The intersection polynomials of a virtual knot
TTHE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT
RYUJI HIGA, TAKUJI NAKAMURA, YASUTAKA NAKANISHI, AND SHIN SATOH
Abstract.
We introduce three kinds of invariants of a virtual knot calledthe first, second, and third intersection polynomials. The definition is basedon the intersection number of a pair of curves on a closed surface. We studyproperties of the intersection polynomials and their applications concerning thebehavior on symmetry, the crossing number and the virtual crossing number, aconnected sum of virtual knots, characterizations of intersection polynomials,finite type invariants of order two, and a flat virtual knot. Introduction
In classical knot theory, we treat a circle embedded in a 3-dimensional sphere S under an ambient isotopy. It is equivalent to study a circle embedded in the product S × I of a 2-sphere S and an interval I instead of S . In this sense, it is naturalto treat a circle in the product Σ g × I of a closed, connected, oriented surface Σ g ofgenus g and I . Kauffman [12] leads the unification of such knot theories in Σ g × I for all g ≥ g for some g ≥ g × I onto Σ g . A virtual knot is regarded as an equivalence class of diagramson closed surfaces under the relation generated by three kinds of Reidemeister movesfor diagrams and (de)stabilizations for surfaces. See Figure 1. stabilizationdestabilization Figure 1
Some invariants of a virtual knot are natural generalizations of those of a classicalknot such as knot groups and Jones polynomials [12], and some vanish for classical
This work was supported by JSPS KAKENHI Grant Numbers JP20K03621, JP19K03492, andJP19K03466.2010
Mathematics Subject Classification . Primary 57M25; Secondary 57M27.
Key words and phrases.
Virtual knot, writhe polynomial, intersection polynomial, connectedsum, flat virtual knot. a r X i v : . [ m a t h . G T ] F e b RYUJI HIGA, TAKUJI NAKAMURA, YASUTAKA NAKANISHI, AND SHIN SATOH knots such as Sawollek polynomials [26] and writhe polynomials. The invariantsintroduced in this paper are of the latter type.The writhe polynomial W K ( t ) of an oriented virtual knot K is a Laurent polyno-mial originally defined in [2, 5, 14, 25] independently, and is extensively studied (cf.[4, 16, 20, 22, 24, 29]). It can be described in terms of homological intersections onΣ g as follows. Let c , . . . , c n be the crossings of an oriented diagram D ⊂ Σ g of K ,and ε i the sign of c i (1 ≤ i ≤ n ). Splicing a crossing c i , we obtain two cycles γ i , γ i on Σ g such that γ i is the part of D from the over-crossing to the under-crossing at c i , and γ i is the one from the under-crossing to the over-crossing. Then the writhepolynomial can be expressed by W K ( t ) = n (cid:88) i =1 ε i ( t γ i · γ i − ∈ Z [ t, t − ] , where γ i · γ i is the intersection number of homological cycles γ i and γ i on Σ g . Thisinvariant does not depend on the genus g of the supporting surface Σ g .In Section 2, we modify the right hand side of this equation and consider fourkinds of Laurent polynomials defined by f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − , f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − ,f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − , f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − . They are invariant under second and third Reidemeister moves, but not invariantunder a first Reidemeister move generally. By studying the difference, we introducethree kinds of invariants of a virtual knot K as follows; I K ( t ) = f ( D ; t ) − ω D W K ( t ) ,II K ( t ) = f ( D ; t ) + f ( D ; t ) − ω D W K ( t ) , and III K ( t ) ≡ f ( D ; t ) (mod W K ( t )) , where ω D = (cid:80) ni =1 ε i is the writhe of D , W K ( t ) = W K ( t )+ W K ( t − ), and f ( t ) ≡ g ( t )(mod h ( t )) means f ( t ) − g ( t ) = mh ( t ) for some m ∈ Z . They are called the first , second , and third intersection polynomials of a virtual knot K , respectively. Weremark that I K ( t ) and II K ( t ) are Laurent polynomials in Z [ t, t − ], and III K ( t ) isan equivalence class of Laurent polynomials.In Section 3, we explain how to calculate the intersection polynomials througha Gauss diagram, which also presents a virtual knot as well as a diagram on aclosed surface. We give the intersection polynomials for all the virtual knots up tocrossing number four in Appendices A and B. By observing the obtained intersectionpolynomials, we see that I K ( t ), II K ( t ), III K ( t ), and W K ( t ) are independent of eachother (Proposition 3.6). The Alexander polynomials ∆ i ( K ) ( i ≥
0) of a virtual knot K are defined in [27] as a generalization of the classical ones. The 0-th Alexanderpolynomial ∆ ( K ) is also called the Sawollek polynomial [26]. It is known in [20]that the writhe polynomial is obtained from ∆ ( K ). On the other hand, we seethat the I K ( t ), II K ( t ), and III K ( t ) are not obtained from ∆ i ( K ) ( i ≥
0) generally(Remark 3.7).
HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 3
In Section 4, we study fundamental properties of the intersection polynomials.The writhe polynomial satisfies W K (1) = W (cid:48) K (1) = 0 [25]. We prove that the sameproperty holds for the first intersection polynomial; that is, I K (1) = I (cid:48) K (1) = 0(Theorem 4.2). On the other hand, the second intersection polynomial is reciprocalwith II K (1) = 0 and II (cid:48)(cid:48) K (1) ≡ III K (1) = 0 and III (cid:48)(cid:48) K (1) ≡ W (cid:48)(cid:48) K (1)(mod 4) (Theorem 4.6). The last congruence is proved by using the forbidden movewhich is an unknotting operation for a virtual knot [11, 23]. These fundamentalproperties characterize a Laurent polynomial to be the intersection polynomial ofsome virtual knot, which will be proved in Section 8.In Section 5, we first study the behaviors of intersection polynomials on symme-try of a virtual knot. For a virtual knot K , we denote by − K , K , and K ∗ thereverse, the vertical mirror image, and the horizontal mirror image of K , respec-tively. Here, the vertical (or horizontal) mirror image of K presented by a circle inΣ g × I is obtained by taking Σ g × ( − I ) (or ( − Σ g ) × I ). By using the intersectionpolynomials, we can construct an infinite family of virtual knots K such that K, − K, K , K ∗ , − K , − K ∗ , K ∗ , and − K ∗ are mutually distinct (Theorem 5.4). In this section, we also give lower bounds ofthe crossing number c( K ) and the virtual crossing number vc( K ) of a virtual knot K . It is known in [25] thatc( K ) ≥ deg W K ( t ) + 1 and vc( K ) ≥ deg W K ( t ) . Similarly to these inequations, we prove thatc( K ) ≥ deg I K ( t ) + 1 , deg II K ( t ) + 1 , deg III K ( t ) + 1 , andvc( K ) ≥ deg I K ( t ) , deg II K ( t ) , deg III K ( t )(Propositions 5.7 and 5.9). We also construct an infinite family of virtual knots K such that c( K ) and vc( K ) are determined by the intersection polynomials, but notby the writhe polynomial (Example 5.10).In Section 6, we study a dotted virtual knot which is also called a long virtualknot. We introduce a pair of invariants W T ( t ) and W T ( t ) of a dotted virtual knot T in a similar way to the definition of the writhe polynomial. The virtual knotobtained from T by ignoring the base point is called the closure of T and denotedby (cid:98) T . We prove that for any virtual knot K , there are infinitely many dotted virtualknots T with (cid:98) T = K by using the invariants W T ( t ) and W T ( t ) (Proposition 6.4).In Section 7, we give the connected sum formulae for the first and second inter-section polynomials (Theorems 7.1 and 7.5). A connected sum K (cid:48)(cid:48) of virtual knots K and K (cid:48) is defined to be the closure of the sum of some dotted virtual knots T and T (cid:48) with (cid:98) T = K and (cid:98) T (cid:48) = K (cid:48) . It is known in [2, 5, 25] that the writhe polynomialis additive with respect to a connected sum; that is, W K (cid:48)(cid:48) ( t ) = W K ( t ) + W K (cid:48) ( t ) . RYUJI HIGA, TAKUJI NAKAMURA, YASUTAKA NAKANISHI, AND SHIN SATOH
On the other hand, we prove that I K (cid:48)(cid:48) ( t ) = I K ( t ) + I K (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) and II K (cid:48)(cid:48) ( t ) = II K ( t ) + II K (cid:48) ( t )+ W T ( t ) W T (cid:48) ( t − ) + W T ( t ) W T (cid:48) ( t − )+ W T ( t − ) W T (cid:48) ( t ) + W T ( t − ) W T (cid:48) ( t ) . Therefore, the first and second intersection polynomials are not additive generally.As an application of the connected sum formulae, we prove that for any virtual knots K and K (cid:48) , there are infinitely many connected sums of K and K (cid:48) (Theorem 7.11).In Section 8, we give characterizations of the intersection polynomials. It isknown in [25] that a Laurent polynomial f ( t ) satisfies f ( t ) = W K ( t ) for somevirtual knot K if and only if f (1) = f (cid:48) (1) = 0. A characterization of the firstintersection polynomial is exactly the same as above (Theorem 8.1). On the otherhand, f ( t ) = II K ( t ) holds for some K if and only if f ( t ) is reciprocal, f (1) = 0, and f (cid:48)(cid:48) (1) ≡ f ( t )and g ( t ) satisfies f ( t ) = W K ( t ) and g ( t ) ≡ III K ( t ) (mod f ( t ) + f ( t − )) for some K if and only if g ( t ) is reciprocal, f (1) = f (cid:48) (1) = g (1) = 0, and f (cid:48)(cid:48) (1) ≡ g (cid:48)(cid:48) (1) (mod 4)(Theorem 8.9). We also give characterizations of the intersection polynomials of aconnected sum of two trivial knots (Propositions 8.11–8.13).In Section 9, we study a finite type invariant of a virtual knot. Dye [4] provesthat the writhe polynomial is a finite type invariant of order 1. On the other hand,we prove that the first and second intersection polynomials are finite type invariantsof order 2 (Theorem 9.2). The definition of a finite type invariant adopted in thispaper is a natural generalization of that in classical knot theory, which is differentfrom the one given by Goussarov, Polyak, and Viro [7].Section 10 proves that I K ( t ) + I K ( t − ) − II K ( t ) is invariant under a crossingchange (Theorem 10.3). The difference W K ( t ) − W K ( t − ) also has the same property[25]. Therefore these polynomials are regarded as invariants of a flat virtual knot,which is obtained from a virtual knot by ignoring the over/under-information ofcrossings. In particular, they are invariants of a homotopy class of a circle immersedin a closed surface. The supporting genus sg ( K ) of a virtual knot K is the minimalgenus of the surface where a diagram of K exists. We see that if W K ( t ) (cid:54) = W K ( t − )or I K ( t ) + I K ( t − ) (cid:54) = II K ( t ), then we have sg ( K ) ≥ Definitions
Let Σ g be a closed, connected, oriented surface of genus g , and α and β closed,oriented curves on Σ g . We often regard these curves as homology cycles of H (Σ g ).The intersection number α · β ∈ Z is defined to be the homology intersection of theordered pair ( α, β ). Geometrically it is calculated as follows. By perturbing α and β if necessary, we may assume that α ∩ β consists of m transverse double points p , . . . , p m . At a double point p k (1 ≤ k ≤ m ), if β intersects α from the left orright as we walk along α , we define e k = +1 or −
1, respectively. Then we have
HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 5 α · β = (cid:80) mk =1 e k . See Figure 2. We remark that α · β = − β · α and α · α = 0 bydefinition. α α α・β=1β・α=1αβ ββ + Figure 2
We consider a circle embedded in Σ g × [0 ,
1] for some g ≥
0. We identify two em-bedded circles up to ambient isotopies and (de)stabilizations. Such an equivalenceclass is called a virtual knot (cf. [3, 10, 12, 17]).More precisely, a virtual knot is described by a diagram on Σ g which is a pro-jection image under the projection Σ g × [0 , → Σ g equipped with over/under-information at double points. A double point with over/under information is calleda crossing . Two diagrams (Σ g , D ) and (Σ g (cid:48) , D (cid:48) ) present the same virtual knot ifand only if there is a finite sequence of diagrams(Σ g , D ) = (Σ g , D ) , (Σ g , D ) . . . , (Σ g s , D s ) = (Σ g (cid:48) , D (cid:48) )such that for each 1 ≤ i ≤ s − g i +1 = g i holds and (Σ g i +1 , D i +1 ) is obtained from (Σ g i , D i ) by an orientation-preserving homeomorphism of Σ g i = Σ g i +1 ,(i) g i +1 = g i holds and D i +1 is obtained from D i by a Reidemeister move onΣ g i = Σ g i +1 , or(ii) g i +1 = g i ± g i +1 is obtained from Σ g i by 1- or 2-handlesurgery missing D i = D i +1 . Such a deformation is called a stabilization or destabilization , respectively.Throughout this paper, we assume that all virtual knots are oriented.Let D = (Σ , D ) be a diagram of a virtual knot K , and c , . . . , c n the crossings of D . We denote by γ D the closed, oriented curve on Σ obtained from D by ignoringover/under-information at c i ’s. Furthermore, we denote by γ i (1 ≤ i ≤ n ) theclosed, oriented curve as a part of γ D from the overcrossing to the undercrossingat c i , and by γ i the curve from the undercrossing to the overcrossing at c i . SeeFigure 3. These curves satisfy γ i + γ i = γ D and γ i · γ i = γ i · ( γ D − γ i ) = γ i · γ D ashomology cycles on Σ. We call γ i and γ i the cycles at c i on Σ. γ i c i c i γ i γ i γ i + Figure 3
Definition 2.1 ([2, 5, 14, 25]) . The Laurent polynomial W D ( t ) = n (cid:88) i =1 ε i ( t γ i · γ i −
1) = n (cid:88) i =1 ε i ( t γ i · γ D − ∈ Z [ t, t − ] RYUJI HIGA, TAKUJI NAKAMURA, YASUTAKA NAKANISHI, AND SHIN SATOH is an invariant of K , where ε i is the sign of c i . It is the writhe polynomial of K anddenoted by W K ( t ). The exponent γ i · γ i is called the index of a crossing c i .Let D be the diagram obtained from D by changing over/under-informationat every crossing of D . Lemma 2.2 ([25]) . For any diagram D on Σ , we have W D ( t ) = − W D ( t − ) .Proof. Let c , . . . , c n be the crossings of D such that c i corresponds to c i , ε i the sign of c i , and γ i the cycle at c i on Σ (1 ≤ i ≤ n ). Then we have ε i = − ε i , γ i = γ i , and γ i = γ i . Therefore it holds that W D ( t ) = n (cid:88) i =1 ε i ( t γ i · γ i −
1) = − n (cid:88) i =1 ε i ( t γ i · γ i −
1) = − W D ( t − ) . (cid:3) Now we consider four kinds of Laurent polynomials as follows; f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − , f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − ,f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − , f ( D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − . Lemma 2.3.
For any diagram D on Σ , we have the following. (i) f ( D ; t ) = f ( D ; t ) . (ii) f ( D ; t ) = f ( D ; t ) .Proof. Let c , . . . , c n be the crossings of D such that c i corresponds to c i , ε i the sign of c i , and γ i the cycle at c i on Σ (1 ≤ i ≤ n ). Then we have ε i = − ε i , γ i = γ i , and γ i = γ i . Since it holds that ε i ε j = ε i ε j , γ i · γ j = γ i · γ j , and γ i · γ j = γ i · γ j , we have the conclusion by definition. (cid:3) In what follows, we often abbreviate f pq ( D ; t ) to f pq ( D ) for p, q ∈ { , } . Lemma 2.4.
