Kyungpook Mathematical Journal 2018; 58(1): 183-202
Published online March 23, 2018
Copyright © Kyungpook Mathematical Journal.
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 : email@example.com, Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 702–701, Korea, e-mail : firstname.lastname@example.org
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
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  and the second coefficient of the Conway polynomial gives a lower bound for the number of Δ-moves in the sequence . 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 . The Kauffman polynomial of virtual knots can be generalized to an arrow polynomial of virtual knots  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  introduced the Δ-
Two virtual link diagrams are said to be
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 . A
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
Forbidden moves unknot all virtual knots. There are two kinds of
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 . 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 , 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. Habiro  introduced
If a Δ-move on a link happens on the same component then it is called a
In  Y. Nakanishi and Y. Ohyama studied
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
2. Numerical Vassiliev Invariants induced from the Arrow Polynomial
A (virtual) link invariant
A (virtual) link invariant
We may get Vassiliev invariants from the quantum polynomial invariants of knots. For example the coefficient
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 .
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 . 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 .
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 .
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 〈
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+〉 = A〈 L0〉+ A−1〈 L∞〉 and 〈 L−〉 = A−1〈 L0〉+ A〈 L∞〉, where L+ , L− , L0, and L∞ are polar link diagrams as shown in Figure 14.
L1〉 = 〈 L2〉, if L1 and L2 can be related by a polar move or a virtual move.
O〉 = 1, 〈 O m〉 = Y m, 〈 L′ ∪ O〉 = (− A2 − A−2) 〈 L′〉 and 〈 L′ ∪ O m〉 = (− A2 − A−2) Y m〈 L′〉 for any polar link diagram L′, where Ois the trivial knot diagram and O mis the polar link diagram with 2 mpoles as shown in Figure 15.
If a function
H. A. Dye and L. H. Kauffman showed that the polynomial 〈
Following their argument we can see that
we get the following
Similarly to the case of virtual links, we define singular polar links and Vassiliev invariants of polar links. Let
We extended the Okada’s theorem for virtual knots by using the normalized bracket polynomial
2. Main Theorem
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
See Figure 18.
From now on
Then we see that
We also see that
we see that
Assume that the polar link diagram
Assume that the polar link diagram
Similarly we can show the statement when
For a singular polar link diagram
By substituting the variable
Now we can evaluate
By applying Lemma 3.3, we see that
By Lemma 3.2,
Then we have
Now we get a necessary condition for two virtual knots to be Δ-homotopic by using the arrow polynomial as following
There are two types of
We have a lower bound for the Δ-Gordian distance of homotopic virtual knots as following
The following example shows that Theorem 3.5 is useful to determine whether two given virtual knots are Δ-homotopic or not.
If two virtual knots are Δ-homotopic then they are homotopic. We see that the knot
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.
If we assume that
We can see that
The work by the first author was supported by the Ministry of Science, ICT and Future Planning.
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