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Kyungpook Mathematical Journal 2018; 58(1): 183-202

Published online March 23, 2018

Delta Moves and Arrow Polynomials of Virtual Knots

Myeong–Ju Jeong* and Chan–Young Park

Department of Mathematics, Korea Science Academy 111 Baekyang Gwanmun–Ro, Busanjin–Gu, Busan 614–822, Korea, e-mail : mjjeong@kaist.ac.kr, Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 702–701, Korea, e-mail : chnypark@knu.ac.kr

Received: March 16, 2017; Accepted: August 9, 2017

Δ-moves are closely related with a Vassiliev invariant of degree 2. For classical knots, M. Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single Δ-move. The first author extended the Okada’s result for virtual knots by using a Vassiliev invariant of virtual knots of type 2 which is induced from the Kauffman polynomial of a virtual knot. The arrow polynomial is a generalization of the Kauffman polynomial. We will generalize this result by using Vassiliev invariants of type 2 induced from the arrow polynomial of a virtual knot and give a lower bound for the number of Δ-moves transforming K1 to K2 if two virtual knots K1 and K2 are related by a finite sequence of Δ-moves.

Keywords: Δ,-move, arrow polynomial, Miyazawa polynomial, virtual knot, Vassiliev invariant

We will generalize a theorem on Δ-moves of virtual knots by using the arrow polynomial. Any knot can be unknotted by a finite sequence of Δ-moves [20] and the second coefficient of the Conway polynomial gives a lower bound for the number of Δ-moves in the sequence [26]. The second coefficient of the Conway polynomial is a Vassiliev invariant and we can extend the result to virtual knots by using a Vassiliev invariant induced from the Kauffman polynomial of virtual knots [11]. The Kauffman polynomial of virtual knots can be generalized to an arrow polynomial of virtual knots [2] and we give a necessary condition for two virtual knots to be related by a Δ-move by using numerical Vassiliev invariants induced from the arrow polynomial of virtual knots.

In 1989 H. Murakami and Y. Nakanishi [20] introduced the Δ-move for link diagrams as illustrated in Figure 1. The Δ-Gordian distancedGΔ(L,L) of links L and L′ is defined to be the minimal number of the Δ-moves to transform a diagram of L to a diagram of L′. They showed that two oriented ordered n-component links L and L′ can be related by a finite sequence of Δ-moves if L and L′ are link-homologous. In particular Δ-moves unknot all knots. The Δ-unknotting number uΔ(K) for a knot K is defined to be dGΔ(K,O), where O denotes the trivial knot. If two links are related by a finite sequence of Δ-moves then they are said to be Δ-homotopic. We will see that the Δ-move is not an unknotting operation for virtual knots.

A virtual link diagram is closed curves generically immersed in the 2–dimensional Euclidean space. A double point of the curve is either a (classical) crossing or a virtual crossing. A virtual crossing is denoted by an encircled singular point. In particular if the virtual link diagram has one component then it is called a virtual knot diagram. See Figure 2 for a virtual knot diagram with six crossings and two virtual crossings.

Two virtual link diagrams are said to be equivalent if there is a finite sequence of Reidemeister moves and virtual moves transforming one diagram to the other diagram. See Figure 3 for Reidemeister moves and Figure 4 for virtual moves. A virtual link is defined to be an equivalence class of a virtual link diagram under the equivalence relation. A virtual link with one circle component is called a virtual knot. If two knot diagrams are equivalent then they can be related by a finite sequence of Reidemeister moves [3, 15].

In this paper all virtual links are assumed to be oriented. It is well-known that any Δ-move on an oriented diagram can be realized by the move as shown in Figure 5. We extend Δ-moves, Δ-homotopy, Δ-Gordian distance and Δ-unknotting number for virtual knots and links naturally.

