TypeX ( or X,
Before a more precise version of the main theorem, we show some plane curves, see
We let
[TypeIX] We add a square along the bottom edge
[TypeX] We add a square along the left edge
We remark that, by the first square addition along an edge
The concrete version of the main theorem (Theorem 1.3) is:
We will also show the following lemmas.
The proof of Theorem 1.5 is divided into two parts: In the first half, starting with Baker’s Dehn surgery description in [6, 8], we study the knots by usual diagrams. In the second half, we will use divide presentations. We will introduce a convenient method, which we call
Next, we remark on the relation between the surgery coefficients and the area of the regions (of the curves). For an Lshaped region ℒ, we let Area(ℒ) denote the area of ℒ, and Conc(ℒ) the number of concave points of ℒ, respectively, see Definition 2.4 for the precise definition of concave points. In the simpler case (Conc(ℒ) = 1) for the knots in the major subfamilies of lens space surgeries, the formula was “Area(ℒ) – Surgery coefficient = 0 or 1”(Theorem 1.4 in [27], see also [25]). It is modified to:
Lemma 1.8 will be verified by Table 2 in Subsection 3.1.
In the next section, we review theory of A’Campo’s divide knots and links briefly and generalize Lshaped plane curve and decide the parametrizing notation. We will introduce a convenient method, which we call
2. Divide Knots and Plane Curves
We review theory of A’Campo’s divide knots and links briefly. We are interested in plane curves constructed as intersection of the
2.1. Torus Knots
We start with a presentation of a (positive) torus knot as a divide knot. Let (
Strictly, the curve
2.2. Basic Facts on Divide Knots
The theory of A’Campo’s
and the original construction is
where
Some characterizations of (general) divide knots and links are known, and some topological invariants of

(1)
L (P )is a knot (i.e., connected) if and only if P is an immersed arc. 
(2)
If L (P )is a knot, the unknotting number, the Seifert genus and the 4genus of L (P )are all equal to the number d (P )of the double points of P. 
(3)
If P =P _{1} ∪P _{2}is the image of an immersion of two arcs, then the linking number of the two component link L (P ) =L (P _{1}) ∪L (P _{2})is equal to the number of the intersection points between P _{1}and P _{2}. 
(4)
If P is connected, then L (P )is fibered. 
(5)
Any divide link L (P )is strongly invertible. 
(6)
A divide P and its mirror image P !present the same knot or link: L (P !) =L (P ). 
(7)
If P _{1}and P _{2}are related by some Δmoves, then the links L (P _{1})and L (P _{2})are isotopic: If P _{1} ~_{Δ}P _{2}then L (P _{1}) =L (P _{2}),see Figure 5. 
(8)
Any divide knot is a closure of a strongly quasipositive braid, i.e., a product of some σ _{ij} in Figure 5.
For theory of divide knots, see also [11, 21] and “transverse ℂlinks” defined by Rudolph [23]. In [12] Couture and Perron pointed out a method to get the braid presentation from the divide in a restricted cases, called “ordered Morse” divides. We can apply their method, see also Hirasawa’s method [20].
Finally we recall an operation “adding a square” on divides
Adding a square is related to “blowdown”. Here, blowdown along
2.3. Curves defined by Regions
In
The lattice
We are interested in curves constructed as intersection of

A region ℛ is a union of a finite number of rectangles.

Each edge of the rectangles in ℛ is horizontal or vertical.

Each vertex of ℛ is at a lattice point.

Difference vectors of any pair of concave points of the region ℛ are even.
Here, a concave point in (iv) is defined as follows: A boundary point
If a concave point
For a region ℛ satisfying the condition (i),(ii),(iii) and (iv), either
We describe a curve
2.4. Lshaped Curves
We generalize
See Figure 8. Let
denote a sequence of lattice points (∈ ℤ^{2}) in
We define a region
We will call this region
If such a region defines a generic immersed curve in the sense of Definition 2.5, we call the curve
An Lshaped region of type [{(
It is easy to see:
2.5. Deformation of Curves, Couture Move
We show a lemma on a deformation of divide presentations of torus knots of type (
Note that
For
For
Note that the case
First, we give a proof in the case
For the case
Definition 2.11.(Couture move)
The deformations of the curves in the proof of Lemma 2.9, especially the first and the second halves in Figure 10, and their generalizations for larger
In the present paper, we will use Couture moves for generalized Lshaped curves in Figures 16, 17 and 20.
3. Details on Sporadic Knots and Proof
We give a precise definition of the plane curves
3.1. Precise Definition of Curves
We define divides
For an integer
(Case
(Case
We define a plane curve
By Lemma 2.7, it is easy to see
The plane curve is constructed by adding a square twice as in Definition 1.4. Since a square addition along an edge
On the other hand, the area of the region of
We calculate them also in the cases
Next, we calculate and verify that the numbers of double points of
Since the number of double points increases by
(Case
(Case
They are equal to the genus of the knots
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
[Case
Second, we decompose the fulltwist at the boxed +1 as two halftwists, denoted by
In the divide presentation of
The linking matrix of
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 Δmoves on divides freely, see Lemma 2.2(7). We have two (or three) steps: (i) Blowdown along
(Step (i)) We take a right handed fulltwist of
This fulltwist is done as a blowdown along the line
(Step (ii)) If
(Step (iii)) The resulting curve is near Lshaped, but the lines are not in the required position. Here we use Couture move, see the second halves of Figure 16 in the case
The sublink
ww [Case
Divide description of
In Figure 20, we show some deformations. Figures 18, 19, 20 (Case
We study a sublink
In [26], a divide knot presentation of cable knots (under some conditions) is studied. Here we use Δmoves on divides freely.
First, assume
The proof in the case
Figures
Tables
Data on Sporadic knots
knot  genus ( 
genus ( 


