Enumeration of ribbon 2-knots presented by virtual arcs with up to four crossings

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1 Enumeration of ribbon 2-knots resented by virtual arcs with u to four crossings Taizo Kanenobu and Seiya Komatsu Deartment of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka , Jaan kanenobu@sci.osaka-cu.ac.j ABSTRACT We enumerate ribbon 2-knots resented by virtual arc diagrams with u to four classical crossings. We use a linear Gauss diagram for a virtual arc diagram. Keywords: Ribbon 2-knot; virtual arc; welded arc; linear Gauss diagram; ribbon handlebody. Mathematics Subject Classification 2000: 57Q45 1. Intoduction A ribbon 2-knot is a knotted 2-shere in R 4 that bounds a ribbon 3-disk, which is an immersed 3-disk with only ribbon singularities. The ribbon crossing number of a ribbon 2-knot is the minimal number of the ribbon singularities of any ribbon 3-disk bounding the knot [24]. The enumeration of the classical knots are based on the crossing number of a knot diagram. So, it is natural to enumerate ribbon 2-knots based on the ribbon crossing number. Yasuda has listed ribbon 2-knots with ribbon crossing number u to four [22,23,25,26,27]. Moreover, a ribbon 2-knot is resented by a virtual arc diagram, which was introduced by Satoh [19]; a virtual arc diagram has classical and virtual crossings and a classical crossing corresonds to a ribbon singularity. If a ribbon 2-knot is resented by a virtual arc diagram with n classical crossings, then its ribbon crossing number is at most n. In this aer we enumerate ribbon 2-knots based on the virtual arc resentation. Our main result is the table of ribbon 2-knots which are resented by virtual arc diagram with u to 4 crossings (Tables 2, 3 and 4). Therefore, the ribbon 2-knots in our table are contained in that of Yasuda. However, in the course of this enumeration we discovered an oversight in Yasuda s table, that is, we found 6 ribbon 2-knots whose Alexander olynomials are not listed in Yasuda s table. From a virtual arc diagram we may obtain a linear Gauss diagram, which is an 1

2 2 T. Kanenobu & S. Komatsu immersing interval with the reimages of each double oint connected with a chord equied with the sign of the crossing. Then an oriented linear Gauss diagram determines an associated ribbon 2-knot (Proosition 3.1). Also there are several moves for a linear Gauss diagram which do not change the isotoy class of the associated ribbon 2-knot (Subsec. 3.2); they are induced from the Reidemeister moves (moves A I, A II, A III ), welded moves (moves D, E II ), negative-amhicheirality of a ribbon 2-knot (reversing move), and comosition of knots (Eq. (3.1)). Furthermore, they yield some simlified moves (moves Ω i, i = 1,..., 8), which are useful for investigating the equivalence of Gauss diagrams. Using these moves, we enumerate linear Gauss diagrams. First, we show a ribbon 2-knot resented by linear Gauss diagrams with u to 4 chords is resented by one of the 13 linear Gauss diagrams Γ 4 l, 1 l 13, given in Fig. 29 (Lemma 5.1). Then, for each diagram we examine the corresonding ribbon 2-knot, where we calculate the knot grou and Alexander olynomial (Sects. 5 and 7). We aly the above mentioned moves for considering the equivalence. However, in some cases we consider a ribbon handlebody (Lemma 7.3) or a virtual arc diagram (Lemma 7.5). This aer is organized as follows: In Sec. 2, we introduce a ribbon 2-knot and its virtual arc resentation, where we use a ribbon handlebody to resent a ribbon 3-disk. In Sec. 3, we introduce a linear Gauss diagram which resents a virtual arc diagram and so a ribbon 2-knot. In Sec. 4, we give our main result, tables of ribbon 2-knots resented by virtual arc diagrams with u to 4 crossings (Tables 2, 3 and 4). Secs. 5, 6, and 7 are devoted to the roof of our main result. Most of the results in this aer are from the master s thesis of one of the authors [14]. 2. Ribbon 2-Knot and Its Virtual Arc Presentation In this section, we introduce a ribbon knot, virtual arc, welded arc, and virtual arc resentation of a ribbon 2-knot Ribbon knot A ribbon (n+1)-disk, n 1, is an immersed (n+1)-disk D n+1 into R n+2 with only transverse double oints such that the singular set consists of ribbon singularities, that is, the reimage of each ribbon singularity consists of a roerly embedded n-disk in D n+1 and an embedded n-disk interior to D n+1. An n-knot is a ribbon n-knot if it bounds a ribbon (n + 1)-disk in R n+2. The ribbon crossing number of a ribbon n-knot K is the smallest number of the ribbon singularities of any ribbon (n + 1)-disk bounding the knot K; cf. [24]. Yasuda [22,23,25,26,27] enumerated ribbon 2-knots with ribbon crossing number u to four. A ribbon handlebody [3] H is a ribbon 2-disk, which is a 2-dimensional handlebody in R 3 consisting of ( + 1) 0-handles D 1, D 2,..., D +1, 1-handles B 1, B 2,..., B and no 2-handle such that the reimage of each ribbon singularity con-

3 Enumeration of ribbon 2-knots resented by virtual arcs 3 sists of an arc in the interior of a 0-handle and a cocore of a 1-handle. We ut D = D 1 D 2 D +1 and B = B 1 B 2 B. Then the set of ribbon singularities in H is the connected comonents of the intersection D B. Note that in order to resent a ribbon handlebody we may use a ribbon regular rojection defined in [6, Sect. 1] or [7, Sect. 3],. We define the associated ribbon 2-knot in R 4 = R 3 R as the ribbon 2-knot that bounds the immersed 3-disk D [ 2, 2] B [ 1, 1]. Conversely, for any ribbon 2-knot K, there exists a ribbon handlebody whose associated ribbon 2-knot is K; see [10,21]. Thus we may reresent a ribbon 2-knot by a ribbon handlebody Virtual arc and welded arc A virtual knot diagram is an immersed circle and a virtual arc diagram is an immersed interval in the lane R 2 such that each crossing is either a classical crossing or a virtual crossing as shown in Fig. 1. (a) (b) Fig. 1. (a) Classical crossing; (b) vitrtual crossing. A virtual knot is an equivalence class of virtual knot diagram under the classical Reidemeister moves A I, A II, A III as shown in Fig. 2 and the virtual Reidemeister moves B I, B II, B III, C, E I as shown in Fig. 3. Similarly, a virtual arc is an equivalence class of virtual arc diagram under the classical Reidemeister moves A I, A II, A III and the virtual Reidemeister moves B I, B II, B III, C as well as E I as shown in Fig. 3. We introduce the welded moves D and E II as shown in Fig. 4; the move D is known as one of the forbidden moves ; cf. [8,17]. Two virtual knot diagrams or virtual arc diagrams A and A are w-equivalent if there exist a finite sequence of the classical Reidemeister moves, virtual Reidemeister moves, and welded moves which transforms A into A. A welded knot is an equivalence class under the w- equivalence of virtual knot diagrams, and a welded arc is an equivalence class under the w-equivalence of virtual arc diagrams. A welded knot or arc is trivial if it is an equivalence class containing a virtual knot or virtual arc diagram without any crossing. Note that a welded knot and a welded arc are called a weakly virtual knot and a weakly virtual knot, resectively, in [9,19].

4 4 T. Kanenobu & S. Komatsu A I A II A III Fig. 2. Reidemeister moves A I, A II, A III. B I B II B III C E I Fig. 3. Virtual Reidemeister moves B I, B II, B III, C, E I Virtual arc resentation of a ribbon 2-knot Given an oriented virtual arc diagram A, we can construct a ribbon handlebody H A as shown in Fig. 5, where an overass at a classical crossing corresonds to a 0-handle containing a ribbon singularity, a virtual crossing corresonds to two 1-

5 Enumeration of ribbon 2-knots resented by virtual arcs 5 D E II Fig. 4. Welded moves D, E II. handles assing over/under each other, and an end oint corresonds to a 0-handle. Then we denote the associated ribbon 2-knot of H A by Tube(A). If two virtual arc diagrams A and A are w-equivalent, then the corresonding ribbon 2-knots Tube(A) and Tube(A ) are ambient isotoic, and thus a welded arc determines a ribbon 2-knot. Conversely, any ribbon 2-knot is associated with some virtual arc diagram; see [19]. Fig. 5. Corresondence of an oriented virtual arc diagram to a ribbon handlebody. Remark 2.1. Satoh [19] constructs the associated ribbon 2-knot Tube(A) using a surface diagram, which is a rojection image of a ribbon 2-knot in R 4 ; see [1], cf. [10]. Furthermore, we define two oriented virtual arc diagrams A and A are r- equivalent, denoted by A r A, if they corresond to the isotoic ribbon 2-knots. Let A be an oriented virtual arc diagram. We denote the horizontal mirror image by A, which is obtained from A by reflecting the diagram across a vertical lane. We denote the orientation-reversion of A by A. See Fig. 6; cf. [5, Sec. 2.2]. Then the both associated ribbon handlebodies H A and H A are obtained from H A by reflecting across the vertical lane and the rojection lane, resectively.