H. A. Dye and L. H. Kauffman introduced the arrow polynomial of a virtual link. They used the state expansion of a crossing as shown in Figure 6. If we resolve a crossing in a virtual link diagram via the state expansion then it may have some nodal cusps. By replacing the nodal cusps with poles as shown in Figure 7, we get a diagram with poles.

Y. Miyazwa introduced a multi-variable polynomial invariant of virtual links by using decorated virtual magnetic graph diagram [17, 18]. A. Ishii simplified the polynomial by using pole diagrams [8]. A pole on a strand of a virtual link diagram is a unit normal vector with the initial point on the strand and it is denoted by a small line segment attached on the diagram. A virtual link diagram allowed to have poles is called a pole diagram or a polar link diagram. The two fragments of the diagram attached to a pole are assumed to be oriented either inward to the pole or outward from the pole as shown in Figure 8. See Figure 9 for polar link diagrams.

We can naturally extend Reidemeister moves and virtual moves of virtual link diagrams to pole diagrams. The local moves on polar link diagrams as shown in Figure 10 are called polar moves. Two polar link diagrams are said to be equivalent if they are related by a finite sequence of Reidemeister moves, virtual moves and polar moves. A polar link is an equivalence class of a polar link diagram under Reidemeister moves, virtual moves and polar moves.

Forbidden moves unknot all virtual knots. There are two kinds of forbidden moves, an FH-move and an FT-move, on virtual knot diagrams as shown in Figure 11. By representing forbidden moves in Gauss diagrams, S. Nelson showed that any virtual knot can be unknotted by applying finitely many forbidden moves [24]. T. Kanenobu showed that any classical crossing of a virtual knot diagram can be transformed to a virtual crossing by using the forbidden moves, the Reidemeister moves and the virtual moves [9]. Since a virtual knot diagram with only virtual crossing represents the trivial knot [15], we see that the family of forbidden moves is an unknotting operation.

T. Kanenobu also showed that a Δ-move can be realized by a finite sequence of the Reidemeister moves, the virtual moves and the forbidden moves [9]. Y. Nakanishi and T. Shibuya studied Δ link homotopy and gave a necessary condition for two links to be Δ link homotopic in terms of Conway polynomials [21, 22].

It can be easily proved that any long virtual knot can be unknotted by a finite sequence of forbidden moves by using Gauss diagrams. In [3], M. Goussarov, M. Polyak and O. Viro introduced finite type invariants of virtual knots via semi-virtual crossings. Then they gave combinatorial representations of several finite type invariants of low degrees by using Gauss diagram formulae. Every finite type invariant is a Vassiliev invariant and there is no non-trivial finite type invariant of degree ≤ 2 for virtual knots. But there is two independent finite type invariants of degree 2 for long virtual knots.

M. Sakurai and the first author independently calculated the difference of the values of a finite type invariant of degree 2 for two long virtual knots K and K′ which are related by a forbidden move and showed that it is 0 or 1 [10, 27]. From this we can get a lower bound for the number of forbidden moves to unknot a long virtual knot by using the two finite type invariants of degree 2.

K. Habiro [7] introduced Cn-moves and gave relationships between Cn-moves and Vassiliev invariants of knots. For each positive integer n, a Cn-move is a local move of knots as shown in Figure 12. He showed that two knots are related by a finite sequence of Cn-moves if and only if they take the same values for all Vassiliev invariants of degree < n. M. N. Goussarov proved similar results independently [4, 5, 6]. In particular a C2-move is equivalent to a Δ-move. Since any knot can be unknotted by a finite sequence of Δ-moves, we see that all Vassiliev invariants of degree < 2 are constants.

If a Δ-move on a link happens on the same component then it is called a self-delta move. T. Kanenobu and R. Nikkuni studied self-delta moves and gave several relationships among the values of Vassiliev invariants induced from the HOMFLY polynomials of a delta skein quadruple [16].

In [23] Y. Nakanishi and Y. Ohyama studied Ck-distances of knots and showed that, for a given knot K, there are infinitely many knots which take the same value for any Vassiliev invariant of degree < n and have some properties for Ck-distance.

