CCF-MOVES FOR VIRTUAL LINKS
KODAI WADA
Abstract.
Oikawa defined an unknotting operation on virtual knots, calleda CF-move, and gave a classification of 2-component virtual links up to CF-moves by the virtual linking number and his n -invariant. In particular, it wasproved that a CF-move characterizes the information contained in the virtuallinking number for 2-component odd virtual links. In this paper, we extendthis result by classifying odd virtual links and almost odd virtual links witharbitrary number of components up to CF-moves, using the virtual linkingnumber. Moreover, we extend Oikawa’s n -invariant and introduce two invari-ants for 3-component even virtual links. Using these invariants together withthe virtual linking number, we classify 3-component even virtual links up toCF-moves. As a result, a classification of 3-component virtual links up toCF-moves is provided. Introduction
Virtual links were introduced by Kauffman in [4] as a generalization of classicallinks in the 3-sphere. For an integer µ ≥
1, a µ -component virtual link diagram is the image of an immersion of µ circles into the plane, whose singularities areonly transverse double points. Such double points are divided into classical cross-ings and virtual crossings as shown in Figure 1.1. A µ -component virtual link isan equivalence class of µ -component virtual link diagrams under generalized Rei-demeister moves , which consist of classical Reidemeister moves R1–R3 and virtualReidemeister moves V1–V4 as shown in Figure 1.2. A 1-component virtual link isalso called a virtual knot . Throughout this paper, all virtual links are assumed tobe ordered and oriented.classical crossing virtual crossing Figure 1.1.
Two types of double points on virtual link diagramsIn virtual knot and link theory, we do not allow to use two local moves as shownin Figure 1.3. These moves are called the forbidden moves . Any virtual knot canbe deformed into the trivial knot by forbidden moves (cf. [2, 3, 8]). Generally, the µ -component virtual links L = K ∪· · ·∪ K µ up to forbidden moves are classified bythe ( i, j )-linking numbers Lk( K i , K j ) ∈ Z (1 ≤ i (cid:54) = j ≤ µ ) completely (cf. [1, 7, 10]). Mathematics Subject Classification.
Primary 57K12; Secondary 57K10.
Key words and phrases. virtual link, crossing change, forbidden move, CF-move, linking num-ber, Gauss diagram.This work was supported by JSPS KAKENHI Grant Number JP19J00006. a r X i v : . [ m a t h . G T ] F e b KODAI WADA
R1 R1 R2 R3V1 V2 V3V4
Figure 1.2.
Generalized Reidemeister moves
Figure 1.3.
Forbidden movesA
CF-move introduced by Oikawa in [9] is a local move on virtual link di-agrams as shown in Figure 1.4. It is considered as a combination of crossingchanges and a forbidden move. Two virtual links are
CF-equivalent if their di-agrams are related by a finite sequence of generalized Reidemeister moves and CF-moves. In [9, Theorem 1.2], Oikawa proved that any virtual knot is CF-equivalentto the trivial knot, i.e. a CF-move is an unknotting operation. Moreover, he de-fined an invariant n ( L ) ∈ N ∪ { } of a 2-component virtual link L = K ∪ K with Lk( K , K ) ≡ Lk( K , K ) (mod 2). Then using the virtual linking num-ber vlk( K , K ) = − (Lk( K , K ) − Lk( K , K )) and n ( L ), a classification of 2-component virtual links up to CF-equivalence was given in [9, Theorem 1.7] asfollows. Figure 1.4.
A CF-move
Theorem 1.1 ([9]) . Let L = K ∪ K and L (cid:48) = K (cid:48) ∪ K (cid:48) be -component virtuallinks. (i) Assume that vlk( K , K ) is not an integer. Then L and L (cid:48) are CF-equivalent if and only if vlk( K , K ) = vlk( K (cid:48) , K (cid:48) ) . (ii) Assume that vlk( K , K ) is an integer. Then L and L (cid:48) are CF-equivalentif and only if vlk( K , K ) = vlk( K (cid:48) , K (cid:48) ) and n ( L ) = n ( L (cid:48) ) . This paper aims to extend this theorem to virtual links with three or morecomponents. To this end, we consider the parity for every component of a µ -component virtual link L = K ∪ · · · ∪ K µ . For 1 ≤ i ≤ µ , the i th component K i of F-MOVES FOR VIRTUAL LINKS 3 L is called odd if one encounters an odd number of classical crossings in going alongthe i th component of a diagram of L ; otherwise it is called even . We say that L is odd if every component of L is odd. On the other hand, we say that L is even ifevery component of L is even (cf. [6]). For example, all virtual knots and classicallinks are even. In the case µ = 2, it follows that L = K ∪ K is odd if and onlyif vlk( K , K ) is not an integer, and that it is even if and only if vlk( K , K ) isan integer. There are virtual links with three or more components that are neitherodd nor even. In particular, we say that L is almost odd if some component of L iseven and the other components are odd. We remark that if L is odd, then µ is aneven integer. On the other hand, if L is almost odd, then µ is an odd integer.For odd virtual links and almost odd virtual links, we have the following. Theorem 1.2.