6 6 T. Kanenobu & S. Komatsu A A A A Fig. 6. Virtual arc diagrams A, A, A, and A. Alternatively, both H A and H A are obtained from H A by reversing the orientation of the immersed surface. This shows that a ribbon 2-knot is negative-amhicheiral; see [20, Theorem 2.18], [19, Proosition 4.1]. Thus we have: Proosition 2.2. Let A be an oriented virtual arc diagram. Then the associated ribbon 2-knots Tube(A ) and Tube( A) are isotoic, which are the mirror images of Tube(A). In other words, A and A are r-equivalent. We define the roduct of two oriented virtual arc diagrams A and A, denoted by A A, as follows: First, we deform the diagrams A and A so that the the terminal oint of A and the initial oint of A lie in the outermost region by the move E I. We denote the resulting diagrams by à and Ã. Then the roduct of A and A is obtained by connecting the terminal oint of à to the initial oint of à as shown in Fig. 7. It is easy to see that the roduct of oriented virtual arc diagrams is well defined u to w-equivalence. T T T T à à A A Fig. 7. Product of virtual arc diagrams. The associated ribbon 2-knot Tube(A A ) is the comosition of Tube(A) and Tube(A ); Tube(A A ) = Tube(A)#Tube(A ). Since the comosition of oriented 2-knots is well defined u to isotoy, we have the following: Proosition 2.3. Let A and A be oriented virtual arc diagrams. Then the associated ribbon 2-knots Tube(A A ) and Tube(A A) are isotoic, A A r A A. Note that A A and A A may not be w-equivalent. We can see Proosition 2.3 by the associated ribbon handlebody as follows: H A A is the boundary connected

7 Enumeration of ribbon 2-knots resented by virtual arcs 7 sum of H A and H A. Then by sliding H A along the boundary of H A we can isotoe H A A into H A ( A ). This imlies Tube(A A ) is isotoic to Tube(A ( A )). From this we have further Tube(A ( A )) is isotoic to Tube(( A ) ( A )). Since the virtual arc ( A ) ( A ) is the same as (A A), by Proosition 2.2 Tube(( A ) ( A )) is isotoic to Tube(A A), comleting the roof. 3. Gauss diagram In this section, we first review a Gauss diagram of a virtual knot or a welded knot, and then introduce a linear Gauss diagram of a virtual arc. We give several moves of a linear Gauss diagram which do not change the isotoy class of the associated ribbon 2-knot Gauss diagrams of virtual and welded knots The Gauss diagram of an oriented virtual knot diagram is an oriented circle as the reimage of the immersed circle with chords connecting the reimages of each classical crossing. We secify crossing information on each chord by directing the chord toward the under crossing equied with the sign of the crossing. We call such a Gauss diagram a circular Gauss diagram. A circular Gauss diagram is considered u to orientation reserving homeomorhism of the underlying oriented circle. A circular Gauss diagram determines a virtual knot diagram u to virtual Reidemeister moves; see [2, Theorem 1.A]; cf. [12]. Namely, if two virtual knot diagrams have the same circular Gauss diagram, then there exist a finite sequence of virtual Reidemeister moves B I, B II, B III, C which transforms one into the other. The classical Reidemeister moves on a knot give rise to moves on circular Gauss diagrams. Deending on orientations of strings involved in the Reidemeister moves, one may distinguish four different versions of each of the A I and A II moves, and eight versions of the A III move as shown in Figs [18, Figs. 2 4]. A Ia A Ib A Ic A Id Fig. 8. Oriented Reidemeister moves of tye I. We give the corresonding moves of circular Gauss diagrams in Figs For each of the moves A IIIa,..., A IIIh for a knot, there are two different ossibilities according to the structure of the diagram outside the local art given in the

8 8 T. Kanenobu & S. Komatsu A IIa A IIb A IIc A IId Fig. 9. Oriented Reidemeister moves of tye II. A IIIa A IIIb A IIIc A IIId A IIIe A IIIf A IIIg A IIIh Fig. 10. Oriented Reidemeister moves of tye III.

9 Enumeration of ribbon 2-knots resented by virtual arcs 9 diagrams in Fig. 10. By abuse of notation, we denote these moves by the same notation; for the moves A IIIa,..., A IIIh, we denote the corresonding moves by A IIIa, A IIIa,..., A IIIh, A IIIh ; cf. [2,17]. Therefore, an oriented virtual knot (that is, an oriented virtual knot diagram modulo Reidemeister moves A Ia, A Ib, A Ic, A Id, A IIa, A IIb, A IIc, A IId, A IIIa,..., A IIIh, and virtual Reidemeister moves B I, B II, B III, C) is equivalent to the corresonding circular Gauss diagram considered u to moves given in Figs A Ia ( = +) or A Id ( = ) A Ib ( = +) or A Ic ( = ) Fig. 11. tye I. Moves of circular Gauss diagrams corresonding to the oriented Reidemeister moves of A IIa ( = +) or A IIb ( = ) A IIc ( = +) or A IId ( = ) Fig. 12. tye II. Moves of circular Gauss diagrams corresonding to the oriented Reidemeister moves of A IIIa A IIIa Fig. 13. Moves of circular Gauss diagrams corresonding to the oriented Reidemeister moves of tye III.

10 10 T. Kanenobu & S. Komatsu A IIIb A IIIb A IIIc A IIIc A IIId A IIId A IIIe A IIIe A IIIf A IIIf A IIIg A IIIg A IIIh A IIIh Fig. 14. Moves of circular Gauss diagrams corresonding to the oriented Reidemeister moves of tye III.

11 Enumeration of ribbon 2-knots resented by virtual arcs 11 Now we consider a circular Gauss diagram for a welded knot. Deending on orientations of strings, there are four different versions of the welded move D as shown in Fig. 15, where, q = ±. q q Fig. 15. Moves of circular Gauss diagrams corresonding to the move D, where, q = ±. Therefore, if we allow the move D, then the moves A IIa and A IIc are equivalent, and the moves A IIb and A IId are equivalent. Also, the moves A IIIa, A IIIa,..., A IIIh, A IIIh are ut together into the moves A IIIx, A IIIy, A IIIz, A IIIw as shown in Fig. 16, where, q = ±. We give the equivalent airs in Table 1, where, for examle, it shows the move A IIIa is equivalent to the move A IVa with (, q) = (+, +). Thus, an oriented welded knot (that is, an oriented virtual knot diagram modulo Reidemeister moves, virtual Reidemeister moves, and welded move D) is equivalent to the corresonding circular Gauss diagram considered u to moves given in Figs. 11, 12, 15, and 16. Table 1. Equivalent moves of circular Gauss diagrams allowing the move D. (, q) (+, +) (+, ) (, +) (, ) A IVa A IIIa A IIIf A IIIe A IIIh A IVb A IIIe A IIIh A IIIa A IIIf A IVc A IIIb A IIId A IIIc A IIIg A IVd A IIIb A IIId A IIIc A IIIg 3.2. Linear Gauss diagram We can define the Gauss diagram for an oriented virtual arc in a similar way to a circular Gauss diagram, which we call a linear Gauss diagram; see Fig. 17. Similarly, a linear Gauss diagram is considered u to orientation reserving homeomorhism of the underlying oriented interval. Then a linear Gauss diagram determines a virtual arc diagram, that is, if two virtual arc diagrams have the same linear Gauss diagram, then there exist a finite sequence of virtual Reidemeister moves B I, B II, B III, C, E I which transforms one into the other. Therefore, since a welded arc determines a ribbon 2-knot [19, Proosition 5.2], we have:

12 12 T. Kanenobu & S. Komatsu q q q q A IIIx A IIIy q q q q A IIIz A IIIw Fig. 16. Moves of circular Gauss diagrams equivalent to the moves A IIIa,..., A IIIh allowing the move D, where, q = ±. Proosition 3.1. A linear Gauss diagram determines an associated ribbon 2-knot. Fig. 17. Virtual arc diagram A in Fig. 6 and its linear Gauss diagram γ. By abuse of notation, we denote by Tube(γ) the ribbon 2-knot obtained from a virtual arc diagram with Gauss diagram γ. Also, we define two Gauss diagrams γ and γ are r-equivalent, denoted by γ r γ, if they corresond to the isotoic ribbon 2-knots; Tube(γ) Tube(γ ). We define the moves A I, A II, A III, D, and E II for a linear Gauss diagrams as follows: A I : One of the 2 tyes of moves as shown in Fig. 18. A II : One of the 4 tyes of moves as shown in Fig. 19. A III : One of the 12 tyes of moves as shown in Fig. 20. D: One of the 3 tyes of moves as shown in Fig. 21. E II : One of the 2 tyes of moves as shown in Fig. 22.