The Jones-Kauffman polynomial of knots can be extended to virtual knots but so far we do not have a virtual knot invariant whose restriction to classical knots is the Conway polynomial. But the second coefficient of the Conway polynomial is a Vassiliev invariant of type 2 and there are infinitely many independent Vassiliev invariants of type 2 for virtual knots. In particular we may obtain many numerical Vassiliev invariants from the arrow polynomial. We show that the Okada’s result on Δ-moves can be extended to virtual knots by using numerical Vassiliev invariants of type 2.

In Section 2, we treat Vassiliev invariants and the arrow polynomial and induce numerical Vassiliev invariants of type n from the arrow polynomial for each n ∈ ℕ. In Section 3, we represent a Δ-move of virtual knots in terms of singular virtual knots. Let n0, n1, … be integers such that n0 ≠ 0. We prove that if two virtual knots K1 and K2 are related by a single Δ-move then hK1(1)-hK2(1)=±96, where hK(A) = fK(An0,An1, …) is the polynomial obtained from the arrow polynomial fK(A, Y1, …) of a virtual knot K by change of variables. We give examples for Δ-homotopy and the Δ-Gordian distance of virtual knots.

2. Numerical Vassiliev Invariants induced from the Arrow Polynomial

A (virtual) link invariant v which takes values in an abelian group can be extended to a singular (virtual) link invariant by using the Vassiliev skein relation: v(L×) = v(L+)−v(L), where L×, L+ and L are singular (virtual) link diagrams which are identical except the indicated local parts as illustrated in Figure 13.

A (virtual) link invariant v is called a Vassiliev invariant of type n if v vanishes on singular (virtual) links with more than n singular points. The smallest such nonnegative integer n is called the degree of v.

We may get Vassiliev invariants from the quantum polynomial invariants of knots. For example the coefficient an(K) of zn in the Conway polynomial ∇K(z) of a knot K is a Vassiliev invariant of type n [1]. In particular the Vassiliev invariant a2(K) of type 2 gives a lower bound for the Δ-Gordian distance of two knots. M. Okada showed that a2(K) − a2(K′) = ±1, if two knots K and K′ are related by a Δ-move [26]. Immediately, we see that

dGΔ(K,K)a2(K)-a2(K).

This is useful to determine the Δ-Gordian distance of knots. In particular the Δ-unknotting numbers for torus knots, some positive knots and positive closed 3-braids are determined by the second coefficient of the Conway polynomial [25].

Let XK(A) be the Kauffman polynomial of a virtual knot K. If we substitute A with ex and expand it in the Maclaurin series then the coefficient of xn in the series is a Vassiliev invariant of type n [15]. Let K2 be a virtual knot obtained from a virtual knot K1 by applying a single Δ-move. Then the coefficient of x2 in the Maclaurin series of XK1 (ex) and that of XK2 (ex) differ by 48 [11].

L. H. Kauffman introduced state models of the Jones polynomial for classical knots and links [12, 14] and then extended it for virtual knots and links [15]. Y. Miyazawa gave several polynomial invariants of virtual knots. In particular he constructed a multi-variable polynomial invariant of virtual links, which generalize the Kauffman polynomial [18].

H. A. Dye and L. H. Kauffman defined the arrow polynomial of a virtual link which is a generalization of the Kauffman polynomial and showed that we can get a lower bound for the virtual crossing number from the arrow polynomial [2].

The multi-variable Miyazawa polynomial and the arrow polynomial were found independently but basically are the same. Actually we may get the arrow polynomial from the multi-variable Miyazawa polynomial and vice versa by a suitable change of variables. Recall that all virtual links are assumed to be oriented in this paper. Although we have interest on virtual links, we will define the arrow bracket polynomial 〈L〉 of a polar link diagram L. This extension will be useful to prove some lemmas.