For two µ -component odd or almost odd virtual links L = K ∪· · · ∪ K µ and L (cid:48) = K (cid:48) ∪ · · · ∪ K (cid:48) µ , the following are equivalent. (i) L and L (cid:48) are CF-equivalent. (ii) Lk( K i , K j ) − Lk( K j , K i ) = Lk( K (cid:48) i , K (cid:48) j ) − Lk( K (cid:48) j , K (cid:48) i ) for any ≤ i < j ≤ µ . By the classification of welded links, which are a quotient of virtual links by the“over” forbidden move in the left of Figure 1.3, up to crossing changes given in [1,Theorem 1], Theorem 1.2 means that CF-equivalence coincides with the equivalencerelation generated by crossing changes and forbidden moves for odd virtual linksand almost odd virtual links. Note that the difference Lk( K i , K j ) − Lk( K j , K i )in Theorem 1.2 is written as vlk ij − vlk ji in [1]. For odd virtual links and almostodd virtual links, a CF-move is indeed a combination of crossing changes and aforbidden move.For even virtual links, we only consider the case of three components. For a 3-component even virtual link L = K ∪ K ∪ K , we define an invariant n ( K i , K j ) ∈ N ∪ { } (1 ≤ i < j ≤
3) as an extension of Oikawa’s n -invariant given in [9](Definition 4.3). Moreover, we introduce two invariants τ ( L ) , τ ( L ) ∈ Z of L (Definition 4.7). Then we have the following. Theorem 1.3.
For two -component even virtual links L = K ∪ K ∪ K and L (cid:48) = K (cid:48) ∪ K (cid:48) ∪ K (cid:48) , the following are equivalent. (i) L and L (cid:48) are CF-equivalent. (ii) Lk( K i , K j ) − Lk( K j , K i ) = Lk( K (cid:48) i , K (cid:48) j ) − Lk( K (cid:48) j , K (cid:48) i ) and n ( K i , K j ) = n ( K (cid:48) i , K (cid:48) j ) for any ≤ i < j ≤ , and τ ( L ) − τ ( L ) = τ ( L (cid:48) ) − τ ( L (cid:48) ) . Since a 3-component virtual link is either almost odd or even, a complete clas-sification of 3-component virtual links up to CF-equivalence is provided by Theo-rems 1.2 and 1.3. 2.
Gauss diagrams up to CF-moves
In this section, we review the definition of Gauss diagrams and give a normalform of a µ -component virtual link up to CF-equivalence in terms of Gauss diagrams(Proposition 2.3).A Gauss diagram is a disjoint union of ordered and oriented circles together withsigned and oriented chords whose endpoints lie disjointly on the circles. Throughoutthis paper, all circles of a Gauss diagram are assumed to be oriented counterclock-wise and arranged in increasing order from left to right. A chord in a Gauss diagramis called a self-chord if both endpoints lie on the same circle of the Gauss diagram;
KODAI WADA otherwise it is called a nonself-chord . In particular, a self-chord is free if its end-points are adjacent on the circle. A nonself-chord is of type ( i, j ) if it is orientedfrom the i th circle to the j th one ( i (cid:54) = j ).Given a µ -component virtual link diagram with n classical crossings, its Gaussdiagram is defined to be the union of µ circles and n chords connecting the preimageof each classical crossing. Each chord is equipped with the sign of the correspond-ing classical crossing, and oriented from the over-crossing to the under-crossing. Bydefinition, the virtual Reidemeister moves V1–V4 on virtual link diagrams do notaffect the corresponding Gauss diagrams. On the other hand, the classical Reide-meister moves R1–R3 change the Gauss diagrams as shown in Figure 2.1. Here,there are several kinds of R3 depending on the signs and orientations of chords, andone of them is shown in the figure. The others are generated by R1, R2, and thisR3 (cf. [11]). Therefore, a virtual link can be considered as an equivalence class ofGauss diagrams under Reidemeister moves R1–R3 (cf. [2, 4]).R1 R1 R2 R2R3 ε ε ε − ε ε − ε Figure 2.1.