13 Enumeration of ribbon 2-knots resented by virtual arcs 13 Fig. 18. Moves A I for a linear Gauss diagram, where = ±. Fig. 19. Moves A II for a linear Gauss diagram, where = ±. q q q q q q Fig. 20. Moves A III for a linear Gauss diagram, where, q = ±. The moves A I, A II, A III, and D are yielded from those of a circular Gauss diagram given in Figs. 11, 12, 16, and 15, resectively, and the move E II corresonds to the welded move E II as shown in Fig. 4. So, we have: Proosition 3.2. If two linear Gauss diagrams are related by the moves A I, A II, A III, D, or E II, then they are r-equivalent. Furthermore, we define the move Ω i, i = 1, 2,..., 8, for a linear Gauss diagram as shown in Fig. 23. Then we have:

14 14 T. Kanenobu & S. Komatsu q q q q q q q q q q q q q q q q q q Fig. 20. Cont d. Proosition 3.3. If two linear Gauss diagrams are related by the move Ω i, i = 1,..., 8, then they are r-equivalent.

15 Enumeration of ribbon 2-knots resented by virtual arcs 15 q q q q q q Fig. 21. Moves D for a linear Gauss diagram, where, q = ±. Fig. 22. Moves E II for a linear Gauss diagram, where = ±. Proof. Each of the moves Ω 1, Ω 2, Ω 3, Ω 4 is realized by a sequence of the moves A I and A III ; for the move Ω 1 see Fig. 24. Next, the move Ω 1 imlies the move Ω 5. In fact, the Ω 5 is realized by a sequence of the moves D and E II as shown in Fig. 25. Similarly, the moves Ω 1, Ω 3, Ω 4 imly the moves Ω 6, Ω 7, Ω 8, resectively. This comletes the roof. Let γ be the Gauss diagram of a virtual arc A. We denote by γ and γ the Gauss diagrams obtained from γ by changing the signs of chords and by reversing the orientation of the underlying interval, resectively. Then γ and γ are the Gauss diagrams of the virtual arcs A and A, resectively. In articular, the Gauss diagram γ is obtained from γ by changing the signs of chords and reversing the orientation of the interval, which is the Gauss diagrams of the virtual arc A. Definition 3.4. We call the change of a linear Gauss diagram γ γ the reversing move; see Fig. 26. Then by Proosition 2.2, we have

16 16 T. Kanenobu & S. Komatsu Ω 1 q q q Ω 2 q q q Ω 3 q q q Ω 4 q q q Ω 5 q q Ω 6 q q Ω 7 q q Ω 8 q q Fig. 23. Moves Ω i, i = 1,..., 8, which reserve the r-equivalence.

17 Enumeration of ribbon 2-knots resented by virtual arcs 17 q q A I q q A III A III q q q q A I q Fig. 24. Proof of Proosition 3.3 for the move Ω 1. E II q q Ω 1 q q E II q Fig. 25. Proof of Proosition 3.3 for the move Ω 5. Fig. 26. Reversion move γ γ, where γ and γ are the linear Gauss diagrams for the virtual arcs A and A given in Fig. 6. Proosition 3.5. The reversing move of a linear Gauss diagram reserves the r-equivalence, γ r γ. Let γ and γ be the Gauss diagrams of virtual arcs A and A. We define the roduct γ γ by connecting the terminal oint of the underlying interval of γ to the initial oint of the underlying interval of γ ; see Fig. 27. Then γ γ is the Gauss diagram of the roduct virtual arc A A.

18 18 T. Kanenobu & S. Komatsu The associated ribbon 2-knots Tube(γ γ ) reresents the comosition of Tube(γ) and Tube(γ ). Since the comosition of oriented 2-knots is well defined u to isotoy, we have the following: γ γ r γ γ r ( γ ) γ. (3.1) γ γ γ γ Fig. 27. Product of Gauss diagrams. By Proosition 2.2 two ribbon 2-knots Tube(γ ) and Tube( γ) are isotoic, which are the mirror images of Tube(γ), and thus, we have γ r γ. (3.2) If a virtual arc diagram A has a nugatory crossing as in Fig. 28(a), then A may be deformed into A as in Fig. 28(b) by the Reidemeister moves and welded moves. Therefore, we have the following: Lemma 3.6. If a Gauss diagram γ has a searated single chord c as in Fig. 28(c) with any orientation and sign, then it is r-equivalent to the Gauss diagram obtained from γ by deleting the chord c. (a) (b) (c) Fig. 28. (a) Virtual arc diagram A with a nugatory crossing; (b) virtual arc diagram A ; (c) Gauss diagram with a searated single chord. 4. Enumeration Result Before stating the main result, we give some convention of a linear Gauss diagram which resents a ribbon 2-knot. Given a linear Gauss diagram we order its chords following the orientation of the underlying interval. More recisely, let γ be a linear

19 Enumeration of ribbon 2-knots resented by virtual arcs 19 Gauss diagram with n chords such that 2n endoints lie in the underlying interval [0, 1], where 0 is the initial oint and 1 is the terminal oint. We order the chords c 1, c 2,..., c n of γ so that h(c i ) < h(c i+1 ), where h(c i ) is the arrowhead oint of c i. If the chord c i has the sign i, i = ±, then we denote γ by γ( 1, 2,..., n ). The following is our main result. Theorem 4.1. A ribbon 2-knot resented by a virtual arc with u to 4 crossings is either the trivial one or one of those listed in Tables 2, 3 and 4. Each column in Tables 2, 3 and 4 shows as follows: The first column, Name, shows the name of the corresonding ribbon 2-knot, where K! denotes the mirror image of the knot K, and K#K the comosition of two knots K and K. Γ 2 1 Γ 3 1 Γ 3 2 Γ 4 1 Γ 4 2 Γ 4 3 Γ 4 4 Γ 4 5 Γ 4 6 Γ 4 7 Γ 4 8 Γ 4 9 Γ 4 10 Γ 4 11 Γ 4 12 Γ 4 13 Fig. 29. Linear Gauss diagrams with u to 4 chords.

20 20 T. Kanenobu & S. Komatsu The second column, Gauss diagram, shows the linear Gauss diagram resenting the ribbon 2-knot, which are shown as in Fig. 29. If there are more than one diagram, then they resent isotoic ribbon 2-knots. The column, Grou, in Tables 2 and 3 shows the grou resentation of the ribbon 2-knot, that is, the fundamental grou of the comlement of the ribbon 2-knot, where G(a 1, b 1,..., a n, b n ) = x, y ; y = wxw 1, w = x a1 y b1 x an y bn. (4.1) Note that x, y are meridians. In Table 4 there is no column listing the knot grou because those ribbon 2-knots are comosite and do not have such a grou resentation. The column, (t), shows the Alexander olynomial of the ribbon 2-knot which is normalized so that (1) = 1 and (d/dt) (1) = 0. We abbreviate (t) as follows: for c i Z n (c m c m+1 c 1 [c 0 ] c 1 c 2 c n ) = c i t i. (4.2) i= m The column, Det, shows the determinant of the ribbon 2-knot which is given by ( 1). In the last column if there is a mark a, then the ribbon 2-knot is amhicheiral. Remark 4.2. The grou G(a 1, b 1,..., a n, b n ) is isomorhic to G( b n, a n,..., b 1, a 1 ). In fact, x, y ; y = wxw 1 = x, y ; x = w 1 yw, where w 1 = y bn x an y b1 x a1 for w = x a1 y b1 x an y bn. This is realized by the reversing move of a linear Gauss diagram. Table 2. Ribbon 2-knots resented by linear Gauss diagrams with 2 or 3 chords. Name Gauss diagram Grou (t) Det R1 2 Γ 2 1 (++), Γ2 1 ( ) G(1, 1) (1 [-1] 1) 3 a R2 2 Γ 2 1 (+ ) G(1, -1) ([0] 2-1) 3 R2 2! Γ2 1 ( +) G(-1, 1) (-1 2 [0]) 3 R1 3 Γ 3 1 (+ + +) G(1, 2) (1-1 [0] 1) 1 R1 3! Γ3 1 ( ) G(-1, -2) (1 [0] -1 1) 1 R2 3 Γ 3 1 (+ ) G(1, -2) ([0] 1 1-1) 1 R2 3! Γ3 1 ( + +) G(-1, 2) ( [0]) 1 R3 3 Γ 3 2 (+ + +), Γ3 2 (+ ) G(1, -1, 1, 1) ([2] -2 1) 5 R3 3! Γ3 2 ( ), Γ3 2 ( + +) G(-1, 1, -1, -1) (1-2 [2]) 5 R4 3 Γ 3 2 (+ + ) G(1, 1, 1, -1) ([1] ) 5 R4 3! Γ3 2 ( +) G(-1, -1, -1, 1) ( [1]) 5 R5 3 Γ 3 2 (+ +), Γ3 2 ( + ) G(1, -1, -1, 1) (-1 [3] -1) 5 a