We modify the original definition of the arrow polynomial given by H. A. Dye and L. H. Kauffman a little bit. We define the arrow bracket polynomial 〈L〉 of a polar link diagram L by using the following relations.

• 1. 〈L+〉 = AL0〉+A−1L〉 and 〈L〉 = A−1L0〉+AL〉, where L+, L, L0, and L are polar link diagrams as shown in Figure 14.

• 2. 〈L1〉 = 〈L2〉, if L1 and L2 can be related by a polar move or a virtual move.

• 3. 〈O〉 = 1, 〈Om〉 = Ym, 〈L′ ∪ O〉 = (−A2A−2) 〈L′〉 and 〈L′ ∪ Om〉 = (−A2A−2)YmL′〉 for any polar link diagram L′, where O is the trivial knot diagram and Om is the polar link diagram with 2m poles as shown in Figure 15.

If a function v from the set of all polar link diagrams to a set takes the same value for any pair of equivalent polar link diagrams, then it is called an invariant of polar links. We define the sign of a crossing of a polar link diagram as shown in Figure 16. The writhe w(L) of a pole diagram L is defined to be the sum of signs of all crossings of L.

H. A. Dye and L. H. Kauffman showed that the polynomial 〈L〉 of a virtual link is invariant under the second Reidemeister moves, the third Reidemeister moves and the virtual moves. Note that w(L) is invariant under all Reidemeister moves and virtual moves except for the first Reidemeister move. They obtained an invariant fL(A, Y1, …) by normalizing 〈L〉 by the formula

fL(A,Y1,)=(-A3)-w(L)L.

Following their argument we can see that fL(A, Y1, …) is a polar link invariant. The two polynomials 〈L〉 and fL(A, Y1, …) take values in the polynomial ring ℤ[A,A−1, Y1, Y2, …]. In particular if we put Y1 = Y2 = … = 1 in fL(A, Y1, Y2, …) we get the Kauffman polynomial XL(A) of a virtual link L.

Let L+, L, L0 and L be polar link diagrams which are identical except for the shown parts in Figure 14. Since fL = (−A3)w(L)L〉 for any polar link diagram L, and

{L+=AL0+A-1L,L-=A-1L0+AL,

we get the following

Lemma 2.1

([2, 13]) For the quadruple (L+, L, L0, L) of polar link diagrams as shown in Figure 14, we get identities

{fL+(A)=-A-2fL0(A)-A-4fLandfL-(A)=-A2fL0(A)-A4fL.

Similarly to the case of virtual links, we define singular polar links and Vassiliev invariants of polar links. Let n0, n1, … be integers such that n0 ≠ 0. For a polar link diagram L, we define hL(A) ∈ ℤ[A,A−1] by

hL(A)=fL(An0,An1,)

where fL(A, Y1, …) ∈ ℤ[A,A−1, Y1, …] is the arrow polynomial of L.

Let vn(L) be the coefficient of xn in the Maclaurin series of hL(ex) for a polar link L. L. H. Kauffman showed that the coefficients in the power expansion of XK(ex) are Vassiliev invariants [15]. We can also get numerical Vassiliev invariants vn(L) of type n from the arrow polynomial as following

Lemma 2.2

For integers n0, n1,with n0 ≠ 0, let hL(A) be obtained from the arrow polynomial by setting hL(A) = fL(An0,An1, …). Then the coefficient vn(L) of xn in the Maclaurin series of hL(ex) is a Vassiliev invariant of type n for polar links.

Proof

Let L+, L, L0, L be polar link diagrams as in Figure 14. By Lemma 2.1, we see that

hL+(A)-hL-(A)=(A2-A-2)hL0(A)+(A4-A-4)hL(A).

We substitute ex for A. Then

A2-A-2=e2x-e-2x=2(2x1!+(2x)33!+(2x)55!+)

and

A4-A-4=e4x-e-4x=2(4x1!+(4x)33!+(4x)55!+).