Reidemeister moves on Gauss diagrams with ε ∈ {± } A CF-move on Gauss diagrams is a local move as shown in Figure 2.2. Two Gaussdiagrams G and G (cid:48) are CF-equivalent if they are related by a finite sequence ofReidemeister moves R1–R3 and CF-moves. It is denoted by G ∼ G (cid:48) . We emphasizethat two virtual links are CF-equivalent if and only if their Gauss diagrams are CF-equivalent. ε ε (cid:48) − ε − ε (cid:48) Figure 2.2.
A CF-move on Gauss diagrams with ε, ε (cid:48) ∈ {± } Lemma 2.1.
If two Gauss diagrams are related by a local move as shown in Fig-ure 2.3, then they are CF-equivalent.Proof.
The proof in the case εε (cid:48) = 1 follows from Figure 2.4, and that in the case εε (cid:48) = − (cid:3) F-MOVES FOR VIRTUAL LINKS 5 ε ε (cid:48) − ε − ε (cid:48) Figure 2.3.
Exchanging two consecutive initial and terminal end-points of a pair of chordsR2 R3 CFR2 CF ε ε ε εε − ε ε − εε ε − ε − ε − ε − ε − εε ε − εεε Figure 2.4.
Proof of Lemma 2.1 in the case εε (cid:48) = 1R2 R3 CFR2 CF ε − ε ε − εε − ε ε − ε − ε ε − ε ε ε − εε − ε ε εεε Figure 2.5.
Proof of Lemma 2.1 in the case εε (cid:48) = − Remark 2.2.
By Lemma 2.1, any two consecutive endpoints of a pair of chords γ and γ (cid:48) can be exchanged up to CF-equivalence, although the signs and orientationsof γ and γ (cid:48) are altered. Therefore, up to CF-equivalence, we can deform everyself-chord in a Gauss diagram into a free chord, and remove it by an R1-move.In particular, any Gauss diagram of a virtual knot is CF-equivalent to the onewithout chords, which represents the trivial knot. This is an alternative proof of [9,Theorem 1.2] in terms of Gauss diagrams.For 1 ≤ i (cid:54) = j ≤ µ and an integer a ij ∈ Z , let G ij ( a ij ) be the Gauss diagramwith µ circles C , . . . , C µ , which consists of only | a ij | horizontal nonself-chords oftype ( i, j ) with sign ε ij , where a ij = ε ij | a ij | . Let G and G (cid:48) be Gauss diagrams with µ circles C , . . . , C µ and C (cid:48) , . . . , C (cid:48) µ , respectively. We put C , . . . , C µ on top and KODAI WADA C (cid:48) , . . . , C (cid:48) µ on bottom. As shown in Figure 2.6, for every i ∈ { , . . . , µ } , we connect C i and C (cid:48) i by an unknotted arc, and then form a new circle C i C (cid:48) i by surgery alongthe unknotted arc. The Gauss diagram consisting of µ circles C C (cid:48) , . . . , C µ C (cid:48) µ is called a connected sum of G and G (cid:48) and denoted by G G (cid:48) . An example is shownin Figure 2.7. C i C (cid:48) i unknottedarc surgery C i C (cid:48) i Figure 2.6.
Forming a new circle C i C (cid:48) i from two circles C i and C (cid:48) i by surgery along an unknotted arc Figure 2.7.
A connected sum G (3) G ( −
2) of two Gauss di-agrams G (3) and G ( −
2) in the case µ = 3 Proposition 2.3.
