On construction of surface-knots

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Figure 1: The pre-image of the partial diagram (left) and its copy (right). 2 ... (3) Fm = Fn if m = n. (4) Dn(Γ) is a trivial surface-knot diagram, where Γ = Jn i=1 γi.
On construction of surface-knots A. Mohamad∗and T. Yashiro†

Abstract In this paper we present a construction of a family of surfaceknot diagrams with cross-exchangeable curves, along which we can change the crossing information to obtain trivial diagrams. These diagrams also satisfy a kind of minimality, called d-minimal surfaceknot diagrams.

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Introduction

A surface-knot is a closed oriented surface embedded in 4-space. A surfaceknot diagram of a surface-knot is the image of the surface-knot in 3-space under the orthogonal projection proj : R4 → R3 defined by proj(x1 , x2 , x3 , x4 ) = (x1 , x2 , x3 ), equipped with crossing information. The image of the projection may have double curves, isolated triple points and isolated branch points [1]. If F is a surface-knot, then we denote its surface-knot diagram by DF . The image proj(F ) without crossing information will be denoted by |DF |. In this paper, we will give an algorithm to obtain a union of double curves in a surface-knot diagram called cross-exchangeable curves (c-e curves), also we will define a crossing change operation along these curves (see Lemma 2.1). The operation is done along c-e curves to exchange the crossing information on the curve (cf. [14]). Therefore, it is natural to ask the following question: ∗

Mathematics Division Department of Mathematical and Physical Sciences, College of Arts and Sciences, University of Nizwa † Department of Mathematics and Statistics, College of Science, Sultan Qaboos University

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Q. For any surface-knot diagram, can we obtain a diagram of a trivial surface-knot by crossing changes along some c-e curves? In this paper we construct surface-knot diagrams with c-e curves (Theorem 1.1 and Theorem 1.2), along which the crossing change operation yields a diagram of a trivial surface-knots (cf. [7][5]). Note that we can always obtain such a surface-knot diagram by the construction given in Satoh’s paper [12]. However, Satoh’s method yields a surface-knot diagram on which we may change the connection of double curves so that some triple points can be eliminated (see [13]). While our construction yields a surface-knot diagram in which one cannot change the connection of double curves easily. This diagram will be called a d-minimal surface-knot diagram (see Section 3 for the definition). The organisation of this paper is the following. In Section 2 we define c-e curves and cross-change operations along the c-e curves. If Γ is a union of c-e curves in a surface-knot diagram DF , then the resulting diagram by applying crossing change operations along Γ will be denoted by DF (Γ). In Section 3, some elementary moves called Roseman moves [10] of surface-knot diagrams will be introduced. One of them, R-6 move changes the connection of double curves. A surface-knot diagram on which one cannot apply R-6 move will be called a d-minimal surface-knot diagram (see Definition 3.1). Sections 4 and 5 are devoted to proving Theorem 1.1 and Theorem 1.2 by constructing the desired surface-knot diagrams. Our construction starts from a special type of surface-knot diagram equipped with some branch points and triple points. The double decker set (the pre-image of singularities) is depicted in Figure 1. The left diagram, if we paste the sides A and B, then an annulus

A

A′

B

B′

Figure 1: The pre-image of the partial diagram (left) and its copy (right). 2

with the double decker set is obtained. If we paste discs to each boundary component, we obtain a surface-knot diagram of a trivial 2-knot. Paste copies of this partial diagram as in Figure 1. For example, paste the side B of the left and the side of A′ of the right and so on. Then we obtain a double decker set on an annulus. This annulus can be embedded in a surface to obtain a diagram of a trivial surface-knot with the double decker set. The diagram has one c-e curve. We obtain the following results. Theorem 1.1. Let F be a closed oriented surface and let δ be a simple closed curve on F . Then F can be embedded in R4 as a tri-colourable surface-knot F satisfying the followings: (1) it has a d-minimal surface-knot diagram DF , and (2) there exists a simple closed curve γ ⊂ DF such that γ is a c-e curve and the corresponding lower decker set to γ is isotopic to δ in F . (3) DF (γ) is a trivial surface-knot diagram. Theorem 1.2. There exist a tri-colourable surface-knot Fn with genus g ≥ 0 and its surface-knot diagram Dn of Fn such that (1) Dn is d-minimal. (2) Dn has n c-e curves γ1 , γ2 , . . . , γn such that (3) Fm 6= Fn if m 6= n. (4) Dn (Γ) is a trivial surface-knot diagram, where Γ =

2

Sn

i=1

γi .