22 
−(11 
11 
11 

22 
−(11 
11 
11 
Area of the plane curve
Curve  Coeff. 
Area (Case 
Area (Case 

22 
22 
22 

22 
22 
22 
References
 N. A’Campo.
Le groupe de monodromie du déploiement des singularité isolées de coubes planes I . Math Ann.,213 (1975), 132.  N. A’Campo.
Generic immersions of curves, knots, monodromy and gordian number . Inst Hautes Etudes Sci Publ Math.,88 (1998), 151169.  N. A’Campo.
Planar trees, slalom curves and hyperbolic knots . Inst Hautes Etudes Sci Publ Math.,88 (1998), 171180.  N. A’Campo.
Real deformations and complex topology of plane curve singularities . Ann Fac Sci Toulouse Math.,8(6) (1999), 523.  VI. Arnold, SM. GuseinZade, and AN. Varchenko.
Singularities of differentiable maps, Volume II. Monodromy and asymptotics of integrals . Monographs in Mathematics,83 , Birkhauser Boston, Inc, Boston, MA, 1988.  Baker. KL.
Knots on Oncepunctured torus fibers , Ph D dissertation, The University of Texas, Austin, 2004.  KL. Baker.
Surgery descriptions and volumes of Berge knots I: Large volume Berge knots . J Knot Theory Ramifications.,17(9) (2008), 10771197.  KL. Baker.
Surgery descriptions and volumes of Berge knots II: Descriptions on the minimally twisted five chain link . J Knot Theory Ramifications.,17(9) (2008), 10991120.  J. Berge.
. Some knots with surgeries yielding lens spaces., (1990):Unpublished manuscript.  J. Berge.
The knots in D^{2}×S^{1} which have nontrivial Dehn surgeries that yield D^{2×S}1 . Topology Appl.,38(1) (1991), 119.  S. Chmutov.
Diagrams of divide links ,131 (2003), 16231627.  O. Couture, and B. Perron.
Representative braids for links associated to plane immersed curves . J Knot Theory Ramifications.,9 (2000), 130.  A. Deruelle, K. Miyazaki, and K. Motegi.
Networking Seifert surgeries on knots. II. The Berge’s lens surgeries . Topology Appl.,156(6) (2009), 10831113.  A. Deruelle, K. Miyazaki, and K. Motegi.
Networking Seifert surgeries on knots . Mem Amer Math Soc.,217(1021) (2012), 130 pp.  R. Fintushel, and R. Stern.
Constructing Lens spaces by surgery on knots . Math Z.,175 (1980), 3351.  H. Goda, M. Hirasawa, and Y. Yamada.
Lissajous curves as A’Campo divides, torus knots and their fiber surfaces . Tokyo J Math.,25(2) (2002), 485491.  C. McA Gordon, Dehn surgery on knots, Proceedings of the International Congress of Mathematicians, Math. Soc. Japan, (1991), 631642.
 C. McA Gordon. Dehn filling: a survey,
42 , Knot theory, Banach Center Publ., , 1998:129144.  JE. Greene.
The lens space realization problem . Ann of Math (2).,177(2) (2013), 449511.  M. Hirasawa.
Visualization of A’Campo’s fibered links and unknotting operation . Topology Appl.,121 (2002), 287304.  CVQ. Hongler, and C. Weber.
The link of an extrovert divide . Ann Fac Sci Toulouse Math (6).,9(1) (2000), 133145.  RP. Osborne, and RS. Stevens.
Group presentations corresponding to spines of 3manifolds III . Trans Amer Math.,234 (1977), 245251.  L. Rudolph.
Knot theory of complex plane curves . Handbook of Knot Theory, Menasco. WW, and Thistlethwaite. MB,, Elsevier B. V, Amsterdam, 2005:349427.  Y. Yamada.
Berge’s knots in the fiber surfaces of genus one, lens space and framed links . J Knot Theory Ramifications.,14(2) (2005), 177188.  Y. Yamada.
A family of knots yielding graph manifolds by Dehn surgery . Michigan Math J.,53(3) (2005), 683690.  Y. Yamada.
Finite Dehn surgery along A’Campo’s divide knots . Singularity Theory and its Applications, Adv Stud Pure Math.,43 (2006), 573583.  Y. Yamada.
Lens space surgeries as A’Campo’s divide knots . Algebr Geom Topol.,9(1) (2009), 397428.  Y. Yamada.
Canonical forms of the knots in the genus one fiber surfaces . Bull Univ ElectroCommun.,22(1) (2010), 2531.