Any Gauss diagram G of a µ -component virtual link is CF-equivalent to ≤ i Let L = K ∪ · · · ∪ K µ be a µ -component virtual link and G its Gauss diagramwith µ circles C , . . . , C µ . For 1 ≤ i (cid:54) = j ≤ µ , the ( i, j ) -linking number of L isdefined to be the sum of signs of all nonself-chords of type ( i, j ) in G . It is denotedby Lk( K i , K j ). This integer Lk( K i , K j ) is an invariant of the virtual link L (cf. [2,Section 1.7]). It can be seen that the difference Lk( K i , K j ) − Lk( K j , K i ) is equal F-MOVES FOR VIRTUAL LINKS 7 to − K i , K j ), where vlk( K i , K j ) denotes virtual linking number of K i and K j given in [9, 10]. Therefore, the following result is obtained from Theorem 1.1. Lemma 3.1. For any ≤ i < j ≤ µ , the difference Lk( K i , K j ) − Lk( K j , K i ) isinvariant under CF-moves. For 1 ≤ i ≤ µ , the i th circle C i of G is called odd if the number of all endpointsof self-/nonself-chords on C i is an odd integer; otherwise it is called even . SinceReidemeister moves and CF-moves do not change the parity of the number ofendpoints on C i , the parity of C i is preserved under CF-equivalence. We emphasizethat the i th component K i of L is odd if and only if C i is odd, and that it is evenif and only if C i is even. Lemma 3.2. Let G be a Gauss diagram with an odd circle C . Let G (cid:48) be a Gaussdiagram obtained from G by altering the sign and orientation of a nonself-chord γ that is attached to C . Then G and G (cid:48) are CF-equivalent.Proof. A CF-equivalence from G to G (cid:48) is given as follows. First, we use Lemma 2.1to slide the endpoint of γ on C along the circle C until it returns to the originalposition. Since C is odd, γ encounters an even number of endpoints during thisprocess. Therefore, γ still has the same sign and orientation. On the other hand,the signs and orientations of the other nonself-chords that are attached to C arealtered. Note that those of all self-chords in C are preserved. Next, we add a freechord in C by an R1-move, and slide one endpoint of the free chord along C untilit is next to the other endpoint by Lemma 2.1. Then the signs and orientations ofall nonself-chords that are attached to C , including γ , are altered, and those of allself-chords in C are preserved. Finally, we delete the free chord by an R1-move.The obtained Gauss diagram is G (cid:48) . (cid:3) Proposition 3.3. Any Gauss diagram G of a µ -component odd or almost oddvirtual link L = K ∪ · · · ∪ K µ is CF-equivalent to ≤ i Throughout this section, we consider a 3-component even virtual link L = K ∪ K ∪ K and its Gauss diagram G with three circles C , C , and C .First we extend Oikawa’s n -invariant given in [9, Definition 1.6] to the 3-componenteven virtual link L from the Gauss diagram point of view. We use the equivalencerelation among the nonself-chords in G connecting C i and C j (1 ≤ i < j ≤ γ and γ (cid:48) be nonself-chords connecting C i and C j . Theendpoints of γ and γ (cid:48) on C i divide the circle C i into two arcs. Let α i be one of thetwo arcs. Similarly, the endpoints of γ and γ (cid:48) on C j divide C j into two arcs, andlet α j be one of the two arcs. See Figure 4.1. We say that γ and γ (cid:48) are equivalent if the number of all endpoints of self-/nonself-chords on α i ∪ α j is an even integer.In particular, γ is equivalent to itself. Since C i and C j are even, the equivalencerelation between γ and γ (cid:48) does not depend on the choice of arcs α i and α j . α i α j C i C j γγ (cid:48) Figure 4.1. A pair of arcs α i and α j for nonself-chords γ and γ (cid:48) connecting C i and C j For 1 ≤ i < j ≤ 3, we fix a nonself-chord γ connecting C i and C j . Let σ ( C i , C j ; γ ) be the sum of signs of all nonself-chords connecting C i and C j , whichare equivalent to γ , including γ itself. On the other hand, let σ ( C i , C j ; γ ) be thesum of signs of all nonself-chords connecting C i and C j , which are not equivalent to γ . We remark that σ ( C i , C j ; γ )+ σ ( C i , C j ; γ ) = Lk( K i , K j )+Lk( K j , K i ). Denoteby n ( C i , C j ; γ ) the absolute value of the difference σ ( C i , C j ; γ ) − σ ( C i , C j ; γ ).Note that n ( C i , C j ; γ ) = n ( C j , C i ; γ ) by definition. Example 4.1. Let G be the Gauss diagram with three circles C , C , and C inFigure 4.2, which represents a 3-component even virtual link. Let γ , . . . , γ bethe nonself-chords in G as shown in the figure. Choose γ as a fixed nonself-chordconnecting C and C . By definition, γ and γ are equivalent to γ . On the otherhand, γ , γ , and γ are not equivalent to γ . Therefore, we have n ( C , C ; γ ) = 3.Similarly, choosing γ as a fixed nonself-chord connecting C and C , we have n ( C , C ; γ ) = 3. Moreover, since γ is the only one nonself-chord connecting C and C , it follows that n ( C , C ; γ ) = 1. Lemma 4.2. For any ≤ i < j ≤ , the nonnegative integer n ( C i , C j ; γ ) is aninvariant of L . Moreover, it is invariant under CF-moves. Since the proof is obtained from the proofs of Proposition 3.1 and Lemma 3.2in [9] by interpreting in terms of Gauss diagrams, we omit it here. By Lemma 4.2,the following definition is well-defined. F-MOVES FOR VIRTUAL LINKS 9 C C C γ γ γ γ γ γ γ γ γ Figure 4.2. A Gauss diagram representing a 3-component evenvirtual link Definition 4.3. The Oikawa invariant of K i and K j in L is the nonnegative integer n ( C i , C j ; γ ) for any nonself-chord γ in G connecting C i and C j (1 ≤ i < j ≤ n ( K i , K j ). Remark 4.4. Consider a Gauss diagram G with three circles C , C , and C associ-ated with a 3-component classical link diagram (i.e. a link diagram without virtualcrossings). Fix a nonself-chord γ in G connecting C i and C j (1 ≤ i < j ≤ C i and C j are equivalent to γ . There-fore, if L = K ∪ K ∪ K is a 3-component classical link (i.e. a link having a classicallink diagram), then n ( K i , K j ) = | Lk( K i , K j ) + Lk( K j , K i ) | = 2 | lk( K i , K j ) | , wherelk( K i , K j ) denotes the classical linking number of K i and K j .Now we introduce two invariants τ ( L ) and τ ( L ) of L . Let γ, γ (cid:48) , and γ (cid:48)(cid:48) benonself-chords in G connecting C and C , C and C , and C and C , respectively.The endpoints of γ and γ (cid:48) on C divide the circle C into two arcs. Let α beone of the two arcs. Similarly, the endpoints of γ and γ (cid:48)(cid:48) on C divide C intotwo arcs, and let α be one of the two arcs. The endpoints of γ (cid:48) and γ (cid:48)(cid:48) on C divide C into two arcs, and let α be one of the two arcs. See Figure 4.3. A triple( γ, γ (cid:48) , γ (cid:48)(cid:48) ) of the three nonself-chords is called even if the number of all endpointsof self-/nonself-chords on α ∪ α ∪ α is an even integer; otherwise it is called odd . Since C , C , and C are even, the parity of ( γ, γ (cid:48) , γ (cid:48)(cid:48) ) does not depend on thechoice of arcs α , α , and α . C C C γγ (cid:48) γ (cid:48)(cid:48) α α α Figure 4.3. A triple of arcs α , α , and α for nonself-chords γ, γ (cid:48) ,and γ (cid:48)(cid:48) Let T ( G ) be the set of even triples ( γ, γ (cid:48) , γ (cid:48)(cid:48) ) in G , and T ( G ) the set of oddtriples ( γ, γ (cid:48) , γ (cid:48)(cid:48) ) in G . For k = 0 , 1, denote by τ k ( G ) the sum of signs of all triplesin T k ( G ), where the sign of a triple ( γ, γ (cid:48) , γ (cid:48)(cid:48) ) ∈ T k ( G ) is the product of signs of γ, γ (cid:48) , and γ (cid:48)(cid:48) . Example 4.5. Consider the Gauss diagram G given in Example 4.1. By definition,we have T ( G ) = (cid:26) ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) (cid:27) and T ( G ) = (cid:26) ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) , ( γ , γ , γ ) (cid:27) . Therefore, it follows that τ ( G ) = 4 and τ ( G ) = − Lemma 4.6. The integers τ ( G ) and τ ( G ) are invariants of L . Moreover, thedifference τ ( G ) − τ ( G ) is invariant under CF-moves.Proof. Let G (cid:48) be a Gauss diagram obtained from G by one of Reidemeister movesand CF-moves.For an R1-move or R3-move, we have T k ( G ) = T k ( G (cid:48) ) ( k = 0 , τ k ( G ) = τ k ( G (cid:48) ).For an R2-move, which adds a pair of chords γ and γ (cid:48) with opposite signs, if γ and γ (cid:48) are self-chords, then T k ( G ) = T k ( G (cid:48) ), and therefore τ k ( G ) = τ k ( G (cid:48) ) ( k = 0 , γ and γ (cid:48) are nonself-chords, then T k ( G (cid:48) ) = T k ( G ) ∪ T k ( G (cid:48) ; γ, γ (cid:48) ) ( k = 0 , T ( G (cid:48) ; γ, γ (cid:48) ) denotes the set of even triples in G (cid:48) that contain γ or γ (cid:48) , and where T ( G (cid:48) ; γ, γ (cid:48) ) denotes the set of odd triples in G (cid:48) that contain γ or γ (cid:48) . Since the sumof signs of all triples in T k ( G (cid:48) ; γ, γ (cid:48) ) is zero, we have τ k ( G ) = τ k ( G (cid:48) ) ( k = 0 , γ and γ (cid:48) , if γ and γ (cid:48) are self-chords, then T k ( G ) = T k ( G (cid:48) ), andtherefore τ k ( G ) = τ k ( G (cid:48) ) ( k = 0 , γ and γ (cid:48) is a nonself-chord, then for a triple t ∈ T k ( G ) that contains γ or γ (cid:48) , it follows that t ∈ T (cid:101) k ( G (cid:48) )( (cid:101) , (cid:101) t is altered. For a triple t ∈ T k ( G ) that does notcontain γ and γ (cid:48) , it follows that t ∈ T k ( G (cid:48) ) and the sign of t is preserved ( k = 0 , τ ( G ) − τ ( G ) = τ ( G (cid:48) ) − τ ( G (cid:48) ). (cid:3) Lemma 4.6 guarantees the well-definedness of the following definition. Definition 4.7. The τ k -invariant of L is the integer τ k ( G ) for any Gauss diagram G of L ( k = 0 , τ k ( L ). Remark 4.8. Consider a Gauss diagram G with three circles C , C , and C as-sociated with a 3-component classical link diagram. It can be seen that for anythree nonself-chords γ, γ (cid:48) , and γ (cid:48)(cid:48) in G connecting C and C , C and C , and C and C , respectively, their triple ( γ, γ (cid:48) , γ (cid:48)(cid:48) ) is even. Therefore, if L is classical,then τ ( L ) = (Lk( K , K ) + Lk( K , K ))(Lk( K , K ) + Lk( K , K ))(Lk( K , K ) +Lk( K , K )) = 8 lk( K , K )lk( K , K )lk( K , K ) and τ ( L ) = 0.For convenience of explanation, let G ( a , a ; a , a ; a , a ) denote the Gaussdiagram ≤ i Let L = K ∪ K ∪ K be the -component virtual link represented by G ( a , a ; a , a ; a , a ) for some a ij ∈ Z . Then L is even if and only if a + a ≡ a + a ≡ a + a (mod 2) . F-MOVES FOR VIRTUAL LINKS 11 Proof. Since the number of endpoints on C is equal to | a | + | a | + | a | + | a | , C is even if and only if a + a ≡ a + a (mod 2). Similarly, C is even if andonly if a + a ≡ a + a (mod 2). Moreover, since the number of endpoints onall circles C , C , and C is an even integer, if C and C are even, then C is alsoeven. Therefore, we have the conclusion. (cid:3) Lemma 4.10. Let L = K ∪ K ∪ K be the -component virtual link repre-sented by