If a diagram D (cid:48) is obtained from D by a second or third Reidemeistermove on Σ , then we have f pq ( D ) = f pq ( D (cid:48) ) for any p, q ∈ { , } .Proof. Since D (cid:48) is obtained from D by a second or third Reidemeister move onΣ, it is sufficient to prove the invariance of f ( D ) and f ( D ) by Lemma 2.3.A second Reidemeister move. Assume that D (cid:48) is obtained from D by a secondReidemeister move removing a pair of crossings c and c of D . For 3 ≤ i ≤ n , let c (cid:48) i be the crossing of D (cid:48) corresponding to c i , ε (cid:48) i the sign of c (cid:48) i , and γ (cid:48) i the cycle at c (cid:48) i on Σ. Then it holds that ε = − ε , γ = γ , and ε (cid:48) i = ε i , γ (cid:48) i = γ i (3 ≤ i ≤ n ) . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 7
See Figure 4. A disk on Σ is bounded by two arcs on D connecting c and c . Since γ D (cid:48) = γ D and γ (cid:48) i = γ i (3 ≤ i ≤ n ), we have f ( D ) − f ( D (cid:48) )= ε ( t γ · γ −
1) + ε ε ( t γ · γ −
1) + ε ε ( t γ · γ −
1) + ε ( t γ · γ − n (cid:88) j =3 ε ε j ( t γ · γ j −
1) + n (cid:88) j =3 ε ε j ( t γ · γ j − n (cid:88) i =3 ε i ε ( t γ i · γ −
1) + n (cid:88) i =3 ε i ε ( t γ i · γ − t γ · γ − − ( t γ · γ − − ( t γ · γ −
1) + ( t γ · γ −
1) = 0 . Therefore f ( D ) is invariant under a second Reidemeister move. On the otherhand, the invariance of f ( D ) is proved by f ( D ) − f ( D (cid:48) ) = ( t γ · γ − − ( t γ · γ − − ( t γ · γ −
1) + ( t γ · γ −
1) = 0 . γ c c γ γ c c γ Figure 4
A third Reidemeister move. Assume that D (cid:48) is obtained from D by a third Rei-demeister move involving three crossings c , c , and c of D . For 1 ≤ i ≤ n , let c (cid:48) i be the crossing corresponding to c i , ε (cid:48) i the sign of c (cid:48) i , and γ (cid:48) i the cycle at c (cid:48) i on Σ.Then it holds that ε (cid:48) i = ε i and γ (cid:48) i = γ i (1 ≤ i ≤ n ) . Figure 5 shows γ (cid:48) i = γ i for i = 1 , ,
3. A disk on Σ is bounded by three arcs of D connecting c and c , c and c , and c and c . Therefore both f ( D ) and f ( D )are invariant under a third Reidemeister move. We remark that γ + γ = γ holdsin this case. (cid:3) γ γ c c c c c c γ γ γ γ Figure 5
RYUJI HIGA, TAKUJI NAKAMURA, YASUTAKA NAKANISHI, AND SHIN SATOH
We use the notation W K ( t ) = W D ( t ) = W D ( t ) + W D ( t − ). It follows byLemma 2.2 that W D ( t ) = W D ( t ) + W D ( t − ) = − W D ( t − ) − W D ( t ) = − W D ( t ) . Lemma 2.5.
If a diagram D (cid:48) is obtained from D by a first Reidemeister move on Σ as shown in Figure 6(a)–(d) , then the difference f pq ( D ) − f pq ( D (cid:48) ) is given asshown in Table 1 . c (a)D D (b) (c) (d) c c c D D D D D D+ +
Figure 6 (a) ε = +1 (b) ε = − ε = − ε = +1 f ( D ) − f ( D (cid:48) ) W K ( t ) − W K ( t ) − W K ( t ) W K ( t ) f ( D ) − f ( D (cid:48) ) W K ( t − ) − W K ( t − ) − W K ( t − ) W K ( t − ) f ( D ) − f ( D (cid:48) ) 0 0 − W K ( t ) W K ( t ) f ( D ) − f ( D (cid:48) ) W K ( t ) − W K ( t ) 0 0 Table 1
Proof.
Assume that c is removed from D by a first Reidemeister move. For 2 ≤ i ≤ n , let c (cid:48) i be the crossing of D (cid:48) corresponding to c i , ε (cid:48) i the sign of c (cid:48) i , and γ (cid:48) i thecycle at c (cid:48) i on Σ. Then it holds that γ (cid:48) i = γ i and ε (cid:48) i = ε i (2 ≤ i ≤ n ). By definition,we have γ = 0 and γ = γ D for (a) and (b), and γ = γ D and γ = 0 for (c) and(d).( p, q ) = (0 , f ( D ) − f ( D (cid:48) ) = ε ( t γ · γ −
1) + n (cid:88) j =2 ε ε j ( t γ · γ j −
1) + n (cid:88) i =2 ε i ε ( t γ i · γ − . For (a) and (b), we have f ( D ) − f ( D (cid:48) ) = ε n (cid:88) i =2 ε i ( t γ i · γ D −
1) = ε n (cid:88) i =2 ε i ( t γ i · γ i −
1) = ε W K ( t ) . For (c) and (d), by using the equation γ D · γ j = ( γ j + γ j ) · γ j = γ j · γ j . we have f ( D ) − f ( D (cid:48) ) = ε n (cid:88) j =2 ε j ( t γ D · γ j −
1) = ε n (cid:88) j =2 ε j ( t γ j · γ j −
1) = ε W K ( t ) . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 9 ( p, q ) = (1 , D (cid:48) is obtained from D by (a), (b), (c), or (d), then D (cid:48) isobtained from D by (c), (d), (a), or (b), respectively. By Lemmas 2.2, 2.3, andthe equation for ( p, q ) = (0 , f ( D ) − f ( D (cid:48) ) = f ( D ) − f ( D (cid:48) ) = ( − ε ) · W D ( t ) = ε W D ( t − ) . ( p, q ) = (0 , f ( D ) − f ( D (cid:48) ) = ε ( t γ · γ −
1) + n (cid:88) j =2 ε ε j ( t γ · γ j −
1) + n (cid:88) i =2 ε i ε ( t γ i · γ − . For (a) and (b), we have f ( D ) − f ( D (cid:48) ) = 0 by γ = 0. For (c) and (d), by using γ = γ D , it holds that f ( D ) − f ( D (cid:48) ) = ε n (cid:88) j =2 ε j ( t γ D · γ j −
1) + ε n (cid:88) i =2 ε i ( t γ i · γ D − ε n (cid:88) j =2 ε j ( t − γ j · γ j −
1) + ε n (cid:88) i =2 ε i ( t γ i · γ i − ε W K ( t − ) + ε W K ( t ) = ε W K ( t ) . ( p, q ) = (1 , p, q ) = (1 , D (cid:48) is obtained from D by (c) and (d), respectively. By the equation for ( p, q ) = (0 , f ( D ) − f ( D (cid:48) ) = f ( D ) − f ( D (cid:48) ) = ( − ε ) · W D ( t ) = ε W D ( t ) . For (c) and (d), since D (cid:48) is obtained from D by (a) and (b), respectively, it holdsthat f ( D ) − f ( D (cid:48) ) = f ( D ) − f ( D (cid:48) ) = 0. (cid:3) The writhe of a diagram D is the sum of the signs of crossings of D , and denotedby ω D = (cid:80) ni =1 ε i . We consider two kinds of Laurent polynomials I D ( t ) = f ( D ; t ) − ω D W K ( t ) and II D ( t ) = f ( D ; t ) + f ( D ; t ) − ω D W K ( t ) . Theorem 2.6.
The Laurent polynomials I D ( t ) and II D ( t ) ∈ Z [ t, t − ] do not dependon a particular choice of a diagram D of a virtual knot K .Proof. Since the intersection numbers among γ i ’s and γ i ’s (1 ≤ i ≤ n ) do notchange by a (de)stabilization, it is sufficient to consider the case that a diagram D (cid:48) is obtained from D by a Reidemeister move on Σ.Assume that D (cid:48) is obtained from D by a second or third Reidemeister moveon Σ. Since ω D (cid:48) = ω D holds, we have I D (cid:48) ( t ) = I D ( t ) and II D (cid:48) ( t ) = II D ( t ) byLemma 2.4.Assume that D (cid:48) is obtained from D by a first Reidemeister move such that acrossing c with the sign ε is removed from D . Since it holds that ω D = ω D (cid:48) + ε and f ( D ) − f ( D (cid:48) ) = ε W K ( t ) by Lemma 2.5, we have I D ( t ) = f ( D ) − ω D W K ( t )= f ( D (cid:48) ) + ε W K ( t ) − ( ω D (cid:48) + ε ) W K ( t )= f ( D (cid:48) ) − ω D (cid:48) W K ( t ) = I D (cid:48) ( t ) . Similarly, since it holds that (cid:0) f ( D ) + f ( D ) (cid:1) − (cid:0) f ( D (cid:48) ) + f ( D (cid:48) ) (cid:1) = ε W K ( t )by Lemma 2.5, we have II D ( t ) = II D (cid:48) ( t ). (cid:3) Definition 2.7.
The Laurent polynomials I D ( t ) and II D ( t ) ∈ Z [ t, t − ] are calledthe first and second intersection polynomials of a virtual knot K , and denoted by I K ( t ) and II K ( t ), respectively. Remark 2.8.
We can also consider the polynomial f ( D ; t ) − ω D W K ( t − ) whichdefines an invariant of K by Lemmas 2.4 and 2.5. However, this is coincident withthe first intersection polynomial I K ( t − ). In fact, we have f ( D ; t ) − ω D W K ( t − ) = (cid:88) ≤ i,j ≤ n ε i ( t γ i · γ j − − ω D W K ( t − )= (cid:88) ≤ i,j ≤ n ε i ( t − γ j · γ i − − ω D W K ( t − ) = I K ( t − ) . For a Laurent polynomial h ( t ), we consider an equivalence relation in Z [ t, t − ]such that f ( t ) ≡ g ( t ) (mod h ( t )) holds if and only if f ( t ) − g ( t ) = mh ( t ) for some m ∈ Z . In the case of h ( t ) = 0, this equivalence relation gives f ( t ) = g ( t ) only. Fora diagram D of a virtual knot K on Σ, we consider the equivalence class f ( D )(mod W K ( t )). The invariance follows by Lemmas 2.4 and 2.5 immediately. Definition 2.9.
The equivalence class f ( D ) (mod W K ( t )) is called the thirdintersection polynomial of a virtual knot K , and denoted by III K ( t ). Remark 2.10.
We can also consider the equivalence class f ( D ) (mod W K ( t ))as an invariant of K . However, since II K ( t ) ≡ f ( D ) + f ( D ) (mod W K ( t )) holdsby definition, we have f ( D ) ≡ II K ( t ) − III K ( t ) (mod W K ( t )) . A virtual knot is classical if it is presented by a diagram on S . By definition, thewrithe polynomial W K ( t ) vanishes for any classical knot [2, 14]. The intersectionpolynomials satisfy the same property as follows. Lemma 2.11.
Any classical knot satisfies I K ( t ) = II K ( t ) = III K ( t ) = 0 . Proof.
All intersection numbers between two cycles on S are zero. (cid:3) Calculations
Let C be a closed, oriented curve on Σ with a finite number of crossings. Whenwe consider C as the image of an immersion S → Σ, the curve C is presented bya Gauss diagram G consisting of the circle S equipped with chords each of whichconnects the preimage of a crossing of C . The endpoints of chords admit signs withrespect to the orientation of C as shown in Figure 7.The endpoints of a chord of G divide the circle S into two arcs. Let α ⊂ S be such an arc, and P ( α ) the set of endpoints of the chords of G in the interior of α . For an endpoint x ∈ P ( α ), we denote by sgn( x ) the sign of x , and by τ ( x ) theother endpoint of the chord incident to x . The arc α ⊂ S presents a cycle on Σ,which is also denoted by α ⊂ Σ. See Figure 8.
HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 11 + Figure 7 αα + ++ +
Figure 8
Lemma 3.1.
Let α be the complementary arc of α ⊂ S . Then we have α · α = (cid:88) x ∈ P ( α ) sgn( x ) . Proof.
Any chord whose endpoints both lie on α does not contribute to the sum inthe right hand side of the equation. Therefore it holds that (cid:88) x ∈ P ( α ) sgn( x ) = (cid:88) x ∈ P ( α ) ,τ ( x ) ∈ P ( α ) sgn( x ) = α · α. (cid:3) Let α and β ⊂ S be arcs for distinct chords a and b of G , respectively. Weconsider an integer S ( α, β ) = (cid:88) x ∈ P ( α ) ,τ ( x ) ∈ P ( β ) sgn( x ) . It follows by definition that S ( α, β ) = − S ( β, α ). We say that the chords a and b of G are linked if their endpoints appear on S alternately, and otherwise unlinked .The number S ( α, β ) is equal to the sum of the signs of the endpoints indicated bydots as shown in Figure 9. α a b a b a abbβ βαβα β α Figure 9
Then the intersection number α · β of the cycles α and β ⊂ Σ is calculated asfollows.
Lemma 3.2. (i) If a and b are unlinked, then α · β = S ( α, β ) . (ii) Assume that a and b are linked as shown in Figure 10 , where ε, δ ∈ {±} .Then it holds that α · β = S ( α, β ) + 12 ( ε + δ ) and β · α = S ( β, α ) −
12 ( ε + δ ) . β εδ abαε δ Figure 10
Proof.
We prove two cases ( ε, δ ) = (+ , +) and (+ , − ) in (ii) as shown in Figure 11.Other cases are similarly proved.( ε, δ ) = (+ , +). We take a parallel copy of the curve α ⊂ Σ (or β ) which lieson the left (or right) side of the original curve. See the left of Figure 11. Thenthe intersections between α and β except one point near the crossing b correspondto the endpoints x ∈ P ( α ) with τ ( x ) ∈ P ( β ). Since the sign of the exceptionalintersection is equal to +, we obtain α · β = S ( α, β ) + 1.( ε, δ ) = (+ , − ). Similarly to the above case, we consider parallel copies of α and β . In this case, there is no exceptional intersection near b . See the right of Figure 11.Therefore we have α · β = S ( α, β ). (cid:3) αβ αβ+++ a b a bβ (ε,δ)=(+,+) (ε,δ)=(+,)ab α + β ab α Figure 11
Example 3.3.
We consider three arcs α , β , and γ of the Gauss diagram as shownin Figure 12. We have S ( α, β ) = − , S ( α, γ ) = 1 , and S ( β, γ ) = 0 . Since α and β are unlinked, it holds that α · β = S ( α, β ) = − α and γ , β and γ are linked, respectively, it holds that α · γ = S ( α, γ ) = 1 and β · γ = S ( β, γ ) + 1 = 1 by Lemma 3.2(ii). (cid:3) HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 13 βα +++++ + γ
Figure 12
Let D ⊂ Σ be a diagram of a virtual knot K with n crossings c , . . . , c n , and G the Gauss diagram of D . We also denote by c i the chord of G corresponding to acrossing c i of D . Each chord of G is oriented from the over-crossing to the under-crossing, and equipped with the same sign as that of the corresponding crossing of D . Then we see that if the sign of a chord is equal to ε , then the initial and terminalendpoints of the chord have the sign − ε and ε , respectively. See Figure 13. εε ε Figure 13
The endpoints of an oriented chord c i of G divide the circle S into two arcs.The arc from the initial endpoint to the terminal corresponds to the cycle γ i ⊂ Σat the crossing c i , and the other arc corresponds to γ i . Therefore we see that γ i · γ i = γ i · γ D = (cid:88) x ∈ P ( γ i ) sgn( x ) . If two chords c i and c j are unlinked, then it follows by Lemma 3.2(i) that γ i · γ j = S ( γ i , γ j ) , γ i · γ j = S ( γ i , γ j ) , and γ i · γ j = S ( γ i · γ j ) . We have similar equations in the case that c i and c j are linked by Lemma 3.2(ii).For example, we consider the case as shown in Figure 14. Then we have γ i · γ j = S ( γ i , γ j ) + ( ε i − ε j ) / ,γ i · γ j = S ( γ i , γ j ) − ( ε i + ε j ) / , and γ i · γ j = S ( γ i , γ j ) + ( ε i + ε j ) / . Let c( K ) denote the crossing number of K , which is the minimal number ofcrossings for all diagrams of K . The virtual knots up to crossing number four aregiven by Green [8]. In what follows, the labels of virtual knots are due to Green’stable. Example 3.4.
We consider the Gauss diagram G of a virtual knot K = 4 .
39 asshown in Figure 15. Table 2 shows the intersection numbers γ i · γ j , γ i · γ j , and εc i c j γ i γ ij ε i ε i ε j εc i c j j ε i ε i ε j γ j γ j εc i c j j ε i ε i ε j γ j γ i Figure 14 γ i · γ j for 1 ≤ i, j ≤
4. Since we have ε = ε = ε = − and ε = +, it holds that W K ( t ) = n (cid:88) i =1 ε i ( t γ i · γ i −
1) = − t + t + 1 − t − and W K ( t ) = − t + t − t + 2 − t − + t − − t − . On the other hand, we have f ( D ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j −
1) = 2 t − t − t − ,f ( D ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j −
1) = t − t − t − + t − , and f ( D ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j −
1) = t − t − t − + t − . Since ω D = − I K ( t ) = − t + 4 t − t,II K ( t ) = − t + 2 t − t + 4 − t − + 2 t − − t − , and III K ( t ) ≡ − t + 2 − t − (mod − t + t − t + 2 − t − + t − − t − ) . (cid:3) ++++ +K=4.39 c c c c Figure 15
Theorem 3.5.
For the virtual knots K with c( K ) ≤ , the intersection polynomials I K ( t ) , II K ( t ) , and III K ( t ) are given in Appendices A and B . (cid:3) By observing the calculations in Appendices A and B, we see that the writhepolynomial and the intersection polynomials are independent of each other in thefollowing sense.
HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 15 γ γ γ γ γ γ − γ − γ γ γ γ γ γ γ − − − γ − − γ − γ γ γ γ γ − − γ γ γ − − Table 2
Proposition 3.6.
There are four pairs of virtual knots K i and K (cid:48) i ( i = 1 , , , which satisfy the following. (i) W K ( t ) (cid:54) = W K (cid:48) ( t ) and X K ( t ) = X K (cid:48) ( t ) ( X = I, II, III ) . (ii) I K ( t ) (cid:54) = I K (cid:48) ( t ) and X K ( t ) = X K (cid:48) ( t ) ( X = W, II, III ) . (iii) II K ( t ) (cid:54) = II K (cid:48) ( t ) and X K ( t ) = X K (cid:48) ( t ) ( X = W, I, III ) . (iv) III K ( t ) (cid:54) = III K (cid:48) ( t ) and X K ( t ) = X K (cid:48) ( t ) ( X = W, I, II ) .Proof. (i) For the virtual knots K = 4 .
36 and K (cid:48) = 4 .
65, it holds that W K ( t ) = t − t − and W K (cid:48) ( t ) = − t + 2 − t − . On the other hand, we have I K ( t ) = I K (cid:48) ( t ) = − t + 2 − t − ,II K ( t ) = II K (cid:48) ( t ) = − t + 4 − t − , and III K ( t ) = III K (cid:48) ( t ) ≡ t − t − (mod 2 t − t − ) . (ii) For the trivial knot K = O and the virtual knot K (cid:48) = 4 .
16, it holds that I K ( t ) = 0 and I K (cid:48) ( t ) = − t + 3 t − t − . On the other hand, we have W K ( t ) = W K (cid:48) ( t ) = 0 ,II K ( t ) = II K (cid:48) ( t ) = 0 , and III K ( t ) = III K (cid:48) ( t ) = 0 . (iii) For the virtual knots K = 3 . K (cid:48) = 4 .
33, it holds that II K ( t ) = − t + 4 − t − and II K (cid:48) ( t ) = t − t + 10 − t − + t − . On the other hand, we have W K ( t ) = W K (cid:48) ( t ) = − t + 2 − t − ,I K ( t ) = I K (cid:48) ( t ) = − t + 2 − t − , and III K ( t ) = III K (cid:48) ( t ) ≡ t − t − (mod 2 t − t − ) . (iv) For the virtual knots K = O and K (cid:48) = 4 .
13, it holds that
III K ( t ) = 0 and III K (cid:48) ( t ) = 2 t − t − . On the other hand, we have W K ( t ) = W K (cid:48) ( t ) = 0 ,I K ( t ) = I K (cid:48) ( t ) = 0 , and II K ( t ) = II K (cid:48) ( t ) = 0 . The Gauss diagrams of the knots are illustrated in Figure 16. (cid:3) (cid:1144) (cid:23)(cid:17)(cid:20)(cid:22)(cid:15913)(cid:15913) (cid:15913) (cid:1144)(cid:1144)(cid:15913) (cid:15913)(cid:15913) (cid:1144)(cid:1144)(cid:1144)(cid:23)(cid:17)(cid:22)(cid:22)(cid:15913)(cid:15913) (cid:15913)(cid:1144)(cid:1144)(cid:15913) (cid:15913)(cid:1144)(cid:1144)(cid:1144) (cid:1144)(cid:23)(cid:17)(cid:25)(cid:24) (cid:15913)(cid:15913)(cid:1144)(cid:15913) (cid:15913)(cid:15913) (cid:1144)(cid:1144)(cid:1144) (cid:1144)(cid:1144) (cid:23)(cid:17)(cid:20)(cid:25)(cid:15913)(cid:15913) (cid:15913) (cid:1144)(cid:1144)(cid:15913) (cid:15913)(cid:1144)(cid:1144)(cid:1144) (cid:1144)(cid:1144)(cid:15913)(cid:15913) (cid:15913)(cid:15913) (cid:15913)(cid:1144)(cid:1144)(cid:15913) (cid:15913)(cid:1144)(cid:1144) (cid:1144) (cid:1144)(cid:1144)(cid:23)(cid:17)(cid:22)(cid:25) (cid:22)(cid:17)(cid:21)(cid:15913) (cid:1144)(cid:15913)(cid:15913) (cid:15913)(cid:1144)(cid:1144) (cid:1144) (cid:1144)
Figure 16
Remark 3.7.
For a virtual knot K , Silver and Williams [27] define a sequence ofAlexander polynomials ∆ i ( K ) as an extension of the classical Alexander polyno-mial. The zero-th Alexander polynomial ∆ ( K )( u, v ) is also defined by Sawollek[26]. Mellor [20] proves that the writhe polynomial is obtained from ∆ ( K )( u, v ) =(1 − uv ) (cid:101) ∆ ( K )( u, v ) by the equation W K ( t ) = − (cid:101) ∆ ( K )( t, t − ) . It is natural to ask whether the intersection polynomials are also obtained from∆ i ( K ). However this does not hold generally; in fact, for the virtual knot K = 4 . ( K ) = 0 and ∆ i ( K ) = 1 ( i ≥
1) which are coincident with those of thetrivial knot. On the other hand, it holds that the intersection polynomials I K ( t ), II K ( t ), and III K ( t ) are all non-trivial.4. fundamental properties The writhe polynomial is characterized by the following property.
Theorem 4.1 ([25]) . Any virtual knot K satisfies W K (1) = W (cid:48) K (1) = 0 . Con-versely, if a Laurent polynomial f ( t ) ∈ Z [ t, t − ] satisfies f (1) = f (cid:48) (1) = 0 , thenthere is a virtual knot K with f ( t ) = W K ( t ) . (cid:3) We remark that the equation W (cid:48) K (1) = 0 is equivalent to n (cid:88) i =1 ε i ( γ i · γ i ) = n (cid:88) i =1 ε i ( γ i · γ D ) = 0by definition.The first intersection polynomial I K ( t ) satisfies the same property as above,which characterizes a Laurent polynomial to be some first intersection polynomial.The characterization will be proved in Section 8. Theorem 4.2.
Any virtual knot K satisfies I K (1) = I (cid:48) K (1) = 0 . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 17
Proof.
We have I K (1) = 0 by definition. Since W (cid:48) K (1) = 0 holds by Theorem 4.1,it holds that I (cid:48) K (1) = f (cid:48) ( D ; 1) − ω D W (cid:48) K (1) = f (cid:48) ( D ; 1)= (cid:88) ≤ i,j ≤ n ε i ε j ( γ i · γ j ) = (cid:32) n (cid:88) i =1 ε i γ i (cid:33) · (cid:32) n (cid:88) j =1 ε j γ j (cid:33) = (cid:32) n (cid:88) i =1 ε i γ i (cid:33) · (cid:32) n (cid:88) j =1 ε j ( γ D − γ j ) (cid:33) = (cid:32) n (cid:88) i =1 ε i γ i (cid:33) · (cid:32) ω D γ D − n (cid:88) j =1 ε j γ j (cid:33) = ω D n (cid:88) i =1 ε i ( γ i · γ D ) − (cid:32) n (cid:88) i =1 ε i γ i (cid:33) · (cid:32) n (cid:88) j =1 ε j γ j (cid:33) = ω D W (cid:48) K (1) = 0 . (cid:3) A Laurent polynomial f ( t ) is reciprocal if it satisfies f ( t − ) = f ( t ). The secondintersection polynomial II K ( t ) satisfies different properties from I K ( t ) as follows.These properties characterize a Laurent polynomial to be some second intersectionpolynomial as shown in Section 8. Theorem 4.3.
For any virtual knot K , II K ( t ) is reciprocal with II K (1) = 0 and II (cid:48)(cid:48) K (1) ≡ . To prove this theorem, we prepare the following two lemmas.
Lemma 4.4.
Let f ( t ) = (cid:80) k ∈ Z a k t k be a Laurent polynomial in Z [ t, t − ] . (i) If f (cid:48) (1) = 0 , then f (cid:48)(cid:48) (1) ≡ (cid:80) k :odd a k (mod 4) . (ii) If f ( t ) is reciprocal, then f (cid:48)(cid:48) (1) ≡ (cid:80) k :odd a k (mod 4) .Proof. (i) It holds that f (cid:48) ( t ) = (cid:88) k ∈ Z ka k t k − and f (cid:48)(cid:48) ( t ) = (cid:88) k ∈ Z k ( k − a k t k − . Then we have f (cid:48)(cid:48) (1) = (cid:88) k ∈ Z k ( k − a k = (cid:88) k ∈ Z k a k − f (cid:48) (1) = (cid:88) k ∈ Z k a k ≡ (cid:88) k :odd a k (mod 4) . (ii) We may take f ( t ) = (cid:80) k ≥ a k ( t k + t − k ) + a . Since f (cid:48) ( t ) = (cid:80) k ≥ ka k ( t k − − t − k − ) holds, we have f (cid:48) (1) = 0. By (i), we have the conclusion. (cid:3) Lemma 4.5.
For a Laurent polynomial f ( t ) , the following are equivalent. (i) f ( t ) is reciprocal, f (1) = 0 , and f (cid:48)(cid:48) (1) ≡ . (ii) f ( t ) = (cid:80) k ≥ a k ( t k + t − k − for some a k ∈ Z ( k ≥ with (cid:80) k :odd ≥ a k ≡ . (iii) There is a Laurent polynomial g ( t ) ∈ Z [ t, t − ] such that g (1) = g (cid:48) (1) = 0 and f ( t ) = g ( t ) + g ( t − ) . Proof. (i) ⇒ (ii). Since f ( t ) is reciprocal, we may take f ( t ) = (cid:80) k ≥ a k ( t k + t − k ) + a . Since f (1) = 0, we have a = − (cid:80) k ≥ a k to obtain f ( t ) = (cid:80) k ≥ a k ( t k + t − k − f ( t ) is equal to2 (cid:80) k :odd ≥ a k , it follows by Lemma 4.4 that2 (cid:88) k :odd ≥ a k ≡ f (cid:48)(cid:48) (1) ≡ . (ii) ⇒ (iii). We have f ( t ) = (cid:88) k ≥ a k ( t k + t − k − (cid:88) k ≥ a k ( t k − kt + k −
1) + (cid:88) k ≥ a k ( t − k − kt − + k − (cid:88) k ≥ ka k ( t + t − − . By assumption, we may put (cid:80) k ≥ ka k = 2 m for some m ∈ Z . Consider the Laurentpolynomial g ( t ) = (cid:88) k ≥ a k ( t k − kt + k −
1) + m ( t + t − − . Then it satisfies that g (1) = g (cid:48) (1) = 0 and f ( t ) = g ( t ) + g ( t − ).(iii) ⇒ (i). Since g (1) = g (cid:48) (1) = 0, we can take g ( t ) = ( t − h ( t ) for some h ( t ) ∈ Z [ t, t − ]. Then the reciprocal polynomial f ( t ) = g ( t ) + g ( t − ) satisfies f (1) = 2 g (1) = 0 and f (cid:48)(cid:48) (1) = 2 g (cid:48)(cid:48) (1) = 4 h (1) ≡ . (cid:3) Proof of
Theorem 4.3 . Since f ( D ; t ), f ( D ; t ), and W K ( t ) are reciprocal, so is II K ( t ). We have II K (1) = 0 by definition.We will prove II (cid:48)(cid:48) K (1) ≡ W K ( t ) = W K ( t ) + W K ( t − ) with W K (1) = W (cid:48) K (1) = 0, we have W (cid:48)(cid:48) K (1) ≡ S be the sumof the coefficients of odd terms of f ( D ; t ) + f ( D ; t ). Since f ( D ; t ) + f ( D ; t )is reciprocal, it is sufficient to prove that S ≡ S = (cid:88) γ i · γ j :odd ε i ε j + (cid:88) γ i · γ j :odd ε i ε j = 2 (cid:32) (cid:88) γ i · γ j :odd ,i On the other hand, we have (cid:88) ≤ i For any virtual knot K , III K ( t ) is reciprocal with III K (1) = 0 and III (cid:48)(cid:48) K (1) ≡ W (cid:48)(cid:48) K (1) (mod 4) . To prove this theorem, we parepare the following two lemmas. An upper (or lower ) forbidden move changes the position of consecutive over-crossings (or under-crossings), respectively, which is known as an unknotting operation [11, 23]. Assumethat a Gauss diagram G (cid:48) is obtained from G by an upper forbidden move involvinga pair of chords c and c of G as shown in Figure 17. G Gʼc c cʼ cʼ Figure 17 For 1 ≤ i ≤ n , let c (cid:48) i be the chord of G (cid:48) corresponding to c i , ε (cid:48) i the sign of c (cid:48) i , and γ (cid:48) i the cycle at c (cid:48) i . Let x and x be the terminal endpoints of c and c , respectively. We classify the chords c , . . . , c n of G into four sets such that P = { c i | neither x nor x lies on γ i } ,Q = { c i | both x and x lie on γ i } ,R = { c i | x does not lie on γ i and x lies on γ i } ,S = { c i | x lies on γ i and x does not lie on γ i } . Lemma 4.7. (i) (cid:80) ≤ i HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 21 Lemma 4.8. (cid:80) γ (cid:48) i · γ (cid:48) i :odd ε (cid:48) i − (cid:80) γ i · γ i :odd ε i = ( − γ · γ ε + ( − γ · γ ε .Proof. We see that γ i · γ i and γ (cid:48) i · γ (cid:48) i have opposite parity for i = 1 , ≤ i ≤ n . Since ε i = ε (cid:48) i holds, we have (cid:88) γ (cid:48) i · γ (cid:48) i :odd ε (cid:48) i − (cid:88) γ i · γ i :odd ε i = ε + ε for γ · γ ≡ γ · γ ≡ ,ε − ε for γ · γ ≡ , γ · γ ≡ , − ε + ε for γ · γ ≡ , γ · γ ≡ , and − ε − ε for γ · γ ≡ γ · γ ≡ . (cid:3) Proof of Theorem 4.6. Since f ( D ; t ) and W K ( t ) are reciprocal, so is III K ( t ). Wehave III K (1) = 0 by f ( D ; 1) = W K (1) = 0.Assume that a Gauss diagram G (cid:48) is obtained from G by an upper forbiddenmove. Then it follows by Lemmas 4.7(iii) and 4.8 that (cid:88) γ (cid:48) i · γ (cid:48) j :odd ε (cid:48) i ε (cid:48) j − (cid:88) γ (cid:48) i · γ (cid:48) i :odd ε (cid:48) i − (cid:88) γ i · γ j :odd ε i ε j − (cid:88) γ i · γ i :odd ε i ≡ γ · γ + 2 γ · γ + ε + ε − ( − γ · γ ε − ( − γ · γ ε ≡ . In other words, (cid:80) γ i · γ j :odd ε i ε j − (cid:80) γ i · γ i :odd ε i (mod 4) is invariant under an upperforbidden move. The invariance under a lower forbidden move can be proved sim-ilarly. Since the forbidden move is an unknotting operation for a virtual knot, weobtain (cid:88) γ i · γ j :odd ε i ε j ≡ (cid:88) γ i · γ i :odd ε i (mod 4) . Since III K ( t ) is reciprocal and W (cid:48) K (1) = 0, this congruence is equivalent to III (cid:48)(cid:48) K (1) ≡ f (cid:48)(cid:48) ( D ; 1) ≡ W (cid:48)(cid:48) K (1) (mod 4)by Lemma 4.4. (cid:3) Remark 4.9. The odd writhe [13] of a virtual knot K is the sum of the coefficientsof odd terms of W K ( t ), and denoted by J ( K ) ∈ Z . We have W (cid:48)(cid:48) K (1) ≡ J ( K )(mod 4) by Lemma 4.4. It is known that J ( K ) is always even [1], which is alsoobtained from W (cid:48) K (1) = 0 immediately.5. Symmetries and crossing numbers For a diagram D on Σ of a virtual knot K , let − D be the diagram by reversing theorientation of D , D the one by changing over/under-information at every crossingof D , and D ∗ the one obtained by an orientation-reversing homeomorphism of Σ.The virtual knots presented by − D , D , and D ∗ are called the reverse , the verticalmirror image , and the horizontal mirror image of K , and denoted by − K , K ,and K ∗ , respectively. The writhe polynomials of these knots are given as follows. Lemma 5.1 ([2, 14, 25]) . Any virtual knot K satisfies W − K ( t ) = W K ( t − ) and W K ( t ) = W K ∗ ( t ) = − W K ( t − ) . Therefore we have W − K ( t ) = W K ( t ) and W K ( t ) = W K ∗ ( t ) = − W K ( t ) . The intersection polynomials of − K , K , and K ∗ are given as follows. Lemma 5.2. For a virtual knot K , we have the following. (i) I − K ( t ) = I K ( t ) = I K ∗ ( t ) = I K ( t − ) . (ii) II − K ( t ) = II K ( t ) = II K ∗ ( t ) = II K ( t ) . (iii) III − K ( t ) = III K ( t ) = II K ( t ) − III K ( t ) and III K ∗ ( t ) = III K ( t ) .Proof. I − K ( t ), II − K ( t ), and III − K ( t ). Let c (cid:48) i be the crossing of − D correspondingto c i , γ (cid:48) i the cycle at c (cid:48) i on Σ, and ε (cid:48) i the sign of c (cid:48) i (1 ≤ i ≤ n ). Then it holds that γ (cid:48) i = − γ i and ε (cid:48) i = ε i . By definition, we have f ( − D ; t ) = (cid:88) ≤ i,j ≤ n ε (cid:48) i ε (cid:48) j ( t γ (cid:48) i · γ (cid:48) j − 1) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − (cid:88) ≤ i,j ≤ n ε i ε j ( t − γ j · γ i − 1) = f ( D ; t − ) ,f ( − D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − 1) = f ( D ; t ) , and f ( − D ; t ) = (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − 1) = f ( D ; t ) . Since ω − D = ω D holds, we have I − K ( t ) = f ( − D ; t ) − ω − D W − D ( t )= f ( D ; t − ) − ω D W D ( t − ) = I K ( t − ) ,II − K ( t ) = f ( − D ; t ) + f ( − D ; t ) − ω − D (cid:0) W − D ( t ) + W − D ( t − ) (cid:1) = f ( D ; t ) + f ( D ; t ) − ω D (cid:0) W D ( t − ) + W D ( t ) (cid:1) = II K ( t ) and III − K ( t ) ≡ f ( − D ; t ) = f ( D ; t ) ≡ II K ( t ) − f ( D ; t ) ≡ II K ( t ) − III K ( t ) .I K ( t ), II K ( t ), and III K ( t ). We use the notations in the proof of Lemma 2.2.By the lemma, we have f ( D ; t ) = f ( D ; t ) and hence f ( D ; t ) = f ( D ; t ).Furthermore it holds that f ( D ; t ) = (cid:88) ≤ i,j ≤ n ( − ε i )( − ε j )( t γ i · γ j − 1) = f ( D ; t − ) . Since it holds that ω D = − ω D and W D ( t ) = − W D ( t − ), we have I K ( t ) = f ( D ; t ) − ω D W D ( t )= f ( D ; t − ) − ω D W D ( t − ) = I K ( t − ) II K ( t ) = f ( D ; t ) + f ( D ; t ) − ω D (cid:0) W D ( t ) + W D ( t − ) (cid:1) = f ( D ; t ) + f ( D ; t ) − ω D (cid:0) W D ( t − ) + W D ( t ) (cid:1) = II K ( t ) , and III K ( t ) ≡ f ( D ; t ) = f ( D ; t ) ≡ II K ( t ) − f ( D ; t ) ≡ II K ( t ) − III K ( t ) .I K ∗ ( t ), II K ∗ ( t ), and III K ∗ ( t ). Let c ∗ i be the crossing of D ∗ corresponding to c i , γ ∗ i the cycle at c ∗ i on Σ, and ε ∗ i the sign of c ∗ i (1 ≤ i ≤ n ). Then it holds that γ ∗ i · γ ∗ j = − γ i · γ j , γ ∗ i · γ ∗ j = − γ i · γ j , γ ∗ i · γ ∗ j = − γ i · γ j and ε ∗ i = − ε i . Since f ( D ; t ) and f ( D ; t ) are reciprocal, we have f ( D ∗ ; t ) = f ( D ; t − ) , f ( D ∗ ; t ) = f ( D ; t ) , and f ( D ∗ ; t ) = f ( D ; t ) . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 23 Since it holds that ω D ∗ = − ω D and W D ∗ ( t ) = − W D ( t − ), we have I K ∗ ( t ) = f ( D ∗ ; t ) − ω D ∗ W D ∗ ( t )= f ( D ; t − ) − ω D W D ( t − ) = I K ( t − ) ,II K ∗ ( t ) = f ( D ∗ ; t ) + f ( D ∗ ; t ) − ω D ∗ (cid:0) W D ∗ ( t ) + W D ∗ ( t − ) (cid:1) = f ( D ; t ) + f ( D ; t ) − ω D (cid:0) W D ( t − ) + W D ( t ) (cid:1) = II K ( t ) , and III K ∗ ( t ) ≡ f ( D ∗ ; t ) = f ( D ; t ) ≡ III K ( t ) . (cid:3) Proposition 5.3. If a virtual knot K satisfies (i) 2 III K ( t ) (cid:54)≡ II K ( t ) (mod W K ( t )) , (ii) W K ( t ) (cid:54) = W K ( t − ) , and (iii) W K ( t ) (cid:54) = − W K ( t − ) or I K ( t ) (cid:54) = I K ( t − ) ,then the eight virtual knots K, − K, K , K ∗ , − K , − K ∗ , K ∗ , and − K ∗ are mutually distinct.Proof. By Lemma 5.2(iii), the virtual knots K, K ∗ , − K , and − K ∗ have the samethird intersection polynomial III K ( t ), and − K, − K ∗ , K , and K ∗ have II K ( t ) − III K ( t ). Therefore, by the condition (i), it holds that { K, K ∗ , − K , − K ∗ } ∩ {− K, − K ∗ , K , K ∗ } = ∅ . Furthermore, by Lemma 5.1, the first four virtual knots K, K ∗ , − K , and − K ∗ have the writhe polynomials W K ( t ), − W K ( t − ), − W K ( t ), and W K ( t − ), respec-tively. Since W K ( t ) (cid:54) = 0 follows by the condition (ii), we have { K, K ∗ } ∩ {− K , − K ∗ } = ∅ . Finally, each of the pairs K and K ∗ , and − K and − K ∗ can be distinguishedby the condition (iii) and Lemma 5.2(i). We can prove that the latter four virtualknots − K, − K ∗ , K , and K ∗ are mutually distinct similarly. (cid:3) Theorem 5.4. There are infinitely many virtual knots K such that K, − K, K , K ∗ , − K , − K ∗ , K ∗ , and − K ∗ are mutually distinct. Proof. Let K n ( n ≥ 1) be the virtual knot presented by the Gauss diagram G n asshown in Figure 19. It holds that W K n ( t ) = t n +1 − ( n + 1) t + ( n + 1) t − − t − n − ,I K n ( t ) = − t n +2 − nt n +1 + t − (2 n − − ( n + 1) t − n − + 2 n (cid:88) i =1 ( t i + t − i ) ,II K n ( t ) = − ( t n +2 + t − n − ) − ( t n +1 + t − n − ) − (2 n + 1)( t n + t − n ) − n ( n + 1)( t + t − ) − (2 n + 3)( t + t − ) + 2( n + n + 2)+4 n +1 (cid:88) i =1 ( t i + t − i ) , and III K n ( t ) = − ( t n +1 + t − n − ) − ( n + 1)( t + t − ) + 2 n +1 (cid:88) i =1 ( t i + t − i ) − n, where we have W K n ( t ) = 0. Since these invariants of K n satisfy the conditions (i),(ii), and (iii) I K n ( t ) (cid:54) = I K n ( t − ) in Proposition 5.3, the eight kinds of virtual knotsassociated with K n are mutually distinct. Furthermore K n (cid:54) = K m ( n (cid:54) = m ) holdsby deg W K n ( t ) = n + 1. We remark that K n satisfies W K n ( t ) = − W K n ( t − ). (cid:3) + ++++++ ++ +nn+1 ………… ………… Figure 19 Example 5.5. We can construct an infinite family of virtual knots K satisfyingthe conditions (i), (ii), and (iii) W K ( t ) (cid:54) = − W K ( t − ) in Proposition 5.3.Let K (cid:48) n ( n ≥ 3) be the virtual knot presented by the Gauss diagram G (cid:48) n as shownin Figure 20. We have W K (cid:48) n ( t ) = − t n − + nt − n + t − ,I K (cid:48) n ( t ) = − t n + ( n − t n − − n − (cid:88) i =1 t i + (2 n − − nt − ,II K (cid:48) n ( t ) = − ( t n + t − n ) + ( n − t n − + t − n +1 ) − n − (cid:88) i =1 ( t i + t − i ) − ( n − n + 1)( t + t − ) + 2 n − , and III K (cid:48) n ( t ) ≡ − n − (cid:88) i =1 ( t i + t − i ) − ( t + t − ) + 2 n − W K (cid:48) n ( t )) , HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 25 where W K (cid:48) n ( t ) = − ( t n − + t − n +1 )+( n +1)( t + t − ) − n . Since these invariants of K (cid:48) n satisfy the conditions (i), (ii), and (iii) W K (cid:48) n ( t ) (cid:54) = − W K (cid:48) n ( t − ) in Proposition 5.3, theeight kinds of virtual knots associated with K (cid:48) n are mutually distinct. Furthermore K (cid:48) n (cid:54) = K (cid:48) m ( n (cid:54) = m ) holds by deg W K (cid:48) n ( t ) = n − 1. We remark that K (cid:48) n satisfies (iii) I K (cid:48) n ( t ) (cid:54) = I K (cid:48) n ( t − ) in Proposition 5.3. (cid:3) ++++ ++− − +− + − − −n ……+ Figure 20 For a Laurent polynomial f ( t ), let deg f ( t ) denote the maximal degree of f ( t ).The writhe polynomial W K ( t ) gives a lower bound of the crossing number c( K ) asfollows. Lemma 5.6 ([25]) . Any non-trivial virtual knot K satisfies c( K ) ≥ deg W K ( t ) + 1 . We remark that the minimal degree of W K ( t ) also gives a lower bound of c( K ) =c( − K ) by the equation to W K ( t − ) = W − K ( t ). The span of W K ( t ) is the differenceof the maximal and minimal degrees of W K ( t ), and denoted by span W K ( t ). ThenLemma 5.6 induces a weaker inequationc( K ) ≥ 12 span W K ( t ) + 1immediately. The intersection polynomials also gives lower bounds of c( K ) asfollows. Here, deg III K ( t ) denotes the maximal number of deg f ( t ) for all f ( t ) with f ( t ) ≡ III K ( t ) (mod W K ( t )). Proposition 5.7. Let K be a non-trivial virtual knot. (i) c( K ) ≥ deg I K ( t ) + 1 . (ii) c( K ) ≥ deg II K ( t ) + 1 . (iii) c( K ) ≥ deg III K ( t ) + 1 .Proof. Assume that a diagram D of K satisfies c ( D ) = c( K ). Since K is non-trivial, it holds that c ( D ) ≥ 2. For ( α, β ) = ( γ i , γ j ), ( γ i , γ j ), and ( γ i , γ j ) with i (cid:54) = j , the intersection number α · β is equal to S ( α, β ) − S ( α, β ), or S ( α, β ) + 1by Lemma 3.2 so that we obtain α · β ≤ S ( α, β ) + 1.Since there are c ( D ) − c i and c j in the Gauss diagram of D ,we have S ( α, β ) ≤ c ( D ) − 2. Therefore it holds thatdeg f pq ( D ) ≤ c ( D ) − K ) − p, q ) = (0 , , (0 , , W K ( t ) ≤ c ( K ) − 1, we have theconclusion. (cid:3) As well as a diagram on Σ or a Gauss diagram, a virtual knot is also presentedby a virtual diagram in R . It is an immersed circle in R with real and virtualcrossings [12]. Here, the real crossings correspond to the crossings on Σ, and thevirtual crossings are surrounded by small circles. The virtual crossing number of avirtual knot K is the minimal number of virtual crossings for all virtual diagramsof K , and denoted by vc( K ).The intersection polynomials are also calculated from a virtual diagram. Forexample, we consider the virtual diagram with three real crossings c , c , and c and two virtual crossings as shown in the leftmost of Figure 21, which presents thevirtual knot K = 3 . 4. To calculate γ · γ , we draw the curves γ and γ equippedwith virtual crossings, and then take the sum of signs of two intersections withignoring virtual crossings to obtain γ · γ = 2. See the second from the left in thefigure. Similarly we obtain γ · γ = 0 and γ · γ = − c c c γ γ γ γ = 2・ γ γ γ γ γ = 1・ γ γ γ = 0・ Figure 21 The writhe polynomial W K ( t ) gives a lower bound of the virtual crossing numbervc( K ) as follows. Lemma 5.8 ([25]) . Any non-trivial virtual knot K satisfies vc( K ) ≥ deg W K ( t ) . We remark that Lemma 5.8 induces a weaker inequationvc( K ) ≥ 12 span W K ( t )immediately, which is proved in [20]. The intersection polynomials also gives lowerbounds of vc( K ) as follows. Proposition 5.9. Let K be a virtual knot. (i) vc( K ) ≥ deg I K ( t ) . (ii) vc( K ) ≥ deg II K ( t ) . (iii) vc( K ) ≥ deg III K ( t ) .Proof. Let D be a virtual diagram of K in R , and α and β cycles on D withcorners at (possibly the same) real crossings of D . By a slight perturbation of β if necessary, we may assume that α and β intersect in a finite number of doublepoints near real and virtual crossings of D as explained as above. By Lemma 5.8, itis sufficient to prove that if the intersection number restricted to the real crossingsbetween α and β in R is equal to n , then the number of virtual crossings of D isgreater than or equal to | n | . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 27 Since the total intersection number between α and β in R is equal to zero, theintersection number restricted to the virtual crossings between α and β is equal to − n .Let v be a virtual crossing of D where two short paths λ and λ (cid:48) ⊂ D intersect.If α and β intersect in virtual crossings near v , then there are four cases as follows.(i) α ⊃ λ , α (cid:54)⊃ λ (cid:48) , and β (cid:54)⊃ λ , β ⊃ λ (cid:48) .(ii) α ⊃ λ, λ (cid:48) , and β (cid:54)⊃ λ , β ⊃ λ (cid:48) .(iii) α ⊃ λ , α (cid:54)⊃ λ (cid:48) , and β ⊃ λ, λ (cid:48) .(iv) α ⊃ λ, λ (cid:48) and β ⊃ λ, λ (cid:48) .See Figure 22. vλ (ii) (iii) (iv)(i)λ α α α α α αβ ββ β ββ Figure 22 In the case (iv), the pair of virtual crossings between α and β does not contributeto the intersection number; in fact, they have opposite signs. On the other hand,each case of (i)–(iii) contains a single virtual crossing. It follows that the numberof virtual crossings of D in the cases (i)–(iii) is greater than or equal to | n | . (cid:3) Example 5.10. (i) Let K n ( n ≥ 1) be the virtual knot presented by the Gaussdiagram and the virtual diagram as shown in Figure 23. We see that c( K n ) = n + 3and vc( K n ) = n + 2 can be detected by I K n ( t ) but not by W K n ( t ), II K n ( t ), and III K n ( t ). In fact, we have W K n ( t ) = t n +1 − t − nt + n + 1 − t − ,I K n ( t ) = − t n +2 + ( n + 1) t n +1 − t n + ( n + 1) t − n (cid:88) i =1 t i ,II K n ( t ) = n ( t n +1 + t − n − ) − ( n + n + 2)( t + t − ) + 2( n + 2 n + 2) − n (cid:88) i =1 ( t i + t − i ) , and III K n ( t ) ≡ n ( t + t − ) − t + t − ) + 4 − n (cid:88) i =1 ( t i + t − i ) (mod W K n ( t )) , where W K n ( t ) = ( t n +1 + t − n − ) − ( t + t − ) − ( n + 1)( t + t − ) + 2 n + 2.