Then hL+ (ex) − hL (ex) is a multiple of x. If L has a singular point then the expansion of hL(ex) is a multiple of x. Inductively we see that hL(ex) is a multiple of xm if L has m singular points. Therefore we see that vn(L) is a Vassiliev invariant of type n.

We extended the Okada’s theorem for virtual knots by using the normalized bracket polynomial XK(A). The crossing change, which transforms a positive crossing to a negative crossing or vice versa, is an unknotting operation for classical links but it is not for virtual ones. So the skein relations which define the Conway polynomial are not enough to define an invariant for virtual links. H. Murakami showed that VK(2)(1)=-6a2(K) for any knot K, where VK(t) = XK(t−1/4) [19]. We will see that any Vassiliev invariant of type < 2 is invariant under the Δ-move. Assume that K1 and K2 are knots related by a Δ-move. We have proved that XK1(1)+XK1(1)2-XK2(1)+XK2(1)2=±48 in [11]. Therefore

XK1(1)-XK2(1)=±96.

For integers n0, n1, … with n0 ≠ 0, let vn(L) be the coefficient of hL(ex) where hL(A) = fL(An0,An1, …). If K1 and K2 are virtual knots related by a Δ-move, then we represent the difference of K1 and K2 as a difference of two singular virtual knots in a Vassiliev skein module. Then by calculating v2(K1) − v2(K2) we will show that hK1(1)-hK2(1)=±96.

Let be the set of singular virtual knot diagrams modulo the equivalence relation of virtual knot diagrams and be the free ℤ-module generated by . We also denote by the submodule of generated by the relations

L×=L+-L-,

where L×, L+ and L are singular virtual knot diagrams which are identical except for the shown parts in Figure 13. We will often denote the equivalence class of a singular virtual knot diagram K in simply by K too. We extend a Vassiliev invariant on linearly and it induces a natural quotient map on .

Lemma 3.1

Let K1and K2be virtual knots related by a Δ-move and K×11×12andK×21×22be 2-singular knots as shown in Figure 17. Then, in we have

K1-K2=K×11×12-K×21×22.
Proof

See Figure 18.

From now on n0, n1, … are assumed to be integers such that n0 ≠ 0, unless otherwise stated. Let hL(A) = fL(An0,An1, …) be obtained from the arrow polynomial fL(A, Y1, Y2, …) by substituting variables. Since vn(L) is defined as the coefficient of xn in the Maclaurin series of hL(ex), vn(L) can be given as vn(L)=HL(n)(0)n!, where HL(x) = hL(ex).

Then we see that

{v0(L)=HL(0)=hL(1),v1(L)=HL(0)=hL(1),v2(L)=HL(0)2=hL(1)+hL(1)2.

We also see that

{hL(1)=v0(L),hL(1)=v1(L),hL(1)=2v2(L)-v1(L).

Lemma 3.2

For any polar link diagram L, v0(L) = (−2)μ(L)−1where μ(L) is the number of components of L.

Proof

Since v0(L) = hL(1), we will show that hL(1) = fL(1, 1, …) = (−2)μ(L)−1. We will apply mathematical induction on the number of crossings of L. If L has no crossing then hL(1) = fL(1, 1, …) = (−2)μ(L)−1. Assume that L has a positive crossing. Let L = L+, L, L0 and L be polar link diagrams as shown in Figure 14. Since

{fL+(A,K1,)=-A-2fL0(A,K1,)-A-4fL(A,K1,),fL-(A,K1,)=-A2fL0(A,K1,)-A4fL(A,K1,),

we see that

hL+(1)=-fL0(1,1,)-fL(1,1,)=hL-(1).

Assume that the polar link diagram L is as shown in Figure 19. Then μ(L0) = μ(L) + 1 and μ(L) = μ(L). By induction hypothesis, we get the equations

{hL0(1)=(-2)μ(L0)-1=(-2)μ(L),hL(1)=(-2)μ(L)-1=(-2)μ(L)-1.