Cross-exchangeable curves

Perturbing a surface-knot F if necessary, we can assume that there are only double curves, isolated triple points and isolated branch points in a surfaceknot diagram of F . In this paper we always deal with this kind of diagram.

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Let h be the height function h : R4 → R defined by h(x, y, z, w) = w. Let A, B be a subset of R4 . Then h(A) < h(B) means for all points x ∈ A and y ∈ B, h(x) < h(y). The cardinality of a set A will be denoted by #A. We recall the definition from [2][1] about the singular set of a surface-knot diagram. Definition 2.1. The singular set of proj|F is S = {x ∈ F | #(proj|F )−1 (proj|F (x)) > 1} The set S is a union of two families Sa = {s1a , s2a , . . . , ska } and Sb = {s1b , s2b , . . . , skb } of immersed open intervals and immersed circles in F such that proj(sia ) = proj(sib ), (i = 1, 2, . . . , k). For xa ∈ sia and xb ∈ sib , (i = 1, 2, . . . , k), if proj(xa ) = proj(xb ), then h(xb ) < h(xa ). We denote the union of the closures of sia by Sa and the union of the closures sib be Sb . We call Sa an upper decker set and Sb a lower decker set and call Sa ∪ Sb the double decker set [2]. The double decker set includes the branch points. Near a triple point of a surface-knot diagram, there are three discs in F called the bottom, middle and top sheets [1] denoted by DB , DM and DT respectively. The bottom sheet DB contains a crossing point of the lower decker set. We add crossing information to the crossing as follows. An upper arc in DB is the arc which is projected onto proj(DB ) ∩ proj(DT ) and a lower arc in DB is the arc which is projected onto proj(DB ) ∩ proj(DM ). Definition 2.2. Let DF be a surface-knot diagram of F . Let Γ be a union of double curves γ0 , γ1 , . . . , γk in DF : Γ=

k [

γi

i=0

The curves in Γ are called a cross-exchangeable curves or c-e curves as an abbreviation, if the crossing information on Γ (upper and lower information) can be exchanged. If we exchange the crossing information along c-e curves in Γ, then the resulting diagram will be denoted by DF (Γ). We call the operation that exchanges the crossing information along c-e curves a crossing change operation (cf. [14]). 4

Note that every surface-knot diagram DF with singularities has c-e curves. For instance, take the union of all double curves in the diagram as Γ. Then DF (Γ) is a surface-knot diagram of the mirror of F .

2.1

Crossing change deformations

Here we describe how a crossing change operation is done. Let γ ⊂ DF be a closed double curve without triple points. Take a point p on γ. There is a small ball neighbourhood B(p) of p in R3 such that there exist two discs DU and DL in F ⊂ R4 such that proj−1 (B(p)) ∩ F = DU ∪ DL , h(DL ) < h(DU ), proj(DU ) ∩ proj(DL ) ⊂ γ By a homotopy move, push DU into DL so that the deformed DU and DL have a pair of singular points above γ. We call this a push down operation (cf. [3]). This local move can be applied at any double point. We use a schematic diagram in Figure 2. In Figure 2, γ is a closed double curve without triple points and p is an initial point on γ. γ

γ

γ

p

p

p

Figure 2: Push down operation and eliminating singular points Apply the push down operation at p so that we have a pair of crossings called singular points depicted as crosses. Since γ does not have any triple point, the singular points bound a simple sub-arc in which the crossing information has changed. We can move both of singular points along γ away from the initial point p so that they meet at some point on γ and the singular points can be eliminated by a push down operation at the shorter sub-arc. If a double curve γ is bounded by a pair of branch points and has no triple points, then we can apply the push down operation to obtain a pair of singular 5