(ii) Let K (cid:48) n ( n ≥ 1) be the virtual knot presented by the Gauss diagram andthe virtual diagram as shown in Figure 24. We see that c( K (cid:48) n ) = n + 3 andvc( K (cid:48) n ) = n +2 can be detected by II K (cid:48) n ( t ) but not by W K (cid:48) n ( t ), I K (cid:48) n ( t ), and III K (cid:48) n ( t ). + + + + +n+ − − − − −−−− − − −… …+ Figure 23 In fact, we have W K (cid:48) n ( t ) = t n +1 − nt + n − − t − + t − ,I K (cid:48) n ( t ) = ( n − t n +1 + (2 n + 1) t − n − − ( n − t − + ( n − t − − n (cid:88) i =1 t i ,II K (cid:48) n ( t ) = ( t n +2 + t − n − ) + ( n − t n +1 + t − n − )+( n − t + t − ) − n ( n + 1)( t + t − ) + 2( n + n + 2) − n (cid:88) i =1 ( t i + t − i ) − n (cid:88) i =1 ( t i − + t − i +1 ) , and III K (cid:48) n ( t ) ≡ − n ( t + t − ) + 4 n − n (cid:88) i =1 ( t i − + t − i +1 ) (mod W K (cid:48) n ( t )) , where W K (cid:48) n ( t ) = ( t n +1 + t − n − ) + ( t + t − ) − ( n − t + t − ) + 2 n − ++ + + +n +− − − −−− − … …+ − − −+ Figure 24 Dotted virtual knots Let ( D, p ) be a pair of a diagram D ⊂ Σ and a point p on D except the crossingsof D . We consider an equivalence relation among all ( D, p )’s generated by stabliza-tions, destabilizations, and Reidemeister moves away from p . Such an equivalenceclass is called a dotted virtual knot . A dotted virtual knot T can be regarded as along virtual knot or a 1-string virtual tangle by cutting a diagram open at p . In HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 29 this sense, the virtual knot presented by a diagram without p is called the closure of T , and denoted by (cid:98) T .Let ( D, p ) be a diagram of a dotted virtual knot T , c , . . . , c n the crossings of D ,and γ i , γ i ⊂ Σ the cycles at c i on Σ. We consider two subsets of indices M ( D ) = { i | p does not lie on γ i } and M ( D ) = { i | p lies on γ i } . We define a pair of Laurent polynomials associated with ( D, p ) as follows; W D,p ( t ) = (cid:88) i ∈ M ( D ) ε i ( t γ i · γ i − 1) and W D,p ( t ) = (cid:88) i ∈ M ( D ) ε i ( t γ i · γ i − . Lemma 6.1. The Laurent polynomials W D,p ( t ) and W D,p ( t ) ∈ Z [ t, t − ] do notdepend on a particular choice of a diagram of a dotted virtual knot T .Proof. Since a (de)stabilization does not change the intersection number γ i · γ i , W D,p ( t ) and W D,p ( t ) are invariant under a (de)stabilization.Assume that ( D (cid:48) , p (cid:48) ) is obtained from ( D, p ) by a Reidemeister move on Σ. If acrossing c of D is removed by a first Reidemeistere move, then we have γ = 0 or γ = 0 and hence ε ( t γ · γ − 1) = 0.If a pair of crossings c and c of D is removed by a second Reidemeister move,then we have γ = γ and ε = − ε as shown in the proof of Lemma 2.4. Fur-thermore the point p lies on γ if and only if p (cid:48) lies on γ . Therefore the sum ε ( t γ · γ − 1) + ε ( t γ · γ − 1) = 0 appears in one of W D,p ( t ) or W D,p ( t ).Assume that a triplet of crossings c , c , and c of D is involved in a thirdReidemeister move. Let c (cid:48) , c (cid:48) , and c (cid:48) be the corresponding crossings of D (cid:48) . Thenwe have γ i = γ (cid:48) i and ε i = ε (cid:48) i ( i = 1 , , 3) as shown in the proof of Lemma 2.4.Furthermore for each i , the point p lies on γ i if and only if p (cid:48) lies on γ (cid:48) i .In all cases, W D,p ( t ) and W D,p ( t ) are invariant under a Reidemeister move. (cid:3) Definition 6.2. The Laurent polynomials W D,p ( t ) and W D,p ( t ) are called the writhe polynomials of a dotted virtual knot T , and denoted by W T ( t ) and W T ( t ),respectively. Example 6.3. A diagram of a dotted virtual knot has a Gauss diagram with apoint on the circle naturally. Let ( G, p ) and ( G, p (cid:48) ) be Gauss diagrams with a pointas shown in Figure 25, and T and T (cid:48) the dotted virtual knots presented by ( G, p )and ( G, p (cid:48) ), respectively. p + +++ ++ +c c c c p + +++ ++ +c c c c Figure 25 For the chords c i ( i = 1 , , , 4) of ( G, p ), it holds that M ( D ) = { , , } and M ( D ) = { } . Since γ · γ = γ · γ = 2 and γ · γ = γ · γ = 0, we have W T ( t ) = t − W T ( t ) = − t + 1.On the other hand, since ( G, p (cid:48) ) is related to the one with no chord by firstand second Reidemeister moves, T (cid:48) is trivial and satisfies W T (cid:48) ( t ) = W T (cid:48) ( t ) = 0.Therefore T and T (cid:48) are different dotted virtual knots. (cid:3) By definition, it holds that W T ( t ) + W T ( t ) = W (cid:98) T ( t ) and W T (1) = W T (1) = 0for any dotted virtual knot T . Conversely we have the following. Proposition 6.4. Let K be a virtual knot, and f ( t ) a Laurent polynomial with f (1) = 0 . Then there is a dotted virtual knot T such that (i) (cid:98) T = K , (ii) W T ( t ) = f ( t ) , and (iii) W T ( t ) = W K ( t ) − f ( t ) .Therefore, for any virtual knot K , there are infinitely many dotted virtual knot T with (cid:98) T = K .Proof. We take a dotted virtual knot T with (cid:98) T = K arbitrarily. Since it holdsthat f (1) − W T (1) = 0, there are integers a n ( n (cid:54) = 0) such that f ( t ) = W T ( t ) + (cid:88) n (cid:54) =0 a n ( t n − . Therefore it is sufficient to prove that for any n (cid:54) = 0 and ε ∈ {±} , there is a dottedvirtual knot T with (cid:98) T = K and W T ( t ) = W T ( t ) + ε ( t n − G , p ) of T . We consider the Gauss diagram ( G , p )as shown in the middle of Figure 26 for n > n < 0. Let T be the dotted virtual knot presented by ( G , p ). Since G without p is related to G by first and second Reidemeister moves, we have (cid:98) T = (cid:98) T = K . Furthermorewe see that W T ( t ) = W T ( t ) + ε ( t n − (cid:3) ++++ ++ +++G G G εε εεp p n εε p n>0 n<0G = ε εε εε ε n……… ……… Figure 26 HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 31 Connected sums of virtual knots Let T and T (cid:48) be dotted virtual knots, and ( D, p ) and ( D (cid:48) , p (cid:48) ) diagrams of T and T (cid:48) on Σ and Σ (cid:48) , respectively. We remove a small open disk neighborhood ∆ of p in Σ (or ∆ (cid:48) of p (cid:48) in Σ (cid:48) ) and identify their boundaries to obtain a diagram ( D (cid:48)(cid:48) , p (cid:48)(cid:48) )on Σ (cid:48)(cid:48) = (Σ \ ∆) ∪ (Σ (cid:48) \ ∆ (cid:48) ), where the point p (cid:48)(cid:48) is taken such that D (cid:48) follows after D with respect to the orientation. See Figure 27. We denote by T + T (cid:48) the dottedvirtual knot presented by the diagram ( D (cid:48)(cid:48) , p (cid:48)(cid:48) ). D ΣD pp pΣ (cid:15468) D Σ (cid:15468) (cid:15468) Figure 27 The writhe polynomial is additive in the sense that W (cid:92) T + T (cid:48) ( t ) = W (cid:98) T ( t ) + W (cid:98) T (cid:48) ( t )holds for any dotted virtual knots T and T (cid:48) (cf. [2, 5, 25]). On the other hand, thefirst intersection polynomial is not additive in general as follows. Theorem 7.1. For dotted virtual knots T and T (cid:48) , it holds that I (cid:92) T + T (cid:48) ( t ) = I (cid:98) T ( t ) + I (cid:98) T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) . To prove this theorem, we prepare several lemmas. Let c , . . . , c n be the crossingsof ( D, p ), and γ i (1 ≤ i ≤ n ) the cycle at c i on Σ. Similarly, let c (cid:48) , . . . , c (cid:48) m be thecrossings of ( D (cid:48) , p (cid:48) ), and γ (cid:48) j (1 ≤ j ≤ m ) the cycle at c (cid:48) j on Σ (cid:48) . The crossings of( D (cid:48)(cid:48) , p (cid:48)(cid:48) ) are identified with c , . . . , c n and c (cid:48) , . . . , c (cid:48) m . Let Γ i (1 ≤ i ≤ n ) be thecycle at c i on Σ (cid:48)(cid:48) , and Γ (cid:48) j (1 ≤ j ≤ m ) the one at c (cid:48) j on Σ (cid:48)(cid:48) . Lemma 7.2. (i) Γ i · Γ j = γ i · γ j (1 ≤ i, j ≤ n ) . (ii) Γ (cid:48) i · Γ (cid:48) j = γ (cid:48) i · γ (cid:48) j (1 ≤ i, j ≤ m ) .Proof. We prove (i) only. The equation (ii) is similarly proved. By definition, itholds thatΓ i = (cid:26) γ i for i ∈ M ( D ) ,γ i + γ D (cid:48) for i ∈ M ( D ) , and Γ j = (cid:26) γ j + γ D (cid:48) for j ∈ M ( D ) ,γ j for j ∈ M ( D ) . Since γ i · γ D (cid:48) = γ D (cid:48) · γ j = γ D (cid:48) · γ D (cid:48) = 0 hold, we have the equation (i). (cid:3) The intersection number Γ i · Γ (cid:48) j (1 ≤ i ≤ n, ≤ j ≤ m ) is given as follows. Lemma 7.3. For any ≤ i ≤ n and ≤ j ≤ m , we have the following. (i) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = γ i · γ D . (ii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = 0 . (iii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = γ i · γ D + γ (cid:48) j · γ D (cid:48) . (iv) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = γ (cid:48) j · γ D (cid:48) . Proof. By definition, it holds thatΓ i = (cid:26) γ i for i ∈ M ( D ) ,γ i + γ D (cid:48) for i ∈ M ( D ) , and Γ (cid:48) j = (cid:26) γ (cid:48) j + γ D for j ∈ M ( D (cid:48) ) ,γ (cid:48) j for j ∈ M ( D (cid:48) ) . (i), (ii) Since γ i · γ (cid:48) j = 0 holds, we have the equations.(iii), (iv) Since it holds that γ D (cid:48) · γ (cid:48) j = − ( γ D (cid:48) − γ (cid:48) j ) · γ D (cid:48) = γ (cid:48) j · γ D (cid:48) and γ D (cid:48) · γ D = 0,we have the equations. (cid:3) The intersection number Γ (cid:48) j · Γ i (1 ≤ i ≤ n, ≤ j ≤ m ) is given as follows. Sincethe proof is similar to that of Lemma 7.3, we leave it to the reader. Lemma 7.4. For any ≤ i ≤ n and ≤ j ≤ m , we have the following. (i) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ (cid:48) j · Γ i = γ (cid:48) j · γ D (cid:48) . (ii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ (cid:48) j · Γ i = γ i · γ D + γ (cid:48) j · γ D (cid:48) . (iii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ (cid:48) j · Γ i = 0 . (iv) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ (cid:48) j · Γ i = γ i · γ D . (cid:3) Proof of Theorem 7.1 . Let ε i be the sign of c i (1 ≤ i ≤ n ), and ε (cid:48) j the sign of c (cid:48) j (1 ≤ j ≤ m ). Put ω kD = (cid:88) i ∈ M k ( D ) ε i and ω kD (cid:48) = (cid:88) j ∈ M k ( D (cid:48) ) ε (cid:48) j ( k = 0 , . By Lemmas 7.2, 7.3, and 7.4, it holds that f ( D (cid:48)(cid:48) ) − f ( D ) − f ( D (cid:48) )= (cid:88) ≤ i,j ≤ n ε i ε j ( t Γ i · Γ j − 1) + (cid:88) ≤ i,j ≤ m ε (cid:48) i ε (cid:48) j ( t Γ (cid:48) i · Γ (cid:48) j − (cid:88) ≤ i ≤ n, ≤ j ≤ m ε i ε (cid:48) j ( t Γ i · Γ (cid:48) j − 1) + (cid:88) ≤ i ≤ n, ≤ j ≤ m ε i ε (cid:48) j ( t Γ (cid:48) j · Γ i − − (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − − (cid:88) ≤ i,j ≤ m ε (cid:48) i ε (cid:48) j ( t γ (cid:48) i · γ (cid:48) j − (cid:88) i ∈ M ( D ) ,j ∈ M ( D (cid:48) ) ε i ε (cid:48) j ( t γ i · γ D − 1) + (cid:88) i ∈ M ( D ) ,j ∈ M ( D (cid:48) ) ε i ε (cid:48) j ( t γ (cid:48) j · γ D (cid:48) − (cid:88) i ∈ M ( D ) ,j ∈ M ( D (cid:48) ) ε i ε (cid:48) j ( t γ i · γ D + γ (cid:48) j · γ D (cid:48) − (cid:88) i ∈ M ( D ) ,j ∈ M ( D (cid:48) ) ε i ε (cid:48) j ( t γ (cid:48) j · γ D (cid:48) − 1) + (cid:88) i ∈ M ( D ) ,j ∈ M ( D (cid:48) ) ε i ε (cid:48) j ( t γ i · γ D − (cid:88) i ∈ M ( D ) ,j ∈ M ( D (cid:48) ) ε i ε (cid:48) j ( t γ i · γ D + γ (cid:48) j · γ D (cid:48) − . The sum of the first and third lines is equal to ω D (cid:48) W T ( t ) + ω D W T (cid:48) ( t ) + ω D W T (cid:48) ( t ) + ω D (cid:48) W T ( t ) . On the other hand, it holds that t γ i · γ D + γ (cid:48) j · γ D (cid:48) − t γ i · γ D − t γ (cid:48) j · γ D (cid:48) − 1) + ( t γ i · γ D − 1) + ( t γ (cid:48) j · γ D (cid:48) − . Therefore the sum of the second and fourth lines is equal to W T ( t ) W T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) + ω D (cid:48) W T ( t ) + ω D (cid:48) W T ( t ) + ω D W T (cid:48) ( t ) + ω D W T (cid:48) ( t ) . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 33 Since it holds that ω D + ω D = ω D and ω D (cid:48) + ω D (cid:48) = ω D (cid:48) , we have f ( D (cid:48)(cid:48) ) − f ( D ) − f ( D (cid:48) )= ω D (cid:48) W (cid:98) T ( t ) + ω D W (cid:99) T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) . Thus we obtain I (cid:92) T + T (cid:48) ( t ) − I (cid:98) T ( t ) − I (cid:98) T (cid:48) ( t )= f ( D (cid:48)(cid:48) ) − f ( D ) − f ( D (cid:48) ) − ( ω D + ω D (cid:48) ) (cid:0) W (cid:98) T ( t ) + W (cid:98) T (cid:48) ( t ) (cid:1) + ω D W (cid:98) T ( t ) + ω D (cid:48) W (cid:98) T (cid:48) ( t )= W T ( t ) W T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t ) . (cid:3) Similarly to the first intersection polynomial, the second intersection polynomialsatisfies the following. Theorem 7.5. For dotted virtual knots T and T (cid:48) , it holds that II (cid:92) T + T (cid:48) ( t ) = II (cid:98) T ( t ) + II (cid:98) T (cid:48) ( t )+ W T ( t ) W T (cid:48) ( t − ) + W T ( t ) W T (cid:48) ( t − )+ W T ( t − ) W T (cid:48) ( t ) + W T ( t − ) W T (cid:48) ( t ) . To prove this theorem, we prepare the following lemmas. Since the proofs aresimilar to those of Lemmas 7.2, 7.3, and 7.4, we omit them. Lemma 7.6. For any ≤ i, j ≤ n , we have the following. (i) Γ i · Γ j = γ i · γ j . (ii) Γ i · Γ j = γ i · γ j . (cid:3) Lemma 7.7. For any ≤ i, j ≤ m , we have the following. (i) Γ (cid:48) i · Γ (cid:48) j = γ (cid:48) i · γ (cid:48) j . (ii) Γ (cid:48) i · Γ (cid:48) j = γ (cid:48) i · γ (cid:48) j . (cid:3) Lemma 7.8. For any ≤ i ≤ n and ≤ j ≤ m , we have the following. (i) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = 0 . (ii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = γ i · γ D . (iii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = − γ (cid:48) j · γ D (cid:48) . (iv) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = γ i · γ D − γ (cid:48) j · γ D (cid:48) .The intersection number Γ (cid:48) j · Γ i is given by Γ (cid:48) j · Γ i = − Γ i · Γ (cid:48) j . (cid:3) Lemma 7.9. For any ≤ i ≤ n and ≤ j ≤ m , we have the following. (i) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = − γ i · γ D + γ (cid:48) j · γ D (cid:48) . (ii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = γ (cid:48) j · γ D (cid:48) . (iii) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = − γ i · γ D . (iv) If i ∈ M ( D ) and j ∈ M ( D (cid:48) ) , then Γ i · Γ (cid:48) j = 0 .The intersection number Γ (cid:48) j · Γ i is given by Γ (cid:48) j · Γ i = − Γ i · Γ (cid:48) j . (cid:3) Proof of Theorem 7.5 . Since the proof is similar to that of Theorem 7.1, we justgive an outline of it. By Lemmas 7.6–7.9, it holds that f ( D (cid:48)(cid:48) ) − f ( D ) − f ( D (cid:48) )= ω D (cid:48) (cid:0) W (cid:98) T ( t ) + W (cid:98) T ( t − ) (cid:1) + ω D (cid:0) W (cid:98) T (cid:48) ( t ) + W (cid:98) T (cid:48) ( t − ) (cid:1) + W T ( t ) W T (cid:48) ( t − ) + W T ( t − ) W T (cid:48) ( t ) , and f ( D (cid:48)(cid:48) ) − f ( D ) − f ( D (cid:48) )= ω D (cid:48) (cid:0) W (cid:98) T ( t − ) + W (cid:98) T ( t ) (cid:1) + ω D (cid:0) W (cid:98) T (cid:48) ( t − ) + W (cid:98) T (cid:48) ( t ) (cid:1) + W T ( t − ) W T (cid:48) ( t ) + W T ( t ) W T (cid:48) ( t − ) . Therefore we have II (cid:92) T + T (cid:48) ( t ) − II (cid:98) T ( t ) − II (cid:98) T (cid:48) ( t )= f ( D (cid:48)(cid:48) ) − f ( D ) − f ( D (cid:48) ) + f ( D (cid:48)(cid:48) ) − f ( D ) − f ( D (cid:48) ) − ( ω D + ω D (cid:48) ) (cid:0) W (cid:98) T ( t ) + W (cid:98) T (cid:48) ( t ) + W (cid:98) T ( t − ) + W (cid:98) T (cid:48) ( t − ) (cid:1) + ω D (cid:0) W (cid:98) T ( t ) + W (cid:98) T ( t − ) (cid:1) + ω D (cid:48) (cid:0) W (cid:98) T (cid:48) ( t ) + W (cid:98) T (cid:48) ( t − ) (cid:1) = W T ( t ) W T (cid:48) ( t − ) + W T ( t ) W T (cid:48) ( t − ) + W T ( t − ) W T (cid:48) ( t ) + W T ( t − ) W T (cid:48) ( t ) . (cid:3) Definition 7.10. For virtual knots K and K (cid:48) , we consider the set of virtual knots C ( K, K (cid:48) ) = { (cid:92) T + T (cid:48) | T , T (cid:48) : dotted virtual knots with (cid:98) T = K and (cid:98) T (cid:48) = K (cid:48) } . A virtual knot in this set is called a connected sum of K and K (cid:48) . Theorem 7.11. For any virtual knots K and K (cid:48) , the set C ( K, K (cid:48) ) is infinite; thatis, there are infinitely many connected sums of K and K (cid:48) .Proof. Let n be an integer. By Proposition 6.4, there are dotted virtual knots T n and T (cid:48) n such that (cid:40) (cid:98) T n = K, W T n ( t ) = t n − , W T n ( t ) = W K ( t ) − t n + 1 , and (cid:98) T (cid:48) n = K (cid:48) , W T (cid:48) n ( t ) = t n − , W T (cid:48) n ( t ) = W K (cid:48) ( t ) − t n + 1 . We denote by K n the virtual knot (cid:92) T n + T (cid:48) n which is a connected sum of K and K (cid:48) .Then by Theorem 7.1, it holds that I K n ( t ) = I K ( t ) + I K (cid:48) ( t )+( t n − (cid:0) W K (cid:48) ( t ) − t n + 1 (cid:1) + (cid:0) W K ( t ) − t n + 1 (cid:1) ( t n − I K ( t ) + I K (cid:48) ( t ) + (cid:0) W K ( t ) + W K (cid:48) ( t ) (cid:1) ( t n − − t n − . For any sufficiently large n , we have deg I K n ( t ) = 2 n and K n ’s are mutually distinct.Therefore C ( K, K (cid:48) ) is an infinite set. (cid:3) Characterizations of intersection polynomials In this section, we give characterizations of the intersection polynomials I K ( t ), II K ( t ), and III K ( t ).Let P denote the set of Laurent polynomials defined by P = { I K ( t ) | K :virtual knots } . The characterization is exactly the same as that of the writhepolynomial as given in Theorem 4.1. Theorem 8.1. For f ( t ) ∈ Z [ t, t − ] , the following are equivalent. HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 35 (i) f ( t ) ∈ P . (ii) f (1) = f (cid:48) (1) = 0 . (iii) f ( t ) = (cid:80) k (cid:54) =0 , a k ( t k − kt + k − for some a k ∈ Z ( k (cid:54) = 0 , . To prove this theorem, we prepare several lemmas. Lemma 8.2. For any f ( t ) , g ( t ) ∈ P , we have the following. (i) f ( t − ) ∈ P . (ii) f ( t ) + g ( t ) ∈ P .Proof. Let K and K (cid:48) be virtual knots with I K ( t ) = f ( t ) and I K (cid:48) ( t ) = g ( t ).(i) It holds that I − K ( t ) = f ( t − ) ∈ P by Lemma 5.2.(ii) By Proposition 6.4, there are dotted virtual knots T and T (cid:48) such that (cid:40) (cid:98) T = K, W T ( t ) = W K ( t ) , W T ( t ) = 0 , and (cid:98) T (cid:48) = K (cid:48) , W T (cid:48) ( t ) = W K (cid:48) ( t ) , W T (cid:48) ( t ) = 0 . Then we have I (cid:92) T + T (cid:48) ( t ) = I K ( t ) + I K (cid:48) ( t ) = f ( t ) + g ( t ) ∈ P by Theorem 7.1. (cid:3) Lemma 8.3. Let n ≥ be an integer. (i) There are integers a k (0 ≤ k ≤ n − such that t n + (cid:80) n − k =0 a k t k ∈ P . (ii) There are integers a (cid:48) k (0 ≤ k ≤ n − such that − t n + (cid:80) n − k =0 a (cid:48) k t k ∈ P .Proof. (i) For n = 2, we have I . ( t ) = ( t − ∈ P .Assume that n ≥ 3. We consider the trivial virtual knot O and the virtual knot3 . W . ( t ) = ( t − and I . ( t ) = 0. By Proposition 6.4, there are dottedvirtual knots T and T (cid:48) such that (cid:40) (cid:98) T = O, W T ( t ) = − ( t − t n − , W T ( t ) = ( t − t n − , and (cid:98) T (cid:48) = 3 . , W T (cid:48) ( t ) = ( t − , W T (cid:48) ( t ) = 0 . Then it holds that I (cid:92) T + T (cid:48) ( t ) = ( t − t n − ∈ P .(ii) For n = 2, we have I . ( t ) = − ( t − ∈ P . Assume that n ≥ 3. We considerthe trivial knot O and the virtual knot − . . We remark that W − . ( t ) = − W . ( t ) = − ( t − and I − . ( t ) = I . ( t ) = 0. Then similarly to the proof of(i), we have − ( t − t n − ∈ P . (cid:3) Lemma 8.4. Let n ≤ − be an integer. (i) There are integers a k ( n + 1 ≤ k ≤ such that t n + (cid:80) k = n +1 a k t k ∈ P . (ii) There are integers a (cid:48) k ( n + 1 ≤ k ≤ such that − t n + (cid:80) k = n +1 a (cid:48) k t k ∈ P .Proof. Assume that n ≤ − 2. By Lemmas 8.2(i) and 8.3, we have t n + (cid:80) k = n +1 a k t k ∈P and − t n + (cid:80) k = n +1 a (cid:48) k t k ∈ P for some a k , a (cid:48) k ∈ Z ( n + 1 ≤ k ≤ n = − 1, we have I . ( t ) = t − − t ∈ P and I . ( t ) = − t − +2 − t ∈ P . (cid:3) Proof of Theorem 8.1. (i) ⇒ (ii). This follows by Theorem 4.2.(ii) ⇒ (iii). Assume that f ( t ) = (cid:80) k ∈ Z a k t k satisfies f (1) = f (cid:48) (1) = 0. Then wehave a = (cid:80) k (cid:54) =0 , ( k − a k and a = − (cid:80) k (cid:54) =0 , ka k .(iii) ⇒ (i). By Lemmas 8.2(ii), 8.3, and 8.4, there are integers a (cid:48) and a (cid:48) such that g ( t ) = (cid:80) k (cid:54) =0 , a k t k + a (cid:48) t + a (cid:48) ∈ P . Since g (1) = g (cid:48) (1) = 0 holds by Theorem 4.2,we have a (cid:48) = (cid:80) k (cid:54) =0 , ( k − a k and a (cid:48) = − (cid:80) k (cid:54) =0 , ka k . Therefore we have g ( t ) = f ( t ) ∈ P . (cid:3) Let P denote the set of Laurent polynomials defined by P = { II K ( t ) | K :virtual knots } . Theorem 8.5. For f ( t ) ∈ Z [ t, t − ] , the following are equivalent. (i) f ( t ) ∈ P . (ii) f ( t ) is reciprocal, f (1) = 0 , and f (cid:48)(cid:48) (1) ≡ . To prove this theorem, we prepare several lemmas. Lemma 8.6. For any f ( t ) , g ( t ) ∈ P , we have f ( t ) + g ( t ) ∈ P .Proof. Let K and K (cid:48) be virtual knots with II K ( t ) = f ( t ) and II K (cid:48) ( t ) = g ( t ).By Proposition 6.4, there are dotted virtual knots T and T (cid:48) such that (cid:40) (cid:98) T = K, W T ( t ) = 0 , W T ( t ) = W K ( t ) , and (cid:98) T (cid:48) = K (cid:48) , W T (cid:48) ( t ) = W K (cid:48) ( t ) , W T (cid:48) ( t ) = 0 . Then we have II (cid:92) T + T (cid:48) ( t ) = II K ( t ) + II K (cid:48) ( t ) = f ( t ) + g ( t ) ∈ P by Theorem 7.5. (cid:3) Lemma 8.7. Let n ≥ be an integer. (i) There are integers a k (0 ≤ k ≤ n − such that ( t n + t − n ) + (cid:80) n − k =1 a k ( t k + t − k ) + a ∈ P . (ii) There are integers a (cid:48) k (0 ≤ k ≤ n − such that − ( t n + t − n ) + (cid:80) n − k =1 a (cid:48) k ( t k + t − k ) + a (cid:48) ∈ P .Proof. (i) We consider the trivial virtual knot O and the virtual knot 4 . 20 with W . ( t ) = ( t − and II . ( t ) = 0. By Proposition 6.4, there are dotted virtualknots T and T (cid:48) such that (cid:40) (cid:98) T = O, W T ( t ) = ( t − t n − , W T ( t ) = − ( t − t n − , and (cid:98) T (cid:48) = 4 . , W T (cid:48) ( t ) = ( t − , W T (cid:48) ( t ) = 0 . Then it holds that II (cid:92) T + T (cid:48) ( t ) = ( t − t − − t n − + ( t − ( t − − t − n +1 = ( t − t n − + ( t − − t − n +3 ∈ P . (ii) We consider the trivial knot O and the virtual knot − . . We remarkthat W − . ( t ) = − W . ( t ) = − ( t − and II − . ( t ) = II . ( t ) = 0. Thensimilarly to the proof of (i), we have − ( t − t n − − ( t − − t − n +3 ∈ P . (cid:3) Lemma 8.8. t − t − ∈ P and − t + 4 − t − ∈ P .Proof. We have II . ( t ) = − t + 4 − t − ∈ P . Furthermore, since II . =4 t − t − ∈ P , we have 2 t − t − = ( − t + 4 − t − ) + (4 t − t − ) ∈ P by Lemma 8.6 (cid:3) Proof of Theorem 8.5. (i) ⇒ (ii). This follows by Theorem 4.3.(ii) ⇒ (i). By Lemma 4.5, we may take f ( t ) = (cid:80) k ≥ a k ( t k + t − k − 2) for some a k ∈ Z ( k ≥ 1) with (cid:80) k :odd ≥ a k ≡ a (cid:48) and a (cid:48) such that g ( t ) = (cid:88) k ≥ a k ( t k + t − k ) + a (cid:48) ( t + t − ) + a (cid:48) ∈ P . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 37 Since (cid:80) k :odd ≥ a k + a (cid:48) ≡ a (cid:48) ≡ a (mod 2).Therefore, by Lemmas 8.6 and 8.8, there is an integer a (cid:48)(cid:48) such that h ( t ) = (cid:88) k ≥ a k ( t k + t − k ) + a (cid:48)(cid:48) ∈ P . Since h (1) = 0 holds, we have a (cid:48)(cid:48) = − (cid:80) k ≥ a k and h ( t ) = f ( t ) ∈ P . (cid:3) Let P denote the set of pairs of Laurent polynomials defined by P = { (cid:0) f ( t ) , g ( t ) (cid:1) | f ( t ) = W K ( t ) and g ( t ) ≡ III K ( t ) for some K } . A pair of Laurent polynomials in the set P is characterized as follows. Theorem 8.9. For f ( t ) and g ( t ) ∈ Z [ t, t − ] , the following are equivalent. (i) (cid:0) f ( t ) , g ( t ) (cid:1) ∈ P . (ii) g ( t ) is reciprocal, f (1) = f (cid:48) (1) = g (1) = 0 , and f (cid:48)(cid:48) (1) ≡ g (cid:48)(cid:48) (1) (mod 4) . To prove this theorem, we prepare the following lemma. Lemma 8.10. Let T and T be dotted virtual knots with (cid:98) T = K and (cid:98) T = O ,respectively. Then we have III (cid:92) T + T ( t ) ≡ III K ( t ) + W T ( t ) W T ( t − ) + W T ( t − ) W T ( t ) (mod W K ( t )) . Proof. Let ( D, p ) and ( D , p ) be diagrams of T and T , respectively. Since W O ( t ) =0 holds, we have f ( D ) = III O ( t ) = 0. Let ( D (cid:48) , p (cid:48) ) be the diagram of T + T ob-tained by taking the sum of ( D, p ) and ( D , p ). By the equation in the proof ofTheorem 7.5, it holds that f ( D (cid:48) ) = f ( D ) + f ( D ) + ω D W K ( t ) + ω D W O ( t )+ W T ( t ) W T ( t − ) + W T ( t − ) W T ( t )= f ( D ) + ω D (cid:48) W K ( t ) + W T ( t ) W T ( t − ) + W T ( t − ) W T ( t ) . Therefore we have we have the conclusion. (cid:3) Proof of Theorem 8 . . (i) ⇒ (ii). This follows by Theorems 4.1 and 4.6.(ii) ⇒ (i). By Theorem 4.1, there is a virtual knot K with W K ( t ) = f ( t ). We takea Laurent polynomial h ( t ) with III K ( t ) ≡ h ( t ) (mod W K ( t )). By Theorem 4.6, h ( t ) is reciprocal, h (1) = 0, and h (cid:48)(cid:48) (1) ≡ f (cid:48)(cid:48) (1) (mod 4).Consider the Laurent polynomial p ( t ) = g ( t ) − h ( t ). Then p ( t ) is reciprocal, p (1) = g (1) − h (1) = 0, and p (cid:48)(cid:48) (1) = g (cid:48)(cid:48) (1) − h (cid:48)(cid:48) (1) ≡ q ( t ) such that p ( t ) = ( t − t − − q ( t ) + ( t − − t − q ( t − ) . It follows by Proposition 6.4 that there are dotted virtual knots T and T suchthat (cid:40) (cid:98) T = K, W T ( t ) = W K ( t ) − ( t − q ( t ) , W T ( t ) = ( t − q ( t ) , and (cid:98) T = O, W T ( t ) = − ( t − , W T ( t ) = t − . Then we have p ( t ) = W T ( t ) W T ( t − ) + W T ( t − ) W T ( t ).By Lemma 8.10, the virtual knot (cid:92) T + T satisfies W (cid:92) T + T ( t ) = W K ( t ) + W O ( t ) = f ( t ) and III (cid:92) T + T ( t ) ≡ h ( t ) + p ( t ) = g ( t ) . (cid:3) By Theorem 7.11, there are infinitely many connected sums of two trivial virtualknots. In particular, we have the following, where 2 P = { f ( t ) | f ( t ) ∈ P } and2 P = { f ( t ) | f ( t ) ∈ P } . Proposition 8.11. For f ( t ) ∈ Z [ t, t − ] , the following are equivalent. (i) There is a virtual knot K ∈ C ( O, O ) with f ( t ) = I K ( t ) . (ii) f ( t ) ∈ P .Proof. (i) ⇒ (ii). Let T and T (cid:48) be dotted virtual knots with K = (cid:92) T + T (cid:48) and (cid:98) T = (cid:98) T (cid:48) = O . By Theorem 4.2, we have f (1) = f (cid:48) (1) = 0. Furthermore, since W T ( t ) = − W T ( t ) and W T (cid:48) ( t ) = − W T (cid:48) ( t ) , we have I K ( t ) = − W T ( t ) W T (cid:48) ( t ) by Theorem 7.1. Therefore all the coefficients of f ( t ) are even.