21 Enumeration of ribbon 2-knots resented by virtual arcs 21 Table 3. Ribbon 2-knots resented by the linear Gauss diagrams Γ 4 l, 1 l 12. Name Gauss diagram Grou (t) Det R1,1 4 Γ 4 1 ( ) G(1, 3) (1-1 0 [0] 1) 3 R1,1 4! Γ4 1 ( ) G(-1, -3) (1 [0] 0-1 1) 3 R1,2 4 Γ 4 1 ( + + +) G(-1, 3) ( [0]) 3 R1,2 4! Γ4 1 (+ ) G(1, -3) ([0] ) 3 R2,1 4 Γ 4 2 ( ), Γ4 2 ( ) G(1, 1, 1, 1) (1-1 [1] -1 1) 5 a R2,2 4 Γ 4 2 (+ + ) G(1, -1, 1, -1) ([0] 0 3-2) 5 R2,2 4! Γ4 2 ( + +) G(-1, 1, -1, 1) ( [0]) 5 R3,1 4 Γ 4 3 ( ), Γ4 3 ( ) G(2, 2) (1 0 [-1] 0 1) 1 a R3,2 4 Γ 4 3 (+ + ) G(2, -2) ([0] ) 1 R3,2 4! Γ4 3 ( + +) G(-2, 2) ( [0]) 1 R4,1 4 Γ 4 4 ( ) G(1, -1, 1, 2) (2 [-2] 0 1) 3 R4,1 4! Γ4 4 ( ) G(-1, 1, -1, -2) (1 0 [-2] 2) 3 R4,2 4 Γ 4 4 (+ + +) G(1, -1, -1, 2) (-1 2 [0]) 3 R4,2 4! Γ4 4 ( + ) G(-1, 1, 1, -2) ([0] 2-1) 3 R4,3 4 Γ 4 4 ( + + +) G(-1, -1, 1, 2) (1-2 1 [1]) 3 R4,3 4! Γ4 4 (+ ) G(1, 1, -1, -2) ([1] 1-2 1) 3 R4,4 4 Γ 4 4 (+ + ) G(1, 1, 1, 2) ([0] ) 3 R4,4 4! Γ4 4 ( + +) G(-1, -1, -1, -2) ( [0]) 3 R5,1 4 Γ 4 5 (+ + ) G(1, -1, -1, -2) ([1] ) 3 R5,1 4! Γ4 5 ( + +) G(-1, 1, 1, 2) ( [1]) 3 R6,1 4 Γ 4 6 ( ) G(1, 1, -1, 1, 1, 1) (1-2 2 [-1] 1) 7 R6,1 4! Γ4 6 ( ) G(-1, -1, 1, -1, -1, -1) (1 [-1] 2-2 1) 7 R6,2 4 Γ 4 6 (+ + + ), Γ4 6 ( +) G(1, 1, -1, -1, 1, 1) (2 [-3] 2) 7 a R6,3 4 Γ 4 6 (+ + +), Γ4 6 ( + ), G(1, -1, -1, 1, 1, 1) (-1 3 [-2] 1) 7 Γ 4 6 ( + +) R6,3 4! Γ4 6 ( + ), Γ4 6 (+ + +), G(-1, 1, 1, -1, -1, -1) (1 [-2] 3-1) 7 Γ 4 6 (+ + ) R6,4 4 Γ 4 6 ( + + +) G(-1, 1, 1, 1, -1, 1) ( [0]) 7 R6,4 4! Γ4 6 (+ ) G(1, -1, -1, -1, 1, -1) ([0] ) 7 R6,5 4 Γ 4 6 (+ + ) G(1, 1, -1, -1, 1, -1) ([0] 3-3 1) 7 R6,5 4! Γ4 6 ( + +) G(-1, -1, 1, 1, -1, 1) (1-3 3 [0]) 7 R6,6 4 Γ 4 6 ( + + ) G(-1, 1, 1, -1, -1, 1) (-2 4 [-1]) 7 R6,6 4! Γ4 6 (+ +) G(1, -1, -1, 1, 1, -1) ([-1] 4-2) 7 R7,1 4 Γ 4 7 ( ) G(-1, 1, -2, -1) ( [2]) 1 R7,1 4! Γ4 7 ( ) G(1, -1, 2, 1) ([2] ) 1 R7,2 4 Γ 4 7 (+ + + ) G(1, 1, -2, -1) (1 [0] -1 1) 1 R7,2 4! Γ4 7 ( +) G(-1, -1, 2, 1) (1-1 [0] 1) 1 R7,3 4 Γ 4 7 ( + + +) G(-1, -1, -2, 1) ( [1]) 1 R7,3 4! Γ4 7 (+ ) G(1, 1, 2, -1) ([1] ) 1 R7,4 4 Γ 4 7 ( + + ) G(1, -1, -2, 1) (-1 1 [2] -1) 1 R7,4 4! Γ4 7 (+ +) G(-1, 1, 2, -1) (-1 [2] 1-1) 1

22 22 T. Kanenobu & S. Komatsu Table 3. Cont d. Name Gauss diagram Grou (t) Det R8,1 4 Γ 4 8 ( ), G(1, 1, 1, -1, -1, 1, 1, 1) (1-2 [3] -2 1) 9 Γ 4 8 ( ) R8,2 4 Γ 4 8 (+ + + ), G(1, -1, 1, 1, -1, 1, 1, -1) ([2] ) 9 Γ 4 8 (+ ) R8,2 4! Γ4 8 ( +), G(-1, 1, -1, -1, 1, -1, -1, 1) ( [2]) 9 Γ 4 8 ( + + +) R8,4 4 Γ 4 8 (+ + ) G(1, -1, 1, -1, -1, -1, 1, -1) ([0] ) 9 R8,4 4! Γ4 8 ( + +) G(-1, 1, -1, 1, 1, 1, -1, 1) ( [0]) 9 R8,5 4 Γ 4 8 (+ + ) G(1, -1, -1, 1, -1, 1, 1, -1) (-2 [5] -2) 9 R8,5 4! Γ4 8 ( + +) G(-1, 1, 1, -1, 1, -1, -1, 1) (-2 [5] -2) 9 R8,6 4 Γ 4 8 ( + + ), G(-1, -1, 1, 1, 1, 1, -1, -1) (1-2 [3] -2 1) 9 a Γ 4 8 (+ +) R9,1 4 Γ 4 9 (+ + ) G(1, 1, -1, 1, -1, -1) (2 [-3] 2) 7 R9,1 4! Γ4 9 ( + +) G(-1, -1, 1, -1, 1, 1) (2 [-3] 2) 7 R9,2 4 Γ 4 9 ( + + ), G(-1, 1, 1, 1, 1, -1) (-1 2 [-1] 2-1) 7 a Γ 4 9 (+ +) R10,1 4 Γ 4 10 ( ) G(1, 1, -1, 1, 1, -1, -1, 1, 1, 1) (1-3 4 [-2] 1) 11 R10,1 4! Γ4 10 ( ) G(-1, -1, 1, -1, -1, 1, 1, -1, -1, -1) (1 [-2] 4-3 1) 11 R10,2 4 Γ 4 10 (+ + + ) G(1, 1, -1, 1, 1, -1, -1, -1, 1, 1) (2 [-4] 4-1) 11 R10,2 4! Γ4 10 ( +) G(-1, -1, 1, -1, -1, 1, 1, 1, 1, 1) (-1 4 [-4] 2) 11 R10,3 4 Γ 4 10 (+ + +) G(1, 1, -1, -1, 1, -1, -1, 1, 1, 1) (-1 4 [-4] 2) 11 R10,3 4! Γ4 10 ( + ) G(-1, -1, 1, 1, -1, 1, 1, -1, -1, -1) (2 [-4] 4-1) 11 R10,4 4 Γ 4 10 (+ + +) G(1, -1, -1, 1, 1, 1, -1, 1, 1, -1) (-1 3 [-3] 3-1) 11 R10,4 4! Γ4 10 ( + ) G(-1, 1, 1, -1, -1, -1, 1, -1, -1, 1) (-1 3 [-3] 3-1) 11 R10,5 4 Γ 4 10 ( + + +) G(-1, 1, 1, 1, -1, -1, 1, 1, -1, 1) ( [0]) 11 R10,5 4! Γ4 10 (+ ) G(1, -1, -1, -1, 1, 1, -1, -1, 1, -1) ([0] ) 11 R10,6 4 Γ 4 10 (+ + ) G(1, 1, -1, -1, 1, -1, -1, -1, 1, 1) (1 [-2] 4-3 1) 11 R10,6 4! Γ4 10 ( + +) G(-1, -1, 1, 1, -1, 1, 1, 1, -1, -1) (1-3 4 [-2] 1) 11 R10,7 4 Γ 4 10 (+ + ) G(1, -1, -1, 1, 1, 1, -1, -1, 1, -1) ([-1] 5-4 1) 11 R10,7 4! Γ4 10 ( + +) G(-1, 1, 1, -1, -1, -1, 1, 1, -1, 1) (1-4 5 [-1]) 11 R10,8 4 Γ 4 10 ( + + ) G(-1, 1, 1, 1, -1, -1, 1, -1, -1, 1) (-2 5 [-3] 1) 11 R10,8 4! Γ4 10 (+ +) G(1, -1, -1, -1, 1, 1, -1, 1, 1, -1) (1 [-3] 5-2) 11 The roof of Theorem 4.1 is given in Sects. 5, 6 and 7, where we attemt to find isotoic ribbon 2-knot airs which are resented by linear Gauss diagrams with u to 4 chords. Actually, Theorem 4.1 does not claim the ribbon 2-knots listed in Tables 2, 3 and 4 are mutually distinct. A air of ribbon 2-knots with the same Alexander olynomial might be isotoic. In the forthcoming aer [11] we discuss the classification of these ribbon 2-knots.

23 Enumeration of ribbon 2-knots resented by virtual arcs 23 Table 3. Cont d. Name Gauss diagram Grou (t) Det R11,1 4 Γ 4 11 ( ) G(1, 1, -1, 1, 1, 1, -1, -1) (1-2 [3] -2 1) 9 R11,1 4! Γ4 11 ( ) G(-1, -1, 1, -1, -1, -1, 1, 1) (1-2 [3] -2 1) 9 R11,2 4 Γ 4 11 (+ + + ) G(1, 1, -1, 1, 1, -1, -1, -1) ([2] ) 9 R11,2 4! Γ4 11 ( +) G(-1,-1, 1,-1,-1, 1, 1, 1) ( [2]) 9 R11,3 4 Γ 4 11 (+ + +), G(1, -1, -1, 1, 1, 1, -1, -1) (-1 [4] -3 1) 9 Γ 4 11 (+ + ) R11,3 4! Γ4 11 ( + ), G(-1, 1, 1, -1, -1, -1, 1, 1) (1-3 [4] -1) 9 Γ 4 11 ( + +) R11,4 4 Γ 4 11 (+ + +) G(-1, 1, 1, 1, -1, 1, 1, -1) ( [3] -1) 9 R11,4 4! Γ4 11 ( + ) G(1, -1, -1, -1, 1, -1-1, 1) (-1 [3] ) 9 R11,5 4 Γ 4 11 ( + + +) G(1, 1, -1, -1, 1, 1, -1, 1) (2-4 [3]) 9 R11,5 4! Γ4 11 (+ ) G(-1, -1, 1, 1, -1, -1, 1, -1) ([3] -4 2) 9 R11,6 4 Γ 4 11 (+ + ) G(1, -1, -1, 1, 1, -1, -1, -1) ([1] ) 9 R11,6 4! Γ4 11 ( + +) G(-1, 1, 1, -1, -1, 1, 1, 1) ( [1]) 9 R11,7 4 Γ 4 11 ( + + ) G(1, 1, -1, -1, 1, -1, -1, 1) (-1 [4] -3 1) 9 R11,7 4! Γ4 11 (+ +) G(-1, -1, 1, 1, -1, 1, 1, -1) (1-3 [4] -1) 9 R12,1 4 Γ 4 12 (+ + + ) G(1, 1, -2, 1) ([1]) 1 R12,1 4! Γ4 12 ( +) G(-1, -1, 2, -1) ([1]) 1 R12,2 4 Γ 4 12 ( + + +) G(-1, 1, 2, 1) (1 [-1] 0 2-1) 1 R12,2 4! Γ4 12 (+ ) G(1, -1, -2, -1) ( [-1] 1) 1 Table 4. Ribbon 2-knots resented by the linear Gauss diagram Γ Name Gauss diagram (t) Det R1 2#R2 1 Γ 4 13 ( ), Γ4 13 ( ), (1-2 [3] -2 1) 9 a Γ 4 13 (+ + ), Γ4 13 ( + +) R1 2#R2 2 Γ 4 13 (+ + + ), Γ4 13 ( + ), ([2] ) 9 Γ 4 13 ( + ), Γ4 13 ( + + +) R1 2#R2 2! Γ4 13 (+ + +), Γ4 13 ( +), ( [2]) 9 Γ 4 13 ( + ), Γ4 13 ( + + +) R2 2#R2 2! Γ4 13 (+ +), Γ4 13 ( + + ) (-2 [5] -2) 9 a R2 2#R2 2 Γ 4 13 (+ + ) ([0] ) 9 R2 2!#R2 2! Γ4 13 ( + +) ( [0]) 9 In the remaining of this section, we exlain the calculation of the knot grou and Alexander olynomial. Let A be an oriented virtual arc diagram. Then we can define the grou of A in the same way as a virtual knot, which uses a generalization of Wirtinger s algorithm; see [12]. It is also the fundamental grou of the comlement of the corresonding ribbon 2-knot Tube(A), π 1 (R 4 Tube(A)) [19, Proosition 5.3]. Furthermore, we can obtain the grou resentation of A (and so of Tube(A)) from a linear Gauss diagram γ which resents A as for a virtual knot exlained in [2,. 1049]; cf. [13,

24 24 T. Kanenobu & S. Komatsu Sec. 2.3]. If we cut the interval at each arrowhead (forgetting arrowtails), the interval of γ is divided into a set of arcs. To each of these arcs there corresonds a generator of the grou, which corresonds to a meridian generator of the grou of Tube(A). Each arrow gives rise to a relation. Suose the sign of an arrow is, its tail lies on an arc labeled x, its head is the final oint of an arc labeled y and the initial oint of an arc labeled z. Then we assign to this arrow the relation z = x yx, meaning: { x x yx 1 yx if = +; = (4.3) xyx 1 if =. The resulting grou is the grou of the virtual arc A and also the grou of the corresonding ribbon 2-knot Tube(A), which we denote by πγ. Examle 4.3. We calculate the grou πγ 4 6(, q, r, s). Cutting the interval at each arrowhead, we obtain 5 arcs. Take generators x, y, z, u, v as shown in Fig. 30. Then we obtain the following relations: y = v xv, (4.4) z = x q yx q, (4.5) u = x r zx r, (4.6) v = z s uz s. (4.7) x y z u v Fig. 30. Generators of the grou πγ 4 6 (, q, r, s). In fact, from the linear Gauss diagram Γ 4 6(+ +) we obtain a virtual arc diagram as shown in Fig. 31, from which we obtain the above grou resentation by Wirtinger s algorithm. Substitute (4.4) and (4.6) into (4.5) and (4.7), resectively. Then we obtain Substitute (4.8) into (4.9). Then we obtain z = x q v xv x q, (4.8) v = z s x r zx r z s. (4.9) v = x q v x s v x q x r x q v xv x q x r x q v x s v x q = (x q v x s v x r v )x(v x r v x s v x q ). (4.10) So, the grou πγ 4 6(, q, r, s) is resented by G( q,, s,, r, ).