G ( a , a ; a , a ; a , a ) for some a ij ∈ Z . If L is even, then wehave Lk( K i , K j ) − Lk( K j , K i ) = a ij − a ji and n ( K i , K j ) = | a ij + a ji | for any ≤ i < j ≤ , and τ ( L ) − τ ( L ) = ( a + a )( a + a )( a + a ) .Proof. Since the sum of signs of all nonself-chords of type ( i, j ) is equal to a ij andthat of type ( j, i ) is equal to a ji , we have Lk( K i , K j ) − Lk( K j , K i ) = a ij − a ji (1 ≤ i < j ≤ γ connecting C i and C j (1 ≤ i < j ≤ C i and C j are equivalent to γ . The sumof their signs is equal to a ij + a ji . Therefore, since σ ( C i , C j ; γ ) = a ij + a ji and σ ( C i , C j ; γ ) = 0, it follows that n ( K i , K j ) = | a ij + a ji | .For any three nonself-chords γ, γ (cid:48) , and γ (cid:48)(cid:48) in G = G ( a , a ; a , a ; a , a )connecting C and C , C and C , and C and C , respectively, their triple ( γ, γ (cid:48) , γ (cid:48)(cid:48) )is even by definition. Therefore, we have τ ( G ) = ( a + a )( a + a )( a + a ) and τ ( G ) = 0, which implies that τ ( L ) − τ ( L ) = ( a + a )( a + a )( a + a ). (cid:3) Lemma 4.11. We have the following CF-equivalent Gauss diagrams of -componentvirtual links. (i) G ( a , a ; a , a ; a , a ) ∼ G ( − a , − a ; − a , − a ; a , a ) . (ii) G ( a , a ; a , a ; a , a ) ∼ G ( − a , − a ; a , a ; − a , − a ) . (iii) G ( a , a ; a , a ; a , a ) ∼ G ( a , a ; − a , − a ; − a , − a ) .Proof. Since the proofs of (i), (ii), and (iii) are similar, we only show (i). Figure 4.4indicates the proof. More precisely, (2) is obtained from (1) by an R1-move addinga free chord γ in C . Applying Lemma 2.1, we slide the terminal endpoint of γ along C with respect to the orientation of C until it is next to the initial endpoint.Then (3) is obtained from (2). We delete γ by an R1-move, and use Lemma 2.1 toexchange the positions of the nonself-chords of type ( i, j ) and those of type ( j, i )for ( i, j ) = (1 , , (1 , (cid:3) Proof of Theorem 1.3. (i) ⇒ (ii): This follows from Lemmas 3.1, 4.2, and 4.6 di-rectly.(ii) ⇒ (i): Let G and G (cid:48) be Gauss diagrams of L and L (cid:48) , respectively. By Proposi-tion 2.3, we have G ∼ G ( a , a ; a , a ; a , a )for some a ij ∈ Z and G (cid:48) ∼ G ( a (cid:48) , a (cid:48) ; a (cid:48) , a (cid:48) ; a (cid:48) , a (cid:48) )for some a (cid:48) ij ∈ Z . Since Lk( K i , K j ) − Lk( K j , K i ) = Lk( K (cid:48) i , K (cid:48) j ) − Lk( K (cid:48) j , K (cid:48) i )and n ( K i , K j ) = n ( K (cid:48) i , K (cid:48) j ), by Lemma 4.10 we have a ij − a ji = a (cid:48) ij − a (cid:48) ji and | a ij + a ji | = | a (cid:48) ij + a (cid:48) ji | (1 ≤ i < j ≤ a = a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = a (cid:48) . (1) G ( a , a ; a , a ; a , a ) (2)R1 C C C | a || a || a || a | ε ε ε ε | a || a | ε ε C C C γ | a || a || a || a | ε ε ε ε | a || a | ε ε (4) G ( − a , − a ; − a , − a ; a , a ) (3)R1 Lem 2.1Lem 2.1 C C C | a || a || a || a | − ε − ε − ε − ε | a || a | ε ε C C C γ | a || a || a || a | − ε − ε − ε − ε | a || a | ε ε Figure 4.4. Proof of Lemma 4.11(i)(2) a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = a (cid:48) , a = a (cid:48) .(3) a = − a (cid:48) , a = − a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = − a (cid:48) , a = − a (cid:48) .(4) a = a (cid:48) , a = a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) .(5) a = − a (cid:48) , a = − a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = a (cid:48) .(6) a = a (cid:48) , a = a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = a (cid:48) , a = a (cid:48) .(7) a = a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = a (cid:48) , a = − a (cid:48) , a = − a (cid:48) .