(ii) ⇒ (i). By Theorem 8.1, there is a Laurent polynomial g ( t ) ∈ Z [ t, t − ] with f ( t ) = 2( t − g ( t ). By Proposition 6.4, there are dotted virtual knots T and T (cid:48) such that (cid:40) (cid:98) T = O, W T ( t ) = − ( t − , W T ( t ) = t − , and (cid:98) T (cid:48) = O, W T (cid:48) ( t ) = ( t − g ( t ) , W T (cid:48) ( t ) = − ( t − g ( t ) . Then the connected sum K = (cid:92) T + T (cid:48) ∈ C ( O, O ) satisfies I K ( t ) = − W T ( t ) W T (cid:48) ( t ) = 2( t − g ( t ) = f ( t ) . (cid:3) Proposition 8.12. For f ( t ) ∈ Z [ t, t − ] , the following are equivalent. (i) There is a virtual knot K ∈ C ( O, O ) with f ( t ) = II K ( t ) . (ii) f ( t ) ∈ P .Proof. (i) ⇒ (ii). Let T and T (cid:48) be dotted virtual knots with K = (cid:92) T + T (cid:48) and (cid:98) T = (cid:98) T (cid:48) = O . By Theorem 4.3, f ( t ) is reciprocal, f (1) = 0, and f (cid:48)(cid:48) (1) ≡ W T ( t ) = − W T ( t ) = ( t − p ( t ) and W T (cid:48) ( t ) = − W T (cid:48) ( t ) = ( t − q ( t )for some p ( t ), q ( t ) ∈ Z [ t, t − ], we have f ( t ) = II K ( t ) = 2 (cid:0) W T ( t ) W T (cid:48) ( t − ) + W T ( t − ) W T (cid:48) ( t ) (cid:1) = 2( t − t − − (cid:0) p ( t ) q ( t − ) + p ( t − ) q ( t ) (cid:1) . Therefore all the coefficients of f ( t ) are even.(ii) ⇒ (i). Put (cid:101) f ( t ) = f ( t ) / ∈ Z [ t, t − ]. By Theorem 4.3, it satisfies that (cid:101) f ( t ) isreciprocal, (cid:101) f (1) = 0, and (cid:101) f (cid:48)(cid:48) (1) ≡ (cid:101) g ( t ) such that (cid:101) f ( t ) = ( t − t − − (cid:101) g ( t ) + ( t − − t − (cid:101) g ( t − ) . By Proposition 6.4, there are dotted virtual knots T and T (cid:48) such that (cid:40) (cid:98) T = O, W T ( t ) = ( t − (cid:101) g ( t ) , W T ( t ) = − ( t − (cid:101) g ( t ) , and (cid:98) T (cid:48) = O, W T (cid:48) ( t ) = t − , W T (cid:48) ( t ) = − ( t − . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 39 Then the connected sum K = (cid:92) T + T (cid:48) ∈ C ( O, O ) satisfies II K ( t ) = 2( t − t − − (cid:0)(cid:101) g ( t ) + (cid:101) g ( t − ) (cid:1) = f ( t ) . (cid:3) Proposition 8.13. For f ( t ) ∈ Z [ t, t − ] , the following are equivalent. (i) There is a virtual knot K ∈ C ( O, O ) with f ( t ) = III K ( t ) . (ii) f ( t ) ∈ P .Proof. By Proposition 6.4 and Lemma 8.10, the condition (i) is equivalent to f ( t ) ∈ { p ( t ) q ( t ) + p ( t − ) q ( t − ) | p ( t ) , q ( t ) ∈ Z [ t, t − ] , p (1) = q (1) = 0 } . This set is coincident with { g ( t ) + g ( t − ) | g ( t ) ∈ Z [ t, t − ] , g (1) = g (cid:48) (1) = 0 } . Therefore (i) is equivalent to (ii) by Lemma 4.5 and Theorem 8.5. (cid:3) Example 8.14. Let T and T (cid:48) be the dotted virtual knots presented by Gaussdiagrams as shown in Figure 28. The closures (cid:98) T and (cid:98) T (cid:48) are trivial virtual knots.Consider the virtual knot K = (cid:92) T + T (cid:48) which is a connected sum of two trivialknots. In [15], Kishino and the fourth author study the virtual knot K = 4 . 55 asshown in Figure 28 and prove that K is not classical by using computer calculationof the Jones polynomial of the 3-parallel of K . Since we have W T ( t ) = − W T ( t ) = − ( t − 1) and W T (cid:48) ( t ) = − W T (cid:48) ( t ) = − ( t − − , it holds that I K ( t ) = 2 t − t − ,II K ( t ) = 2 t − t + 4 − t − + 2 t − , and III K ( t ) = t − t + 2 − t − + t − by Theorems 7.1, 7.5, and Lemma 8.10. This also induces that K is not classicalby Lemma 2.11. T+ ++ T+ ++ K+ +++ = ++ Figure 28 Finite type invariants of order D be a diagram on a closed, connected, oriented surface Σ with m doublepoints (with no over/under-information) and some crossings. For ε i ∈ {± } (1 ≤ i ≤ m ), we denote by D ε ...ε m the diagram by giving over/under-information at p i with sign ε i , and K ε ...ε m the virtual knot presented by D ε ...ε m . An invariant v ( K )of a virtual knot K is called a finite type invariant of order m [12] if it satisfies(i) (cid:80) ε i = ± ε . . . ε m +1 v ( K ε ...ε m +1 ) = 0 for any diagram D with m + 1 doublepoints, and(ii) (cid:80) ε i = ± ε . . . ε m v ( K ε ...ε m ) (cid:54) = 0 for some diagram D with m double points.We remark that this definifion of a finite type invariant is different from the onegiven by Goussarov, Polyak, and Viro [7].It is known in [4] that the writhe polynomial is a finite type invariant of order1. Namely any diagram with two double points satisfies W K ++ ( t ) − W K + − ( t ) − W K − + ( t ) + W K −− ( t ) = 0 . Lemma 9.1. For any diagram D with three double points and an integer ω , wehave (cid:88) ε ,ε ,ε = ± ( ω + ε + ε + ε ) ε ε ε W K ε ε ε ( t ) = 0 . Proof. Since the writhe polynomial is a finite type invariant of order 1, it holdsthat (cid:88) ε ,ε ,ε = ± ωε ε ε W K ε ε ε ( t ) = ω (cid:88) ε = ± ε (cid:32) (cid:88) ε ,ε = ± ε ε W K ε ε ε ( t ) (cid:33) = 0and (cid:88) ε ,ε ,ε = ± ( ε + ε + ε ) ε ε ε W K ε ε ε ( t )= (cid:88) ε = ± (cid:32) (cid:88) ε ,ε = ± ε ε W K ε ε ε ( t ) (cid:33) + (cid:88) ε = ± (cid:32) (cid:88) ε ,ε = ± ε ε W K ε ε ε ( t ) (cid:33) + (cid:88) ε = ± (cid:32) (cid:88) ε ,ε = ± ε ε W K ε ε ε ( t ) (cid:33) = 0 . (cid:3) Let D be a diagram with three double points p , p , p and n − c , . . . , c n . For each p i and ε i ∈ {± } ( i = 1 , , γ i ( ε i ) the cycle at p i in the diagram D ε ε ε on Σ. The cycle γ ( ε ) on Σ is determined independentlyof ε and ε and so on. It holds that γ i (+) + γ i ( − ) = γ D ( i = 1 , , 3) by definition.For 4 ≤ i ≤ n , the cycle γ i is defined at c i on Σ independently of ε , ε , and ε . Theorem 9.2. The first and second intersection polynomials are finite type invari-ants of order . HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 41 Proof. I K ( t ). Let ω be the sum of signs of c , . . . , c n of D . Since the writhe of D ε ε ε is equal to ω + ε + ε + ε , we have (cid:88) ε ,ε ,ε = ± ε ε ε I K ε ε ε ( t ) = (cid:88) ε ,ε ,ε = ± ε ε ε f ( D ε ε ε ) − (cid:88) ε ,ε ,ε = ± ( ω + ε + ε + ε ) ε ε ε W K ε ε ε ( t ) . By Lemma 9.1, the second sum is equal to zero. On the other hand, the firstsum is divided into five terms as follows; (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) i =1 ε i ( t γ i ( ε i ) · γ i ( ε i ) − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i (cid:54) = j ≤ ε i ε j ( t γ i ( ε i ) · γ j ( ε j ) − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i ≤ , ≤ j ≤ n ε i ε j ( t γ i ( ε i ) · γ j − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i ≤ n, ≤ j ≤ ε i ε j ( t γ i · γ j ( ε j ) − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − (cid:33) . We see that each of the five terms is equal to zero. For example, the second termis equal to (cid:88) ≤ i (cid:54) = j ≤ (cid:32) (cid:88) ε ,ε ,ε = ± ε ε ε ε i ε j ( t γ i ( ε i ) · γ j ( ε j ) − (cid:33) = (cid:88) ≤ i (cid:54) = j ≤ (cid:32) (cid:88) ε p = ± ε p (cid:32) (cid:88) ε i ,ε j = ± ( t γ i ( ε i ) · γ j ( ε j ) − (cid:33)(cid:33) = 0 , where p is taken satisfying { i, j, p } = { , , } . Similarly, the third term is equal to (cid:88) ≤ i ≤ , ≤ j ≤ n (cid:32) (cid:88) ε ,ε ,ε = ± ε ε ε ε i ε j ( t γ i ( ε i ) · γ j − (cid:33) = (cid:88) ≤ i ≤ , ≤ j ≤ n (cid:32) (cid:88) ε p ,ε q = ± ε p ε q (cid:32) (cid:88) ε i = ± ε j ( t γ i ( ε i ) · γ j − (cid:33)(cid:33) = 0 , where p , q are taken satisfying { i, p, q } = { , , } . The first, fourth, and fifth termsare similarly calculated to be zero. II K ( t ). Similarly to the case of I K ( t ), we have (cid:88) ε ,ε ,ε = ± ε ε ε II K ε ε ε ( t )= (cid:88) ε ,ε ,ε = ± ε ε ε f ( D ε ε ε ) + (cid:88) ε ,ε ,ε = ± ε ε ε f ( D ε ε ε ) − (cid:88) ε ,ε ,ε = ± ( ω + ε + ε + ε ) ε ε ε (cid:0) W K ε ε ε ( t ) − W K ε ε ε ( t ) (cid:1) . By Lemma 9.1, the third sum is equal to zero. On the other hand, the first sumis equal to (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) i =1 ε i ( t γ i ( ε i ) · γ i ( ε i ) − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i (cid:54) = j ≤ ε i ε j ( t γ i ( ε i ) · γ j ( ε j ) − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i ≤ , ≤ j ≤ n ε i ε j ( t γ i ( ε i ) · γ j − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i ≤ n, ≤ j ≤ ε i ε j ( t γ i · γ j ( ε j ) − (cid:33) + (cid:88) ε ,ε ,ε = ± ε ε ε (cid:32) (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − (cid:33) . Each of the five terms is equal to zero. We omit the proof which is similar to thatfor I K ( t ). The second sum is obtained from the first by replacing D ε ε ε with D ε ε ε , and is also equal to zero.On the other hand, we consider the diagram D with two double points as shownin Figure 29. It is easy to see that (cid:88) ε ,ε = ± ε ε I K ε ε ( t ) = I . ( t ) − I O ( t ) − I O ( t ) + I . ( t ) = − t + 4 − t − and (cid:88) ε ,ε = ± ε ε II K ε ε ( t ) = II . ( t ) − II O ( t ) − II O ( t ) + II . ( t ) = − t + 8 − t − . Since they are not equal to zero, we have the conclusion. (cid:3) (cid:15913) (cid:15913)(cid:21)(cid:17)(cid:20)(cid:39) (cid:50) (cid:50) (cid:21)(cid:17)(cid:20)(cid:6)(cid:1144) (cid:1144)(cid:1144) (cid:1144) (cid:15913) (cid:15913) Figure 29 HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 43 Flat virtual knots Let D be a diagram with a double point p and n − c , . . . , c n . For ε ∈ {± } , we denote by γ ( ε ) the cycle at p in the diagram D ε on Σ. Lemma 10.1. It holds that (cid:88) ε = ± ε (cid:0) f ( D ε ) + f ( D ε ) − f ( D ε ) − f ( D ε ) (cid:1) = 0 . Proof. Since γ ( − ε ) = γ ( ε ) and γ ( − ε ) = γ ( ε ) hold by definition, we have (cid:88) ε = ± ε f ( D ε )= (cid:88) ε = ± ε ( t γ ( ε ) · γ ( ε ) − 1) + (cid:88) ε = ± ε n (cid:88) j =2 ε ε j ( t γ ( ε ) · γ j − (cid:88) ε = ± ε n (cid:88) i =2 ε i ε ( t γ i · γ ( ε ) − 1) + (cid:88) ε = ± ε (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − (cid:88) ε = ± ε ( t γ ( ε ) · γ ( ε ) − (cid:88) ε = ± n (cid:88) j =2 ε j ( t γ ( ε ) · γ j − 1) + (cid:88) ε = ± n (cid:88) i =2 ε i ( t γ i · γ ( ε ) − , (cid:88) ε = ± ε f ( D ε )= (cid:88) ε = ± ε ( t γ ( ε ) · γ ( ε ) − 1) + (cid:88) ε = ± ε n (cid:88) j =2 ε ε j ( t γ ( ε ) · γ j − (cid:88) ε = ± ε n (cid:88) i =2 ε i ε ( t γ i · γ ( ε ) − 1) + (cid:88) ε = ± ε (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − − (cid:88) ε = ± ε ( t γ ( ε ) · γ ( ε ) − (cid:88) ε = ± n (cid:88) j =2 ε j ( t γ ( ε ) · γ j − 1) + (cid:88) ε = ± n (cid:88) i =2 ε i ( t γ i · γ ( ε ) − , (cid:88) ε = ± ε f ( D ε )= (cid:88) ε = ± ε ( t γ ( ε ) · γ ( ε ) − 1) + (cid:88) ε = ± ε n (cid:88) j =2 ε ε j ( t γ ( ε ) · γ j − (cid:88) ε = ± ε n (cid:88) i =2 ε i ε ( t γ i · γ ( ε ) − 1) + (cid:88) ε = ± ε (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − (cid:88) ε = ± n (cid:88) j =2 ε j ( t γ ( ε ) · γ j − 1) + (cid:88) ε = ± n (cid:88) i =2 ε i ( t γ i · γ ( ε ) − , and (cid:88) ε = ± ε f ( D ε )= (cid:88) ε = ± ε ( t γ ( ε ) · γ ( ε ) − 1) + (cid:88) ε = ± ε n (cid:88) j =2 ε ε j ( t γ ( ε ) · γ j − (cid:88) ε = ± ε n (cid:88) i =2 ε i ε ( t γ i · γ ( ε ) − 1) + (cid:88) ε = ± ε (cid:88) ≤ i,j ≤ n ε i ε j ( t γ i · γ j − (cid:88) ε = ± n (cid:88) j =2 ε j ( t γ ( ε ) · γ j − 1) + (cid:88) ε = ± n (cid:88) i =2 ε i ( t γ i · γ ( ε ) − . Therefore we have the conclusion. (cid:3) Lemma 10.2 ([25]) . The Laurent polynomial W K ( t ) − W K ( t − ) is invariant undera crossing change. (cid:3) For the intersection polynomials, we have the following. Theorem 10.3. The Laurent polynomial I K ( t ) + I K ( t − ) − II K ( t ) is invariantunder a crossing change.Proof. Let ω be the sum of signs of c , . . . , c n of D . By Lemma 10.1, (cid:88) ε = ± ε (cid:0) I D ε ( t ) + I D ε ( t − ) − II D ε ( t ) (cid:1) = (cid:88) ε = ± ε (cid:110) f ( D ε ) − ( ω + ε ) W D ε ( t ) + f ( D ε ) − ( ω + ε ) W D ε ( t − ) − f ( D ε ) − f ( D ε ) + ( ω + ε ) (cid:0) W D ε ( t ) + W D ε ( t − ) (cid:1)(cid:111) = 0 . This induces I K + ( t ) + I K + ( t − ) − II K + ( t ) = I K − ( t ) + I K − ( t − ) − II K − ( t ). (cid:3) A flat virtual knot F is an equivalence class of virtual knots under a crossingchange. Equivalently, it is presented by an immersed circle in a closed, connected,oriented surface which has double points with no over/under-information up tostabilizations, destabilizations, and flattened Reidemeister moves [12, 28]. Let c( F )denote the crossing number of F , which is the minimal number of double pointsfor all immersed circles presenting F . It is known that there are one flat virtualknot F (3 , 1) of c( F ) = 3 and eleven flat virtual knots F (4 . F (4 . 