points. If a singular point and a branch point are joined by a simple double curve without triple points, then move the singular point towards the branch point so that the crossing information on the double curve is changed and the singular point can be eliminated. Suppose that there is a triple point p on γ. There is a small ball neighbourhood B(p) of p in R3 and there are three discs DB , DM and DT in F ⊂ R4 above p such that proj−1 (B(p)) ∩ F = DB ∪ DM ∪ DT , h(DB ) < h(DM ) < h(DT ), proj(DB ) ∩ proj(DM ) ∩ proj(DT ) = {p}. Near the triple point, there are three types of double edges meeting together: (b/m)-edge, (m/t)-edge and (b/t)-edge; the (x/y)-edge is formed by the proj(DX ) and proj(DY ) for X, Y ∈ {B, M, T } [11]. We call (b/m)-edge and (m/t)-edge nice-edges. Note that we cannot apply the push down operation at a triple point along (b/t)-edge as the deformed DT intersects DM . On the other hand, we can apply the push down operation at a triple point on nice-edges. In order to apply the push down operation along (b/t)-edge, we need to apply the operation on one of two nice-edges first, so that it changes (b/t)-edge into either (b/m)-edge or (m/t)-edge. In Figure 3, schematically we draw the possible situations around a triple point when the push down operations are applied. (m/t)

(b/t) q (b/m)

(m/t) q

q (m/t)

(b/m)

(b/t)

(b/t)

(b/t)

q (b/m)

(m/t)

(b/m)

Figure 3: Situations of a triple point. Let DF be a surface-knot diagram of a surface-knot F . We can construct a set of c-e curves as follows: take a double curve γ0 in DF . Then there are three cases: 6

Case 0: γ0 has no triple points. Case 1: γ0 has triple points whose (b/t)-edges are not on γ0 . Case 2: γ0 has triple points {t1 , t2 , . . . , tk } whose (b/t)-edges are on γ0 . For Case 0 and Case 1, put Γ = γ0 . One can apply the push down operation at any double point or any triple points on γ0 and move them away from the initial point to eliminate the singular points. For Case 2, apply a push down operation at each triple point ti (i = 1, 2, . . . , k) on one of two nice edges, say γi crossing γ0 at ti to obtain a pair of singular points. Then set k [ Γ= γi i=0

For each γi , (i = 1, 2, . . . , k), check if it meets Cases 0 or 1, then apply the push down operation according to the cases. If it meets Case 2, then we will find double curves γi1 , γi2 , . . . , γil , (i = 1, 2, . . . , k) that the push down operations will be applied. Then add these double curves to Γ: Γ=

k [

γi ∪

i=0

k [ l [

γij

i=1 j=1

We continue this process until we meet the stage that we do not need to add any double curves. Since the number of double curves is finite, this procedure will finally stop. Then each arc joining triple points in the double curves in Γ will have a pair of singular points. They can be eliminated by the the push down operation. The singular point between a triple point and a branch point can be eliminated by moving the singular point to the branch point so that the crossing information will be exchanged along the arc. This exchanges the crossing information on Γ. The authors understand that S. Kamada pointed out the existence of c-e curves in a surface-knot diagram. Lemma 2.1. Suppose that a triple point t exists in a surface-knot diagram DF and there exists cross-exchangeable curve c in DF such that c passes t once. Then the followings hold:

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(a) If the type of double curve near t along c is an (m/t)-edge, then the crossing change operation along c exchanges the middle sheet and the top sheet as well as it changes the crossing information at t of the lower decker set. (b) If the type of double curve near t along c is (b/m)-edge, then the crossing change operation along c exchanges the bottom sheet and the middle sheet. Proof. The crossing change operation along (m/t)-edge exchanges the roles of the middle and top sheets also the crossing change operation along (b/m)edge exchanges the roles of the bottom and middle sheets. Thus we need

(a)

(b)

Figure 4 to show the second part of (a). The bottom sheet at t is penetrated by the top sheet and the middle sheet, which form a (m/t)-edge at t. By the assumption this short edge lies along the cross-exchangeable curve c and thus we can change the crossing information at t along c. Since c passes t once, this operation changes the crossing information of the lower decker set at t.

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3

Pinch discs and descendent discs

D. Roseman [10] introduced seven types of elementary moves for surfaceknot diagrams. It is known that a surface-knot diagram is deformed into an equivalent diagram by a finite sequence of Roseman moves [10]. In Figure 5 and Figure 6 we present six moves instead of seven (cf. [4]). Each move from left to right is denoted by R-X + and right to left by R-X − except R-6 move. In [15] it is shown that Roseman’s seven moves are described by these six moves (see also [8]). In this paper we call these six moves also Roseman moves.

R-1±

R-2±

R-3±

Figure 5

R-4±

R-5±

R-6

Figure 6 A generic disc equipped with a pair of branch points and a simple double segment between them, is called a bug. This is produced by R-4+ move (see Figure 6).