25 Enumeration of ribbon 2-knots resented by virtual arcs 25 Fig. 31. Virtual arc diagram with Gauss diagram Γ 4 6 (+ +) Alternatively, substitute (4.9) into (4.4). Then we obtain y = z s x r z x r z s xz s x r z x r z s (4.11) Then substitute (4.11) into (4.5). Then we obtain z = (x q z s x r z x r z s )x(z s x r z x r z s x q ). (4.12) So, the grou πγ 4 6(, q, r, s) is also resented by G( q, s, r,, r, s). In Table 3 we list the former resentation. In articular, the grou πγ 4 6(+ +) has two resentations, G(1, 1, 1, 1, 1, 1) and G(1, 1, 1, 1, 1, 1). It is not trivial that they resent the same grou. Examle 4.4. We calculate the grou πγ 4 11(, q, r, s). Take generators x, y, z, u, v as shown in Fig. 32. Then we obtain the following relations: Removing x, y and u, we obtain: y = z xz, (4.13) z = v q yv q, (4.14) u = y r zy r, (4.15) v = x s ux s. (4.16) v = (z v q z s v q z )(v q z r v q zv q z r v q )(z v q z s v q z ). (4.17) So, the grou πγ 4 11(, q, r, s) is resented by G(, q, s, q,, q, r, q), which is isomorhic to G(q, r, q,, q, s, q, ). In Table 3 we list this resentation. Alternatively, removing y, u and v, we obtain: z = (x s z x r z q x r z x s z )x(z x s z x r z q x r z q x s ). (4.18) So, the grou πγ 4 11(, q, r, s) is also resented by G( s,, r, q, r,, s, ). From the grou resentation Eq. (4.1) we obtain the Alexander olynomial as follows: 1 t a1 + t a1+b1 t a1+b1+a2 + t a1+b1+a2+b2 t a1+b1+ +bn 1+an + t a1+b1+ +bn 1+an+bn ; (4.19)

26 26 T. Kanenobu & S. Komatsu x y q z ru s v Fig. 32. Generators x, y, z, u, v of the grou πγ 4 11 (, q, r, s). see [15]. If (t) is the normalized Alexander olynomial of a ribbon 2-knot, then the normalized Alexander olynomial of its mirror image is given by (t 1 ), and thus their determinants coincide. Further, if (t) (t 1 ), then it is not amhicheiral. Remark 4.5. We can define a sequence of invariants for a ribbon 2-knot by α n (K) = (l/n!)d n /dt n K (1), which are finite tye invariants [3, Theorem2.2]; see also [4,10]. 5. Ribbon 2-Knots Presented by Linear Gauss Diagrams with u to 3 Chords In this section, we rove Theorem 4.1 for a linear Gauss diagram with u to 3 chords Linear Gauss diagrams with u to 2 chords A ribbon 2-knot resented by a linear Gauss diagram with one chord is trivial by the move A I. In order to enumerate ribbon 2-knots resented by linear Gauss diagrams with 2 chords, by Lemma 3.6 and the move E II we have only to consider the diagram Γ 2 1 as shown in Fig. 29. Then since Γ 2 1(++) r Γ 2 1( ) by the reversing move, we obtain the ribbon 2-knots R 2 1, R 2 2, R 2 2! as listed in Table 2. The grou is resented as follows: πγ 2 1(, q) = G(, q). (5.1) 5.2. Linear Gauss diagrams with 3 chords We enumerate ribbon 2-knots resented by linear Gauss diagrams with 3 chords. Lemma 5.1. Any linear Gauss diagram with 3 chords is r-equivalent to either the trivial diagram or the diagrams Γ 2 1, Γ 3 1, or Γ 3 2 given in Fig. 29 with some signs. Proof. By Lemma 3.6 we have only to consider three linear Gauss diagrams β i, i = 1, 2, 3, as shown in Fig. 33, where we ignore the orientation of the interval, and the orientations and signs of the chords. Then, by the move E II and reversing move we have only to consider the linear Gauss diagrams Γ 3 1, Γ 3 2, β 21, β 31 as shown in Fig. 34. By the move D we have

27 Enumeration of ribbon 2-knots resented by virtual arcs 27 β 1 β 2 β 3 Fig. 33. Unoriented linear Gauss diagrams with 3 chords. Γ 3 1 Γ 3 2 β 21 β 31 Fig. 34. Oriented linear Gauss diagrams with 3 chords. β 21 (, q, r) r Γ 3 1(, q, r), and by the moves D and A I we have β 31 (, q, r) r Γ 2 1(q, r). This comletes the roof. Now, we consider the diagrams Γ 3 1(, q, r) and Γ 3 2(, q, r). By the moves A II and A I the diagram Γ 3 1(, q, q) is deformed into the trivial diagram. By the move Ω 5 and reversing move the diagram Γ 3 2(, q, r) is deformed into Γ 3 2(, r, q); see Fig. 35(a). This imlies Γ 3 2(+ + +) r Γ 3 2(+ ). Furthermore, by the reversing move and the move Ω 2 the diagram Γ 3 2(,, q) is deformed into Γ 3 2( q, q, ); see Fig. 35(b). This imlies Γ 3 2(+ +) r Γ 3 2( + ). Then we obtain the ribbon 2-knots R 3 i, R3 i! (i = 1, 2, 3, 4) and R3 5 as listed in Table 2. The grous are resented as follows: πγ 3 1(, q, r) = G(, q + r); (5.2) πγ 3 2(, q, r) = G(, r, q, r). (5.3) (a) (b) q r qr r q Reversing Ω 5 q q q q Reversing Ω 2 Fig. 35. r-equivalent linear Gauss diagrams with 3 chords.

28 28 T. Kanenobu & S. Komatsu 6. Linear Gauss diagrams with 4 chords In this section, we rove the following lemma. Lemma 6.1. Any linear Gauss diagram with 4 chords is r-equivalent to either a linear Gauss diagram with u to 3 chords or Γ 4 l, 1 l 13, given in Fig. 29 with some signs. Proof. First we note there are seven tyes of circular Gauss diagrams with 4 chords that has no searated single chord as shown in Fig. 36. γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 γ 7 Fig. 36. Circular Gauss diagrams with 4 chords. From the circular Gauss diagram γ i, 1 i 7, we obtain linear Gauss diagrams γ ij ignoring the orientation of the interval, and the orientations and signs of the chords as shown in Fig. 37. From γ ij we obtain oriented Gauss diagrams γ ijk as shown in Fig. 38 u to r-equivalence, where: γ ijk = Γ 4 l means that the diagram γ ijk is the same as Γ 4 l given in Fig. 29. γ ijk Γ 3 means that by the move D the diagram γ ijk (, q, r, s) with any signs (, q, r, s = ±) is deformed into a linear Gauss diagram having a single searated chord, which is r-equivalent to a Gauss diagram with u to 3 chords by Lemma 3.6. γ ijk ±Γ 4 l means that by the move D the diagram γ ijk (, q, r, s) is deformed into the Gauss diagram ±Γ 4 l (, q, r, s ) for some l and some signs, q, r, s = ±, where Γ 4 l is given in Fig. 29. γ ijk ±γ i j k means that by the move D the diagram γ ijk(, q, r, s) is deformed into the Gauss diagram ±γ i j k (, q, r, s ). The following are roved by the move Ω i, i = 5, 6, 7, 8: γ 112 (, q, r, s) r Γ 4 7(q, r, s, ) (by the move Ω 6 ), (6.1) γ 113 (, q, r, s) r Γ 4 6(q, r, s, ) (by the move Ω 5 ), (6.2) γ 114 (, q, r, s) r Γ 4 5(q, r, s, ) (by the move Ω 5 ), (6.3) γ 441 (, q, r, s) r Γ 4 6(, s, q, r) (by the move Ω 8 ), (6.4) γ 442 (, q, r, s) r Γ 4 2(, s, q, r) (by the move Ω 8 ), (6.5)

29 Enumeration of ribbon 2-knots resented by virtual arcs 29 γ 11 γ 12 γ 21 γ 22 γ 23 γ 24 γ 25 γ 31 γ 32 γ 41 γ 42 γ 43 γ 44 γ 51 γ 52 γ 53 γ 61 γ 62 γ 63 γ 71 Fig. 37. Linear Gauss diagrams γ ij obtained from γ i, 1 i 7. γ 531 (, q, r, s) r Γ 4 8(r,, q, s) (by the move Ω 7 ), (6.6) γ 423 (, q, r, s) r γ 243 (, q, r, s) (by the move Ω 7 ), (6.7) γ 622 (, q, r, s) r γ 113 (, q, s, r) (by the move Ω 8 ), (6.8) γ 713 (, q, r, s) r γ 442 (q,, r, s) (by the move Ω 7 ). (6.9) Thus, each linear Gauss diagram γ ijk (, q, r, s) is r-equivalent to either a diagram with u to 3 chords or Γ 4 l (, q, r, s ) for some l and some signs, q, r, s. This comletes the roof. 7. Ribbon 2-Knots Obtained from the Linear Gauss Diagrams Γ 4 l In this section, we rove Theorem 4.1 for a linear Gauss diagram with 4 chords using Lemma 6.1; we examine the associated ribbon 2-knot for each Γ 4 l (, q, r, s), 1 l Ribbon 2-knots obtained from Γ 4 1 By the move A II if either q = r or r = s, then Γ 4 1(, q, r, s) is r-equivalent to a diagram with 2 chords. So, we consider the case q = r = s. Then we obtain the

30 30 T. Kanenobu & S. Komatsu γ 111 = Γ 4 1 γ 112 γ 113 γ 114 γ 121 Γ 3 γ 122 Γ 3 γ 123 Γ 3 γ 124 Γ 3 γ 211 Γ 3 γ 212 Γ 3 γ 213 Γ 4 13 γ 221 = Γ 4 10 γ 222 = Γ 4 12 γ 223 Γ 3 γ 224 Γ 3 γ 231 γ 232 γ 233 Γ 3 γ 234 Γ 3 γ 241 Γ 3 γ 242 Γ 3 γ 243 γ 531 γ 244 Γ 3 γ 251 = Γ 4 6 γ 252 = Γ 4 2 γ 253 = Γ 4 5 γ 311 Γ 3 γ 312 Γ 4 12 γ 321 = Γ 4 7 γ 322 = Γ 4 3 γ 323 Γ 4 5 Fig. 38. Oriented linear Gauss diagrams γ ijk obtained from γ ij.

31 Enumeration of ribbon 2-knots resented by virtual arcs 31 γ 411 Γ 4 1 γ 412 = Γ 4 4 γ 413 = Γ 4 11 γ 414 γ 114 γ 421 Γ 3 γ 422 Γ 3 γ 423 γ 424 Γ 3 γ 431 Γ 4 6 γ 432 Γ 4 2 γ 433 Γ 3 γ 434 Γ 3 γ 441 γ 231 γ 442 γ 443 γ 112 γ 444 Γ 4 1 γ 511 = Γ 4 13 γ 521 = Γ 4 8 γ 531 γ 611 Γ 3 γ 612 Γ 4 2 γ 613 = Γ 4 9 γ 621 Γ 4 1 γ 622 γ 623 Γ 4 3 γ 624 Γ 4 7 γ 631 Γ 4 1 γ 632 Γ 4 4 γ 711 Γ 4 1 γ 712 Γ 4 3 γ 713 Fig. 38. Cont d. ribbon 2-knots R1,j 4, R4 1,j!, j = 1, 2, as listed in Table 3. The grou is resented as

32 32 T. Kanenobu & S. Komatsu follows: πγ 4 1(, q, r, s) = G(, q + r + s). (7.1) 7.2. Ribbon 2-knots obtained from Γ 4 2 By the move D the diagram Γ 4 2(, q, r, s) is deformed into γ 612 (, q, r, s) given in Fig. 38. If either s = q or r =, then by the move A III the diagram γ 612 (, q, r, s) is deformed into a diagram with a single searated chord; see Fig. 39. So, we consider the case s = q and r =. Note that Γ 4 2( ) r Γ 4 2( ) by the reversing move. Then we obtain the ribbon 2-knots R 4 2,1, R 4 2,2, R 4 2,2! as listed in Table 3. The grou is resented as follows: πγ 4 2(, q, r, s) = G(, q, r, s). (7.2) γ 612 (, q, r, q) A III γ 612 (, q,, s) A III Fig. 39. The diagrams γ 612 (, q, r, q) and γ 612 (, q,, s) are deformed into diagrams with a single searated chord Ribbon 2-knots obtained from Γ 4 3 If either = q or q = r, then by the move A II the diagram Γ 4 3(, q, r, s) is r-equivalent to a diagram with 2 chords. So, we consider the case = q and r = s. Note that Γ 4 3( ) r Γ 4 3( ) by the reversing move. Then we obtain the ribbon 2-knots R 4 3,1, R 4 3,2, R 4 3,2! as listed in Table 3. The grou is resented as follows: πγ 4 3(, q, r, s) = G( + q, r + s). (7.3) 7.4. Ribbon 2-knots obtained from Γ 4 4 By the move D the diagram Γ 4 4(, q, r, s) is deformed into γ 632 (, q, r, s) given in Fig. 38. Then if s = r, by the move A III the diagram γ 632 (, q, r, s) is deformed into a Gauss diagram with a single searated chord; see Fig. 40. So, we consider

33 Enumeration of ribbon 2-knots resented by virtual arcs 33 the cases r = s. Then we obtain the ribbon 2-knots R4,j 4, R4 4,j!, j = 1, 2, 3, 4, as listed in Table 3. The grou is resented as follows: πγ 4 4(, q, r, s) = G(, s, q, r + s). (7.4) γ 632 (, q, r, r) A III Fig. 40. γ 632 (, q, r, r) is r-equivalent to a diagram with a single searated chord Ribbon 2-knots obtained from Γ 4 5 By the move Ω 5 the diagram Γ 4 5(, q, r, s) is r-equivalent to γ 114 ( s,, q, r) given in Fig. 38; see Fig. 41. If = s, then by the move A II the diagram γ 114 ( s,, q, r) is r-equivalent to a diagram with 2 chords. So, we consider the case = s. Note that by the reversing move we have Γ 4 5(+ ) r Γ 4 5(+++ ) and Γ 4 5( ++ +) r Γ 4 5( +). Γ 4 5(, q, r, s) Ω 5 γ114 ( s,, q, r) Fig. 41. The diagram Γ 4 6 (, q, r, s) is deformed into γ 114( s,, q, r). Furthermore, we have the following r-equivalent airs. Lemma 7.1. Γ 4 4(+ + +) r Γ 4 5( + +); (7.5) Γ 4 4( + ) r Γ 4 5(+ + ); (7.6) Γ 4 4( + + +) r Γ 4 5( +); (7.7) Γ 4 4(+ ) r Γ 4 5(+ + + ). (7.8) Proof. Figure 42 shows the diagrams Γ 4 4(,, +, +) and Γ 4 5(,,, +) are r- equivalent to the same diagram, roving Eqs. (7.5) and (7.7); Eqs. (7.6) and (7.8) are their mirror images.

34 34 T. Kanenobu & S. Komatsu Then the remaining diagrams are Γ 4 5( + +) and Γ 4 5(+ + ), which resent the ribbon 2-knots R 4 5,1 and R 4 5,1!, resectively, as listed in Table 3. The grou is resented as follows: πγ 4 5(, q, r, s) = G( r, q, r, s ). (7.9) 7.6. Ribbon 2-knots obtained from Γ 4 6 First we show the following lemma. Lemma 7.2. Γ 4 6(, q, r, q) r Γ 4 9(, r,, q); (7.10) Proof. See Fig. 43. Now we show the following r-equivalence. Lemma 7.3. Γ 4 6(+ + + ) r Γ 4 6( +); (7.11) Γ 4 6( + ) r Γ 4 6(+ + ) r Γ 4 6(+ + +); (7.12) Γ 4 6(+ + +) r Γ 4 6( + +) r Γ 4 6( + ). (7.13) Proof. First, we show Eq. (7.11). By Lemma 7.2 we have Γ 4 6(+ + + ) r Γ 4 9( ) and Γ 4 6( +) r Γ 4 9( ). Since Γ 4 9( ) r Γ 4 9( ) by the reversing move, we obtain Eq. (7.11). Next, we show Eq. (7.13); Eq. (7.12) is its mirror image. We start with the ribbon handlebody H 0 as shown in Fig. 44, which is consisting of four 0-handles and three 1-handles. We deform this into three ribbon handlebodies H 1, H 2, H 3 as shown in Fig. 44 by the stably equivalent move introduced by Marumoto [16, Fig. 2-2], and so these four ribbon handlebodies reresent the same ribbon 2-knot. Then H 1, H 2, H 3 are reresented by the virtual arcs A 1, A 2, A 3, and their diagrams are δ 1, δ 2, δ 3, resectively, as shown in Fig. 45. Note that δ 1 = γ 441 (+ + +), δ 2 = γ 441 (+ +), δ 3 = γ 441 ( +). Then δ 1 is r-equivalent to Γ 4 6(+ + +) by the move Ω 8 ; δ 2 is r-equivalent to γ 441 ( + + ) by the reversing move, which is r-equivalent to Γ 4 6( + +) by the move Ω 8 ; δ 3 is r-equivalent to Γ 4 6( + ) by the move Ω 8 ; see Eq. (6.4). Thus we obtain Eq. (7.13). This comletes the roof. Therefore, we obtain the ribbon 2-knots R 4 6,j R4 6,j!, j = 1, 3, 4, 5, 6, and R4 6,2 as listed in Table 3. For the grou πγ 4 6(, q, r, s), see Examle 4.3. Remark 7.4. In [24, Examle 4.1] Yasuda has shown the ribbon 2-knot reresented by H 0 has three different ribbon handlebodies consisting of two 0-handles and one 1-handle. The ribbon handlebodies H 0 and H 1 in Fig. 44 are given in [24, Fig. 7].