11) of c( F ) = 4up to mirror images and reversion (cf. [6, 18]), where Gauss diagrams of these flatvirtual knots are illustrated in Figure 30.There is a natural map ϕ : { virtual knots } → { flat virtual knots } defined byignoring the over/under-information of crossings. For each flat virtual knot F ofc( F ) ≤ 4, we give the list of virtual knots K of c( K ) ≤ ϕ ( K ) = F asshown in Table 3, where we abbreviate K = 4 .n to n for 1 ≤ n ≤ (cid:0) ϕ ( K ) (cid:1) ≤ c( K ) holds for any virtual knot K .Lemma 10.2 and Theorem 10.3 imply that the Laurent polynomials W K ( t ) − W K ( t − ) and I K ( t )+ I K ( t − ) − II K ( t ) are invariants of a flat virtual knot F = ϕ ( K ). Theorem 10.4. For the flat virtual knots F of c( F ) ≤ , the Laurent polynomials W K ( t ) − W K ( t − ) and I K ( t ) + I K ( t − ) − II K ( t ) are given as in Table 4 , where K is a virtual knot with F = ϕ ( K ) . (cid:3) HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 45 + + + + ++++ ++++ ++ + +++ ++ ++ +++++++ + ++++ + + ++ +++ ++ ++++F(3.1) F(4.1) F(4.2) F(4.3)F(4.4) F(4.5) F(4.6) F(4.7)F(4.8) F(4.9) F(4.10) F(4.11) −−− −− −− − −− − − −− −− − −− − − −− − − −− −−−−− −− −− − −− − − −− − −−− Figure 30 c( K ) ≤ K ) = 4 O . 1, 2 . 1, 3 . . 5, 3 . 6, 3 . F (3 . 1) 3 . 1, 3 . 3, 3 . F (4 . 1) 83, 87, 88, 93, 97, 103 F (4 . 2) 26, 28, 45, 47, 80, 81 F (4 . 3) 22, 24, 39, 42, 62, 66, 78, 79 F (4 . 4) 10, 15, 20, 23, 32, 35, 38, 50, 63, 67 F (4 . 5) 85, 89, 90, 98, 107 F (4 . 6) 11, 17, 19, 29, 34, 49, 57, 70 F (4 . 7) 13, 18, 31, 51, 60, 69 F (4 . 8) 14, 21, 30, 48, 59, 71 F (4 . 9) 9, 16, 33, 52, 58, 72 F (4 . 10) 1, 4, 8, 55, 77 F (4 . 11) 2, 5, 7, 56, 76 Table 3 Here, we use the following notations; (cid:104) a + a + · · · = a ( t − t − ) + a ( t − t − ) + · · · and[ b + b + b + · · · = b + b ( t + t − ) + b ( t + t − ) + · · · . W K ( t ) − W K ( t − ) I K ( t ) + I K ( t − ) − II K ( t ) O F (3 . (cid:104) − − − F (4 . (cid:104)− − − − F (4 . (cid:104) − − 16 + 9 + 0 − F (4 . (cid:104)− − − − F (4 . (cid:104) − F (4 . 5) 0 [ − − F (4 . (cid:104) − − − F (4 . 7) 0 0 F (4 . 8) 0 [ − 10 + 5 + 1 − F (4 . 9) 0 [ − − F (4 . 10) 0 [ − 12 + 8 − F (4 . 11) 0 [12 − Table 4 By observing Table 4, we see that these invariants classify the flat virtual knots F of c( F ) ≤ { O, F (4 . } and { F (3 . , F (4 . } .The supporting genus of a virtual knot K is the minimal genus of Σ for alldiagrams (Σ , D ) of K , and denoted by sg ( K ). We remark that sg ( K ) = 0 holdsif and only if K is a classical knot. Similarly, we can define the supporting genus sg ( F ) of a flat virtual knot F . Lemma 10.5 ([9, 19]) . There is no flat virtual knot F of sg ( F ) = 1 .Proof. Let C be an immersed circle in the torus Σ = T . By using flat Reidemeistermoves and orientation-preserving homeomorphisms of T , C can be deformed intothe curve C n for some n ≥ T so that C n presents the trivial flat virtual knot. (cid:3) n Figure 31 Corollary 10.6. If a virtual knot K satisfies W K ( t ) − W K ( t − ) (cid:54) = 0 or I K ( t ) + I K ( t − ) − II K ( t ) (cid:54) = 0 , then we have sg ( K ) ≥ .Proof. If sg ( K ) ≤ F = ϕ ( K ) satisfies sg ( F ) ≤ F is the trivial flat virtual knot and hence the invariants vanish. (cid:3) HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 47 Example 10.7. We consider the virtual knot K (cid:48) n ( n ≥ 3) given in Example 5.5.It holds that W K (cid:48) n ( t ) − W K (cid:48) n ( t − ) = − ( t n − − t − n +1 ) + ( n − t − t − ) and I K (cid:48) n ( t ) + I K (cid:48) n ( t − ) − II K (cid:48) n ( t ) = − ( t n − + t − n +1 )+( n − ( t + t − ) − n + 4 n. Then we have sg ( K (cid:48) n ) ≥ G (cid:48) n isrealized by a diagram on a closed surface of genus two, we have sg ( K (cid:48) n ) ≤ 2, andhence sg ( K (cid:48) n ) = 2. Remark 10.8. It is known that the writhe polynomial is invariant under a Deltamove [25]. 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Knot TheoryRamifications (2001), no. 6, 931–935.[24] Y. Ohyama and M. Sakurai, A construction of infinitely many virtual knots with propertiesof Kishino’s knot , talk at the autumn meeting of the Mathematical Society of Japan, 2020.[25] S. Satoh and K. Taniguchi, The writhes of a virtual knot , Fund. Math. (2014), 327–342.[26] J. Sawollek, On Alexander-Conway polynomials for virtual knots and links , available atarXiv:math/9912173.[27] D. Silver and S. Williams, Polynomial invariants of virtual links , J. Knot Theory Ramifica-tions (2003), 987–1000.[28] V. Turaev, Virtual strings , Ann. Inst. Fourier (Grenoble) (2004), no. 7, 2455–2525.[29] M. Xu, Writhe polynomial for virtual links , available at arXiv:1812.05234. Appendix A. Table of W K ( t ) , I K ( t ) , and II K ( t ) . Table 5 shows W K ( t ), I K ( t ), and II K ( t ) of a virtual knot K up to crossingnumber four according to Green’s table [8] with a choice of orientations. We usethe following notations; { n } ( a + a + · · · + a m ) = a t n + a t n +1 + · · · + a m t n + m and[ b + b + b + · · · = b + b ( t + t − ) + b ( t + t − ) + · · · , where m ≥ a (cid:54) = 0. W K ( t ) I K ( t ) II K ( t )2 . − − − . {− } ( − − { } ( − − 1) [4 − . − − − . {− } ( − − {− } ( − − 2) [10 − − . { } (1 − − . − − − . . − − − . − − − 12 + 24 . − − − . − {− } ( − − − . − { } ( − − 1) [10 − . − {− } ( − − − − . {− } ( − − 1) [2 + 0 − . − − − . − − − . − − − . {− } ( − − {− } ( − − 2) [0 + 2 − . {− } ( − − {− } ( − − 2) [10 − − . 12 0 0 04 . 13 0 0 04 . 14 [0 − − − − . {− } ( − − {− } (1 − − 3) [4 + 0 − . 16 0 {− } (1 − − 1) 04 . {− } ( − − { } ( − − 1) [4 − . 18 [2 − − − HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 49 . {− } ( − − { } ( − − 1) [4 − . { } (1 − . 21 [ − {− } ( − − − 1) [8 − − . {− } ( − − 1) 0 [4 − − . {− } ( − − {− } ( − − 2) [0 + 2 − . {− } (1 + 0 − − {− } ( − − − 2) [0 + 3 − − . 25 [4 − {− } ( − − − . {− } ( − − {− } (1 − − 1) [10 − − . 27 [2 − {− } ( − − 1) [6 − − . {− } (2 − − { } ( − − 2) [10 − − . {− } ( − − {− } ( − − 2) [10 − − . 30 [2 − { } ( − − 1) [10 − . 31 0 0 04 . {− } ( − − 1) [2 − − . 33 [2 − − − . { } (1 − − . {− } ( − − 1) [2 − − . 36 [ − − − . 37 [4 − − − − − − . { } (1 − { } (2 − − . {− } ( − − { } ( − − 2) [4 − − . 40 [2 − − − . 41 0 0 04 . { } (1 − − { } ( − − 1) [4 − . 43 [4 − − − . 44 [2 − { } (1 − − . {− } ( − − { } ( − − 1) [12 − . 46 0 {− } ( − − 1) [2 + 0 − . {− } (1 + 0 − {− } (1 − − 1) [8 − . 48 [4 − − − − − . { } (1 − − . {− } ( − − {− } ( − − 2) [0 + 2 − . 51 0 0 04 . 52 [2 − − − . 53 [4 − − − . 54 [2 − {− } ( − − 1) [6 − − . 55 0 [ − − . 56 0 { } (2 − − . {− } ( − − { } ( − − 1) [4 − . 58 0 {− } ( − − 1) [8 − . 59 0 { } ( − − 1) [8 − . 60 [2 − − − . 61 [2 − − − . {− } ( − − − {− } ( − − 1) [12 − − . {− } ( − − {− } (1 − − 1) [8 − . 64 [0 − − − . 65 [2 + 0 − − − . {− } (1 + 0 − − {− } (1 − − 1) [8 − − . {− } (1 − − − − . 68 0 0 04 . 69 [2 − − − . {− } ( − − { } ( − − 1) [4 − . 71 0 {− } ( − − − − 1) [8 − . 72 0 {− } ( − − 1) [8 − . 73 [4 − − − . 74 [2 − { } (1 − − . 75 0 [2 + 0 − − . 76 0 [4 − − − . 77 0 { } ( − − 2) [8 − . {− } ( − − − {− } ( − − 3) [4 + 2 − − . { } (1 − − { } (1 − − . {− } ( − − {− } ( − − 3) [20 − 10 + 2 − . { } (2 − { } (1 − − − . 82 [4 − − − − − − . {− } ( − − − {− } ( − − − 2) [4 − − . 84 [0 + 1 − − − . 85 [2 + 0 − {− } ( − − 1) [8 − − . 86 [2 + 0 − − − . {− } ( − − − {− } ( − − − 3) [12 + 0 − − . { } (1 − { } ( − − 1) [4 − . 89 [4 + 0 − − − 12 + 24 . 90 0 { } ( − − 2) [8 − . 91 [4 − − − − − − . 92 [2 + 0 + 0 − − − − − . {− } ( − − {− } ( − − − − . 94 [2 − − − . 95 [2 + 0 + 0 − − − . 96 [2 + 0 − − − . {− } (1 − − − − . 98 0 [0 − − − . 99 0 [8 − − . 100 [4 − − − . 101 [2 + 0 + 0 − − − . 102 [0 − − − − − . {− } ( − − {− } ( − − 1) [8 + 0 − . 104 [ − − − − − . 105 0 [ − − 16 + 84 . 106 [2 + 0 − − − . 107 0 [ − − . 108 0 0 0 Table 5 HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 51 Appendix B. Table of W K ( t ) , f ( D ; t ) , f ( D ; t ) , and III K ( t ) . Table 6 shows W K ( t ), f ( D ; t ), f ( D ; t ), and III K ( t ) of a virtual knot K up tocrossing number four with a choice of orientations. We remark that these polyno-mials are all reciprocal. W K ( t ) f ( D ; t ) f ( D ; t ) III K ( t )2 . − − − − . − − − . − − − − . − − − − − . − − − − . − − − − . . − − − − . − − − − . − − − . − − − 10 + 4 + 1 04 . − − − − . − − − − − . − − . − − 12 + 6 [ − 12 + 6 [ − . − − − . − − 10 + 5 [ − − . 10 [2 + 0 − − . 11 [6 − − − − − . 12 0 0 0 04 . 13 0 [ − − − . 14 [0 − − − − − . 15 [6 − − − − 12 + 5 + 1 [ − . 16 0 0 0 04 . 17 [2 + 0 − − . 18 [4 − − − − . 19 [2 + 0 − − . 20 [2 − − − − . 21 [ − − − − − . 22 [2 − − − − − − . 23 [2 + 0 − − − . 24 [ − − − − − − . 25 [8 − − 10 + 4 + 1 [ − − . 26 [0 + 1 + 0 − − − − − − . 27 [4 − − − − . 28 [ − − − − . 29 [6 − − − − − . 30 [4 − − − − . 31 0 [4 − − − . 32 [2 + 0 − − − . 33 [4 − − − − . 34 [2 − − − − . 35 [2 + 0 − − . 36 [ − − − − . 37 [8 − − − 10 + 3 + 2 [ − 10 + 3 + 2 [ − . 38 [2 − − − − . 39 [2 − − − − − . 40 [4 − − − − . 41 0 0 0 04 . 42 [2 − − − . 43 [8 − − − . 44 [4 − − − − . 45 [4 − − − − − − . 46 0 0 [2 + 0 − . 47 [0 − − − − . 48 [8 − − − − 12 + 3 + 3 [ − . 49 [2 − − − − . 50 [2 + 0 − − . 51 0 [ − − − . 52 [4 − − − − . 53 [8 − − 10 + 4 + 1 [ − 10 + 4 + 1 [ − . 54 [4 − − − − . 55 0 [2 − − − . 56 0 [ − − − . 57 [2 + 0 − − − . 58 0 [4 − − − . 59 0 [4 − − − . 60 [4 − − − − . 61 [4 − − − − . 62 [6 − − − − − − − . 63 [6 − − − − − . 64 [0 − − − − . 65 [4 + 0 − − − − . 66 [ − − − − − − . 67 [ − − − . 68 0 0 0 04 . 69 [4 − − − − . 70 [2 + 0 − − − − . 71 0 [4 − − − . 72 0 [4 − − − . 73 [8 − − − . 74 [4 − − − − . 75 0 [2 + 0 − − − . 76 0 [ − − − − − − . 77 0 [4 − − − . 78 [6 − − − − 12 + 4 + 1 + 1 [ − − HE INTERSECTION POLYNOMIALS OF A VIRTUAL KNOT 53 . 79 [2 − − − − − − − . 80 [8 − − − − − . 81 [4 − − − − − − − . 82 [8 − − − − 12 + 3 + 3 [ − . 83 [4 − − − − − . 84 [0 + 2 − − − − − − . 85 [4 + 0 − − − . 86 [4 + 0 − − − − − − . 87 [8 − − − − 10 + 2 + 2 + 1 [ − 10 + 2 + 2 + 1 [ − . 88 [0 + 1 − − − − . 89 [8 + 0 − − − − . 90 0 [4 − − − . 91 [8 − − − 12 + 3 + 2 + 1 [ − 12 + 3 + 2 + 1 [ − 12 + 3 + 2 + 14 . 92 [4 + 0 + 0 − − 10 + 2 + 2 + 1 [ − 10 + 2 + 2 + 1 [ − 10 + 2 + 2 + 14 . 93 [4 − − . 94 [4 − − − − . 95 [4 + 0 + 0 − − − − . 96 [4 + 0 − − . 97 [0 − − − − − . 98 0 [4 − − − . 99 0 [8 − − − . 100 [8 − − − . 101 [4 + 0 + 0 − − − − . 102 [0 − − − − − − − . 103 [4 + 1 − − − − − . 104 [ − − − − − − − . 105 0 [ − − − . 106 [4 + 0 − − − − − . 107 0 [0 + 2 − − − . 108 0 0 0 0 Table 6Appendix C. Ohyama-Sakurai’s Theorem Recall the virtual knot K = 4 . 55 in Example 8.14. It is known that K satisfiesthe properties(i) W K ( t ) = 0,(ii) V K ( t ) = 1, where V K ( t ) is the Jones polynomial of K , and(iii) u v ( K ) = 1, where u v ( K ) is the virtual unknotting number of K .Ohyama and Sakurai [24] construct an infinite family of virtual knots with thesame properties (i)–(iii) as K , which can be distinguished by using their Miyazawapolynomials [21]. In this appendix, we construct another infinite family of virtualknots with the property(iv) R K ( A, (cid:126)x ) = − A − A − , where R K ( A, (cid:126)x ) is the Miyazawa polynomial of K in addition to (i)–(iii). We will distinguish the virtual knots by using their inter-section polynomials. Let K n ( n ≥ 0) be the virtual knot presented by the Gaussdiagram G n as shown in Figure 32. ++− − +−+ ++ ++++ ++− −2n 2n………… − −−− − −− −+ − + −++ −+−+− Figure 32 The writhe polynomial and the intersection polynomials of K n are given asfollows; W K n ( t ) = 0 ,I K n ( t ) = − t n +2 − t n +1 + 2 t + 2 − t − n − t − n − ,II K n ( t ) = − ( t n +2 + t − n − ) − t n +1 + t − n − ) − ( t n + t − n )+2( t + t − ) + 4 , and III K n ( t ) = − ( t n +1 + t − n − ) − ( t n + t − n ) + ( t + t − ) + 2 , where we have W K n ( t ) = 0. Therefore K n satisfies the property (i) and K n (cid:54) = K m for any n (cid:54) = m .By removing the horizontal chord of G n , the obtained diagram presents thetrivial knot so that K n satisfies the property (iii).Since the Jones polynomial is derived from the Miyazawa polynomial, it is suf-ficient to prove the property (iv). By applying a skein relation for the chord in themiddle of the Gauss diagram G n , we have R K ( A, (cid:126)x ) = − A − A − immediately. Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan Email address : [email protected] Faculty of Education, University of Yamanashi, Takeda 4-4-37, Kofu, Yamanashi,400-8510, Japan Email address : [email protected] Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan Email address : [email protected] Department of Mathematics, Kobe University, Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan Email address ::