3.1

Pinch disc

For a double point in a surface-knot diagram, the vector n1 normal to the upper sheet and the vector n2 normal to the lower sheet form a 2-frame 9

{n1 , n2 }. We can choose the third vector v such that the 3-frame {n1 , n2 , v} determines the right handed orientation. The orientation of the vector v determines the orientation of the double curve [1]. We put an orientation to the double curve of the bug so that the orientation will be denoted by an arrow from one branch point. Let l be an immersed interval α : [0, 1] → R2 with a single crossing in α((0, 1)). The product l × [0, 1] forms an immersed disc in R3 with single double arc. The loop l × {a}, a ∈ (0, 1) bounds a disc in R3 . Then we can shrink the loop along the disc into a point then create a pair of branch points. This operation can be done by the Roseman movesR-4+ and R-6 shown in Figure 7 [4][15]. The disc shrunk into a point will be called a pinch disc. We R-4±

R-6

Figure 7: The pinch move. will call the deformation depicted in Figure 7 a pinch move (see [10][8]). It is possible that we obtain an isolated bug after a pinch move is applied, and the bug can be eliminated by R-4− move (see Figure 8). We will call this pinch disc a trivial pinch disc.

R-4−

Figure 8: A trivial pinch disc.

3.2

Descendent disc

Let α and β be arcs on an oriented surface F parametrized by 0 ≤ t ≤ 1 and 0 ≤ u ≤ 1. We define a sign at a crossing point of α ∩ β. Let v1 = α′ (t), and let v2 = β ′ (u). Then if the pair {v1 , v2 } at the crossing point forms the same orientation on F , then the sign is +1, otherwise −1. 10

Let γ be a simple arc on a surface-knot diagram DF of a surface-knot F . Then the lift of γ is an arc on F such that the projected image of the arc coincides γ. We denote the lift by γ e.

For a surface-knot diagram DF of a surface-knot F , the closure of a complementary region of |DF | in R3 may contain a disc D which satisfies the following conditions: (1) ∂D consists of a union of two arcs γ1∗ ∪ γ2∗ . (2) γ1∗ and γ2∗ can be extended in DF in small neighbourhoods of end points along DF denoted by γ1 and γ2 respectively. (3) The lifts γ ei (i = 1, 2) are disjoint.

(4) The both pre-images of double points on one of the lift γei∗ (i = 1, 2), are on Sa and the other are on Sb . (5) γei∗ does not meet with Sa ∪ Sb other than the pre-images of crossings. (6) Signs of crossing points of γ1 and γ2 are opposite. The disc D satisfying the conditions above is called a descendent disc [4]. If a descendent disc D exists, then R-6 Roseman move can be applied as γ1 D γ e1 e1 E

γ e2

E2

γ2 E1

e2 E

Figure 9 follows: Take a small closed neighbourhood N (D) of D in R3 so that we can suppose that N (D) ∩ E1 is a small disc neighbourhood of γ1 . We denote this neighbourhood by N1 .

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Note that N1 is embedded in R3 and it separates N (D) into two pieces V and V ′ , where D ⊂ V , V is homeomorphic to a closed 3-ball and N1 ⊂ ∂V . The boundary ∂V is the union of N1 and the closure of the complement of N1 in ∂V , denoted by N2 : ∂V = N1 ∪ N2 . We deform the embedded disc N1 into the embedded disc N2 keeping the boundary in V so that the double curves are merged along the arc γ1 (see Figure 10, cf. [6][4]).

E1 E2 e1 E

e2 E

Figure 10 Note that for every double curve, there is a descendent disc (see Figure 11). For instance, let c be a double arc formed by two sheets E1 and E2 in a surface-knot diagram. Take two distinct points a and b on c. Let N (c) be a small neighbourhood of c in R3 . Let γ1 be a simple arc from a to b in N (c) ∩ E1 , and let γ2 be a simple arc from b to a in N (c) ∩ E2 . Then there exists an embedded disc D in N (c) such that γ1 , γ2 and D satisfy the conditions (1) to (6) of the definition of descendent disc. Along this descendent disc, we can apply the Roseman move R-6 so that it makes a simple closed double curve bounded by two discs. Then we can apply R1− to eliminate the simple closed double curve. We call the disc a trivial descendent disc. Otherwise a non-trivial descendent disc. Definition 3.1. Let F be a surface-knot and let DF be a surface-knot diagram of F . A surface-knot diagram is said to be d-minimal if it has neither non-trivial descendent discs nor non-trivial pinch discs in the complement of the diagram. 12