(8) a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) , a = − a (cid:48) .Since τ ( L ) − τ ( L ) = τ ( L (cid:48) ) − τ ( L (cid:48) ), Lemma 4.10 implies that( a + a )( a + a )( a + a ) = ( a (cid:48) + a (cid:48) )( a (cid:48) + a (cid:48) )( a (cid:48) + a (cid:48) ) . F-MOVES FOR VIRTUAL LINKS 13 This indicates that Cases (5)–(8) do not occur. Hence, it is enough to considerCases (1)–(4).In Case (1), we have G ∼ G ( a , a ; a , a ; a , a ) = G ( a (cid:48) , a (cid:48) ; a (cid:48) , a (cid:48) ; a (cid:48) , a (cid:48) ) ∼ G (cid:48) . In Case (2), it follows from Lemma 4.11(i) that G ∼ G ( a , a ; a , a ; a , a )= G ( − a (cid:48) , − a (cid:48) ; − a (cid:48) , − a (cid:48) ; a (cid:48) , a (cid:48) ) ∼ G ( a (cid:48) , a (cid:48) ; a (cid:48) , a (cid:48) ; a (cid:48) , a (cid:48) ) ∼ G (cid:48) . Cases (3) and (4) are shown similarly. (cid:3) We conclude this paper with a result that gives a relation among the classifyinginvariants Lk( K i , K j ) − Lk( K j , K i ), n ( K i , K j ), and τ ( L ) − τ ( L ) of Theorem 1.3. Proposition 4.12. Let L = K ∪ K ∪ K be a -component even virtual link, and let x ij , y ij , and z be integers (1 ≤ i < j ≤ . Assume that Lk( K i , K j ) − Lk( K j , K i ) = x ij , n ( K i , K j ) = | y ij | , and τ ( L ) − τ ( L ) = z . Then we have x ≡ x ≡ x ≡ y ≡ y ≡ y (mod 2) and | z | = | y y y | . Proof. By Proposition 2.3, L is CF-equivalent to a 3-component even virtual linkrepresented by G ( a , a ; a , a ; a , a ) for some a ij ∈ Z . By Lemmas 3.1, 4.2,4.6, and 4.10, we have Lk( K i , K j ) − Lk( K j , K i ) = a ij − a ji , n ( K i , K j ) = | a ij + a ji | (1 ≤ i < j ≤ τ ( L ) − τ ( L ) = ( a + a )( a + a )( a + a ). Theassumption implies that x ij = a ij − a ji , | y ij | = | a ij + a ji | (1 ≤ i < j ≤ , and z = ( a + a )( a + a )( a + a ) . Since L is even, we have a + a ≡ a + a ≡ a + a (mod 2) by Lemma 4.9.Therefore, it follows that x ≡ x ≡ x ≡ y ≡ y ≡ y (mod 2) . Moreover, since | y ij | = | a ij + a ji | (1 ≤ i < j ≤ 3) and z = ( a + a )( a + a )( a + a ), we have | z | = | y y y | . (cid:3) References [1] B. Audoux, P. Bellingeri, J.-B. Meilhan, and E. Wagner, Extensions of some classical localmoves on knot diagrams , Michigan Math. J. (2018), no. 3, 647–672.[2] M. Goussarov, M. Polyak, and O. Viro, Finite-type invariants of classical and virtual knots ,Topology (2000), no. 5, 1045–1068.[3] T. Kanenobu, Forbidden moves unknot a virtual knot , J. Knot Theory Ramifications (2001), no. 1, 89–96.[4] L. H. Kauffman, Virtual knot theory , European J. Combin. (1999), no. 7, 663–690.[5] J.-B. Meilhan, S. Satoh, and K. Wada, Classification of 2-component virtual links up to Ξ -moves , preprint (2020), arXiv:2002.08305.[6] H. A. Miyazawa, K. Wada, and A. Yasuhara, Linking invariants of even virtual links , J. KnotTheory Ramifications (2017), no. 12, 1750072, 12 pp.[7] T. Nasybullov, Classification of fused links , J. Knot Theory Ramifications (2016), no. 14,1650076, 21 pp. [8] S. Nelson, Unknotting virtual knots with Gauss diagram forbidden moves , J. Knot TheoryRamifications (2001), no. 6, 931–935.[9] T. Oikawa, On a local move for virtual knots and links , J. Knot Theory Ramifications (2009), no. 11, 1577–1596.[10] T. Okabayashi, Forbidden moves for virtual links , Kobe J. Math. (2005), no. 1–2, 49–63.[11] M. Polyak, Minimal generating sets of Reidemeister moves , Quantum Topol. (2010), no. 4,399–411. Department of Mathematics, Graduate School of Science, Osaka University, 1-1Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan Email address ::