35 Enumeration of ribbon 2-knots resented by virtual arcs 35 Γ 4 4(, q, +, +) A II (, q, +,, +, +) = (, q, +,, +, +) A III (,, q, +, +) A I (,, q, +, +) D (,, q, +, +) Γ 4 5(,,, +) Ω 6 (,, +, +) A I (,,,, +, +) = (,,,, +, +) A III (,,,, +, +) D (,,,, +, +) A I (,,, +, +) D (,,, +, +) Fig. 42. Proof of Lemma 7.1.

36 36 T. Kanenobu & S. Komatsu q r q Γ 4 6(, q, r, q) Ω 5 γ113 (q,, q, r) = γ 113 (q,, q, r) r Ω 8 γ622 (q,, r, q) = γ 622 (q,, r, q) Ω 3 Γ 4 9 (, r,, q) Fig. 43. Proof of Lemma 7.2. H 0 H 1 H 2 H 3 Fig. 44. Ribbon handlebodies Ribbon 2-knots obtained from Γ 4 7 If q = r, then the diagram Γ 4 7(, q, r, s) is r-equivalent to a diagram with 2 chords by the move A II. So, we consider the case q = r. Then we obtain the ribbon 2-knots

37 Enumeration of ribbon 2-knots resented by virtual arcs 37 A 1 A 2 A δ 1 δ 2 δ 3 Fig. 45. Virtual arcs A 1, A 2, A 3 reresenting ribbon handlebodies H 1, H 2, H 3, and their diagrams δ 1, δ 2, δ 3. R7,j 4, R4 7,j!, j = 1, 2, 3, 4, as listed in Table 3. The grou is resented as follows: πγ 4 7(, q, r, s) = G( s,, q r, ). (7.14) 7.8. Ribbon 2-knots obtained from Γ 4 8 Note that Γ 4 8(, q, r, s) r Γ 4 8( s, r, q, ) by the reversing move. Then we obtain the ribbon 2-knots R8,1, 4 R8,6, 4 and R8,j 4, R4 8,j!, j = 2, 3, 4, 5, as listed in Table 3. The grou is resented as follows: πγ 4 8(, q, r, s) = G(, s, q, s,, r,, s). (7.15) 7.9. Ribbon 2-knots obtained from Γ 4 9 If either = r or q = s, then Γ 4 9(, q, r, s) is r-equivalent to Γ 4 6(, q, r, s ) for some signs, q, r, s. In fact, if = r, then by Lemma 7.2 we have Γ 4 9(, q,, s) r Γ 4 7(, s, q, s). Next, suose q = s. By the reversing move we have Γ 4 9(, q, r, q) r Γ 4 9( q, r, q, ), which is r-equivalent to Γ 4 6( q,, r, ) by Lemma 7.2. Then we obtain the ribbon 2-knots R 4 9,1, R 4 9,1!, and R 4 9,2 as listed in Table 3. The grou is resented as follows: πγ 4 9(, q, r, s) = G(, q,, s, r, s). (7.16)

38 38 T. Kanenobu & S. Komatsu Ribbon 2-knots obtained from Γ 4 10 From the diagram Γ 4 10 we obtain the ribbon 2-knots R10,j 4, R4 10,j!, j = 1,..., 8, as listed in Table 3. The grou is resented as follows: πγ 4 10(, q, r, s) = G(, q,, r,, q,, s,, q). (7.17) Ribbon 2-knots obtained from Γ 4 11 We have the following r-equivalent airs. Lemma 7.5. Γ 4 11(+ + +) r Γ 4 11(+ + ); (7.18) Γ 4 11( + ) r Γ 4 11( + +). (7.19) Proof. We show Eq. (7.18). The diagram Γ 4 11(+ + +) reresents the virtual arc diagram A 4 as shown in Fig. 46, which is r-equivalent to A 4. Then it is deformed into A 4 by the moves A II, A III, D, E II as shown in Fig. 46, which has the diagram δ 4 as shown in Fig. 48. A 4 A 4 A 4 Fig. 46. Virtual arc diagram A 4 having diagram Γ 4 11 (+ + +), and r-equivalent virtual arcs. The diagram Γ 4 11(+ + ) reresents the virtual arc diagram A 5 as shown in Fig. 47. It is deformed into A 5 by the moves A II, A III, B II, B III, D, E I, E II as shown in Fig. 47, which has the Gauss diagram δ 5 as shown in Fig. 48. Both the Gauss diagrams γ 4 and γ 5 are deformed into γ 6 as shown in Fig. 48 by the move D, which comletes the roof. Then we obtain the ribbon 2-knots R 4 11,j, R4 11,j!, j = 1,..., 7, as listed in Table 3. For the grou πγ 4 11(, q, r, s), see Examle 4.4

39 Enumeration of ribbon 2-knots resented by virtual arcs 39 A 5 A 5 Fig. 47. Virtual arc diagram A 5 having diagram Γ 4 11 (+ + ), and r-equivalent virtual arcs. δ 4 δ 5 δ 6 Fig. 48. Gauss linear diagrams for the virtual arcs A 4 and A Ribbon 2-knots obtained from Γ 4 12 If = r, then the diagram Γ 4 12(, q, r, s) is r-equivalent to a linear Gauss diagram with less than 4 chords. In fact, by the move A III the diagram Γ 4 12(, q,, s) is r-equivalent to a linear Gauss diagram with a searated chord; see Fig. 49. Moreover, if s = q, then we have Γ 4 12(, q, r, q) r Γ 4 7(, r,, q) as shown in Fig. 50. By the reversing move Γ 4 7(, r,, q) r Γ 4 7(q,, r, ). So, we consider Γ 4 12(, q, r, s) with = r and q = s. Then we obtain the ribbon 2-knots R12,j 4, R12,j 4!, j = 1, 2, as listed in Table 3. The grou is resented as follows: πγ 4 12(, q, r, s) = G(, q, r, s) (7.20)

40 40 T. Kanenobu & S. Komatsu q r Γ 4 12(, q,, s) A III q Fig. 49. Γ 4 12 (, q,, s) is deformed into a diagram with a single searated chord. Γ 4 12(, q, r, q) D γ 312 (, q, r, q) Ω 2 Γ 4 7 (, r,, q) Fig. 50. Γ 4 12 (, q, r, q) is r-equivalent to Γ4 7 (, r,, q) Ribbon 2-knots obtained from Γ 4 13 Since the diagram Γ 4 13(, q, r, s) is the roduct Γ 2 1(, q) Γ 2 1(r, s) and Γ 2 1(++) r Γ 2 1( ), by Eq. (3.1) we have: Γ 4 13( ) r Γ 4 13( ) r Γ 4 13(+ + ) r Γ 4 13( + +), (7.21) Γ 4 13(+ + + ) r Γ 4 13(+ + +) r Γ 4 13( + ) r Γ 4 13(+ ), (7.22) Γ 4 13(+ + +) r Γ 4 13( + + +) r Γ 4 13( +) r Γ 4 13( + ), (7.23) Γ 4 13(+ +) r Γ 4 13( + + ), (7.24) which resent the comosite ribbon 2-knots R 2 1#R 2 1, R 2 1#R 2 2, R 2 1#R 2 2!, R 2 2#R 2 2!, resectively. Additionally, we have R 2 2#R 2 2 and R 2 2!#R 2 2!, which are resented by Γ 4 13(+ + ) and Γ 4 13( + +), resectively. For the comosition of two ribbon 2-knots K and K, K#K, we have K#K (t) = K (t) K (t), and so we obtain Table Final Remarks (i) In Tables 2, 3 and 4 there are several airs and triles sharing the same Alexander olynomials. Excet for the air (R 4 8,1, R 4 8,6) we can distinguish using the knot grou and the twisted Alexander olynomial; see the forthcoming aer [11] for the detail. The grous of this air are actually isomorhic. (ii) The following ribbon 2-knots are missing in Yasuda s table [22,23,25,26,27]. In fact, in Yasuda s table there do not exist ribbon knots having the Alexander olynomials of these knots. R 4 11,4, R 4 11,4!, R 4 11,5, R 4 11,5!, R 4 11,6, R 4 11,6!. (8.1)