E2

R-6

b

R-1−

E2

E2

a E1

E1

E1

Figure 11: A trivial descendent disc. In the rest of this paper, a pinch disc and a descendent disc mean a non-trivial pinch disc and a non-trivial descendent disc respectively

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Proof of Theorem 1.1

In order to prove Theorem 1.1 we use the following construction of a family of surface-knot diagrams of non-trivial surface-knots. Let S 2 = {(x, y, z) | x2 +y 2 +z 2 = 1 }. Let E, SE and SW denote the equator, the eastern hemisphere and the western hemisphere of S 2 respectively: E = {(x, y, 0) ∈ S 2 }, SE = {(x, y, z) ∈ S 2 : x ≥ 0}, SW = {(x, y, z) ∈ S 2 : x ≤ 0}.

(1) (2) (3)

We orient the equator E so that the orientation is consistent with the orientation of R3 . First, put a bug on the eastern sphere SE so that the double arc of the bug transversely intersects the equator E and one branch point is in the northern hemisphere and the other branch point is in the southern hemisphere. Then put another bug which is a copy of the first bug rotating π radian around the z axis. Now two bugs are in SE and SW . Viewing bugs from the positive z-axis and denote branch points of the first bug by b1 and b2 , and of the second bug by b3 and b4 such that b1 and b3 are in the northern hemisphere and b2 and b4 are in the southern hemisphere (see the left diagram of Figure 12). Make each double segment ‘zig-zag’ as the right diagram of Figure 12. Put additional points r1 , r2 on the zig-zag double arc between b3 and b4 and r3 , r4 between b1 and b2 . Take arcs γ1 from b1 to r1 , γ2 from b2 to r2 , γ3 from b3 to r3 and γ4 from b4 to r4 . Move the branch point bi along γi (i = 1, · · · , 2) 13

SE

SE

b4

b2 b3 SW

b1 SW

Figure 12 to the point ri and apply R-5 and R-6 Roseman moves to create a triple point (see [8][15]) and a double loop based on the triple point. γ2 γ3 b3

r4

r1

τ2

r3

r2

b1

b4

b4

b1

b2

b2

τ1

τ3

τ4

b3

γ1 γ4

Figure 13 We obtain a generic surface with four triple points and four branch points. We denote the triple point created at ri by τi (i = 1, · · · , 4) (see the right diagram in Figure 13 ). Here the resulting generic surface is liftable and by the construction it is a surface-knot diagram of a trivial 2-knot. Figure 14 shows the complication described above and it is constructed on an annulus. In Figure 14a the outer most dotted top arc is part of the boundary and the inner most dotted arc is part of another boundary of the annulus. Figure 14b depicts the pre-image of the complication of the annulus. The edges A and B are identified as an edge to obtain an annulus. The double looped cross section in Figure 14a corresponds to the thicken line in the diagram depicted in Figure 14b; that is, if we cut the annulus along this line and construct the complication, we obtain the diagram shown in Figure 14a.

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γ1

C

D γ2

b4

b1

A

b3

B

b2

(b) Pre-image of the diagram in (a).

(a) Branch points and Triple points.

Figure 14: The resulting diagram and the pre-image. We have used only Roseman moves to construct the generic annulus. Therefore, the resulting diagram is liftable. Thus we will call the closure of preimage of the multiple point set the double decker set which is the union of upper decker sets and lower decker sets (see [1][2] for details). In order to obtain a cross-exchangeable curve, we find descendent discs along each of which R-6 move is applied. One descendent disc D appears in the diagram in Figure 14a. The boundary of D is the union of two arcs γ1 and γ2 . The arc γ1 joins two double loops based at triple points τ2 and τ3 . Comparing Figure 14a and Figure 9, the reader can see that the conditions for descendent disc are satisfied. In the pre-image, the boundary curve of D will be the disjoint union of simple arcs: γe1 ∪ γe2 (see Figure 15a). Here γe1 joins upper decker curves and γe2 joins lower decker curves. Now we apply R-6 move along the descendent disc D so that these two double loops will be joined along γ2 or along γ1 . In the pre-image, upper arcs are joined along γe1 , and lower arcs are joined along γe2 (see Figure 15b).