Therefore

hL(1)=hL+(1)=-hL0(1)-hL(1)=-(-2)μ(L)-(-2)μ(L)-1=(-2)μ(L)-1.

Assume that the polar link diagram L is as shown in Figure 20. Then μ(L0) = μ(L) − 1 and μ(L) = μ(L) − 1. By induction hypothesis, we get the equations

{hL0(1)=(-2)μ(L0)-1=(-2)μ(L)-2,hL(1)=(-2)μ(L)-1=(-2)μ(L)-2.

Therefore

hL(1)=hL+(1)=-hL0(1)-hL(1)=-(-2)μ(L)-2-(-2)μ(L)-2=(-2)μ(L)-1.

Similarly we can show the statement when L has a negative crossing.

For a singular polar link diagram L× we can calculate v1(L×) and v2(L×) as following

Lemma 3.3

Let L×, L+, L, L0and Lbe polar link diagrams as shown in Figure 21. Then vi(L×) = 4vi−1(L0) + 8vi−1(L) for i = 1, 2.

Proof

Since fL+fL = (A2A−2)fL0 + (A4A−4)fL by Lemma 2.1 and hL(A) = fL(An0,An1, …), we see that

hL×(A)=hL+(A)-hL-(A)=(A2-A-2)hL0(A)+(A4-A-4)hL(A).

By substituting the variable A with ex, we have

hL×(ex)=hL+(ex)-hL-(ex)=(e2x-e-2x)hL0(ex)+(e4x-e-4x)hL(ex)=(4x+83x3+)hL0(ex)+(8x+643x3+)hL(ex).

Since vn(L) is the coefficient of xn in the Maclaurin series of hL(ex), we see that vi(L×) = 4vi−1(L0) + 8vi−1(L) for i = 1, 2.

Now we can evaluate v2(L××) for a polar link diagram L×× with two singular points as following

Lemma 3.4

Let L××be a polar link diagram with two singular points and v2(L) be the coefficient of x2in the Maclaurin series of hL(ex). Then

v2(L××)=16((-2)μ(L00)-1+2(-2)μ(L0)-1+2(-2)μ(L0)-1+4(-2)μ(L)-1).
Proof

By applying Lemma 3.3, we see that

v2(L××)=4v1(L0×)+8v1(L×)=4(4v0(L00)+8v0(L0))+8(4v0(L0)+8v0(L))=16(v0(L00)+2v0(L0)+2v0(L0)+4v0(L))

By Lemma 3.2, v0(L) = (−2)μ(L)−1 for any polar link diagram L.

Then we have

v2(L××)=16((-2)μ(L00)-1+2(-2)μ(L0)-1+2(-2)μ(L0)-1+4(-2)μ(L)-1).

Now we get a necessary condition for two virtual knots to be Δ-homotopic by using the arrow polynomial as following

Theorem 3.5

Let n0, n1,be integers such that n0 ≠ 0. Let hL(A) be obtained from the arrow polynomial by setting hL(A) = fL(An0,An1, …). If K1and K2are virtual knots related by a single Δ-move, then

hK1(1)-hK2(1)=±96.
Proof

Let K1 and K2 be virtual knots related by a Δ-move and K×11×12 and K×21×22 be singular knots as shown in Figure 17. By applying Lemma 3.1 and Lemma 3.4, we see that

v2(K1)-v2(K2)=v2(K×11×12)-v2(K×21×22)=16((-2)μ(K00)-1+2(-2)μ(K0)-1+2(-2)μ(K0)-1+4(-2)μ(K)-1)-16((-2)μ(K00)-1+2(-2)μ(K0)-1+2(-2)μ(K0)-1+4(-2)μ(K)-1).

There are two types of K1 as shown in Figure 22.