By symmetry, we can find another descendent disc D′ bounded by γ1′ and γ2′ between points on double loops based at τ1 and τ4 . Then the pre-image will be shown in the Figure 16a. Along the descendent disc, R-6 move is applied so that the pair of upper decker curves and the pair of lower decker curves are joined in the pre-image (see Figure 16b). 15

γ1

γ2

(b)

(a) γe1 and γe2

Figure 15: Deformations of double decker sets.

γ e1′

γ e2′

(a) γe1′ and γe2′

(b)

Figure 16: Deformations of pre-images. By an isotopy we can deform the double decker set as depicted as diagrams in Figure 17a. Now we find an exchangeable curve c. In the pre-image in Figure 17, there is a pair of parallel double decker curves; one is lower the other is upper. It is not too difficult to see that the projected image c of the pair is a crossexchangeable curve. We apply the operation on c so that the upper and lower information of the double decker set will be exchanged. We have obtained a surface-knot diagram of an annulus. Suppose that the boundary of the diagram is filled with surfaces with boundaries so that the diagram is a surface-knot diagram of a surface-knot. We shall prove the resulting diagram is a diagram of a non-trivial surface16

(a) The double decker set.

(b)

Figure 17: Crossing change along the cross-exchangeable curve. knot. We consider the half of the rectangle with two crossings of lower decker set, and call this rectangle a primitive set (see Figure 14b). Pasting the both sides of the primitive set, an annulus with double decker set is obtained. This annulus gives a surface-knot diagram with two boundary components. We can add a surface with two boundaries to the diagram. The resulting diagram yields the trivial surface-knot group [9]. Thus with this construction, in order to obtain a surface-knot diagram with non-trivial surface-knot group, we need to paste at least two copies of the primitive set. Figure 18 depicts the union of two copies of the primitive set.

V0

V3

V2

V4

V1 R

Figure 18: The lower decker set. Let R be the set formed with two copies of the primitive sets. Let A be an annulus such that it is formed by pasting the both outer sides of R. Let 17

e → A be a finite cyclic covering space over A, where A e is constructed p:A by pasting copies of R. Let F ′ be a closed oriented surface embedded in R3 × {0} ⊂ R4 . Let δ ′ e ∼ e in be a simple closed curve on F ′ . Since A = S1 × I, we can embed A ′ ′ 1 the tubular neighbourhood of δ in F so that the centre line S × {1/2} is mapped onto δ ′ . This gives a closed surface with double decker set in R4 . This defines a surface-knot F , embedded F ′ in R4 with a simple closed curve δ, embedded δ ′ . From the construction, δ is isotopic to the lower decker curve corresponding to the c-e curve. Therefore, (2) holds. We can colour this surface-knot diagram with the dihedral quandle of order 3; X = {0, 1, 2} (see [1] for definition). For example, in Figure 18, V0 7→ 0, V1 7→ 0, V2 7→ 1, V3 7→ 2, V4 7→ 1, . . . gives a non-trivial colouring. Therefore, the surface-knot diagram is a diagram of a non-trivial surface-knot. From the construction, DF (γ) is a diagram of a trivial surface-knot and thus (3) holds. We will show that the resulting diagram is d-minimal. There are pinch discs near a branch point. However, from the construction, these pinch discs are trivial ones. In order to have a descendent disk, the boundary consists of two arcs in DF and one of their lifts joins the lower decker set at its boundary. At the end points of the lifted arc, the orientations of the lower decker set must be opposite with respect to the orientation of the lifted arc. However, from the construction, we cannot find such an arc in the complementary region of the lower decker set. Therefore, there is no descendent disc. Therefore, the surface-knot diagram is d-minimal thus (1) holds. This completes the proof.

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Proof of Theorem 1.2

For a 2-sphere, apply the construction of Theorem 1.1 to obtain a surfaceknot diagram D with the tri-colouring number 32 . For a non-negative integer g, take a trivial surface-knot diagram D0 of genus g embedded in R3 × {0} ⊂ R4 . Let Dn be a connected sum of D0 and n copies of D. Then each Dn is d-minimal and represents a surface-knot of genus g with tri-colouring number 3n+1 . This implies that (1) and (3) hold and by the construction, (2) and (4) hold.

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Acknowledgements The authors thank to Professors A. Kawauchi, S. Kamada and S. Satoh for having valuable discussions on the topic for the early version of this paper. Also we would like to thank to the referees for giving us valuable suggestions to improve the paper.

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