Assume that K1 is of type 1. Then K00, K0∞, K∞0, K∞∞, K00,K0,K0,K are as shown in Figure 23. Since μ(K00) = 3, μ(K0∞) = μ(K∞0) = 2, μ(K∞∞) = 1, μ(K00)=μ(K0)=μ(K0)=1, and μ(K)=2, we have

v2(K1)-v2(K2)=48.

Assume that K1 is of type 2. Then μ(K00) = μ(K0∞) = μ(K∞0) = 1, μ(K∞∞) = 2, μ(K00)=3,μ(K0)=2,μ(K0)=2,μ(K)=1. So we see that

v2(K1)-v2(K2)=-48.

Since K1 and K2 are related by a Δ-move, by Lemma 3.1 v1(K1) = v1(K2). Since hL(1)=2v2(L)-v1(L) for any polar link L, we see that

hK1(1)-hK2(1)=2v2(K1)-v1(K1)-2v2(K2)+v1(K2)=±96.

Take hL(A) = fL(A, 1, …) = XL(A) in Theorem 3.5. If K1 and K2 are related by a Δ-move then by Lemma 3.1, v1(K1) = v1(K2). Since v1(L)=hL(1), we have the following

Corollary 3.6

([11]) Let K2be a virtual knot obtained from a virtual knot K1by applying a single Δ-move. Then

vx2(K1)-vx2(K2)=±48,

wherevx2(K)=XK(1)+XK(1)2.

We have a lower bound for the Δ-Gordian distance of homotopic virtual knots as following

Corollary 3.7

Let n0, n1,be fixed integers such that n0 ≠ 0. Let hK(A) = fK(An0, An1, …) be obtained from the arrow polynomial fL(A, Y1, …) by change of variables. If K1and K2are Δ-homotopic virtual knots then

dGΔ(K1,K2)hK1(1)-hK2(1)96.

The following example shows that Theorem 3.5 is useful to determine whether two given virtual knots are Δ-homotopic or not.

Example 3.8

Let K be the virtual knot as shown in Figure 24. Then by calculating the arrow polynomial of K we see that

fK(A,Y1,)=A8+(1-A8)Y12and XK(A)=1.

Take hL(A) = fL(A, A, 1, 1, …) for any virtual link L. Then hK(1)=-32 and XK(1)=0. Although we may not determine whether K is Δ-homotopic to the trivial knot or not by XK(A), we see that K is not Δ-homotopic to the trivial knot by applying Theorem 3.5.

If two virtual knots are Δ-homotopic then they are homotopic. We see that the knot K in Figure 24 is neither Δ-homotopic nor homotopic to the trivial knot. Therefore the Δ-move is not an unknotting operation for virtual knots.

In the following example we can see that Corollary 3.7 is useful to find the Δ-Gordian distance between two Δ-homotopic virtual knots.

Example 3.9

Let K1 and K2 be the virtual knots as shown in Figure 25. Then

fK1(A,Y1,)=2-2A4+2A8-A12+(-A-6+2A-2-2A2+A6)Y1

and

fK2(A,Y1,)=A-8-A-4+2-2A4+A8+(-A6+A10)Y1.

Take hL(A) = fL(A,A6, 1, 1, …) for any virtual link L. Since

hK1(A)=1

and

hK2(A)=A-8-A-4+2-2A4+A8-A12-A10,

we have

hK1(1)-hK2(1)=-192.

If we assume that K1 and K2 are Δ-homotopic, then by Corollary 3.7, we get a lower bound for the Δ-Gordian distance dGΔ(K1,K2) between K1 and K2 as following

dGΔ(K1,K2)2.

We can see that K1 and K2 are Δ-homotopic and dGΔ(K1,K2)=2 from Figure 26.

The work by the first author was supported by the Ministry of Science, ICT and Future Planning.

Fig. 1. A Δ-move.
Fig. 3. Reidemeister moves.
Fig. 4. Virtual moves.
Fig. 6. State expansion.
Fig. 11. Forbidden moves.
Fig. 12. A Cn-move
Fig. 16. The sign of a crossing.
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