Article
Kyungpook Mathematical Journal 2021; 61(1): 205212
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
Forbidden Detour Number on Virtual Knot
Shun Yoshiike, Kazuhiro Ichihara*
Nihon University Buzan Junior & Senior High School, 54010 Otsuka, Bunkyoku, Tokyo 1120012, Japan
email : s6115m15@math.chs.nihonu.ac.jp
College of Humanities and Sciences, Nihon University, 32540 Sakurajosui, Setagayaku, Tokyo 1568550, Japan
email : ichihara.kazuhiro@nihonu.ac.jp
Received: August 30, 2019; Revised: June 2, 2020; Accepted: June 4, 2020
Abstract
We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour moves necessary to transform a diagram of the virtual knot into the trivial knot diagram. Some upper and lower bounds on the forbidden detour number are given in terms of the minimal number of real crossings or the coefficients of the affine index polynomial of the virtual knot.
Keywords: virtual knot, forbidden move, forbidden number.
1. Introduction
As a generalization of (classical) knots in 3space, Kauffman introduced

Figure 1. Forbidden moves
F and forbidden detour moveF_{d}
In the studies of forbidden moves in [3], Kanenobu introduced and used several moves for virtual knot diagrams. Two of them, called
In this paper, we study this move, and obtain the following.
Theorem 1.1.
Let
Remark 1.2.
We note that the

Figure 2.
F_{2} move
In virtue of the theorem above, we can introduce the following notion.
Definition 1.3.
Let
We next consider lower bounds on the forbidden detour numbers of virtual knots. To obtain lower bounds, the variation of an invariant, called the affine index polynomial, under a forbidden detour move, plays a key role. In fact, we have the following.
Theorem 1.4.
Let
In the following, our terminology about virtual knot and Gauss diagram follows from those in [1].
2. Forbidden Detour Number
A
On the other hand, forbidden moves and forbidden detour moves can change virtual knots by modifying Gauss diagrams. In fact, as claimed in [1, Section 2], the forbidden detour move gives the effect on Gauss diagrams of switching the head of one arrow with the tail of an adjacent arrow. See Figure 3.

Figure 3. The effect of an
F_{d} move on Gauss diagrams
In the following, we call the move on Gauss diagrams corresponding to a forbidden detour move also a forbidden detour move on Gauss diagrams.
Let us remove

Figure 4. Sequences of forbidden detour moves (
F_{d} moves)
Here, since
Let
Then, since
On the other hand, when
Then, since
Consequently,
3. Lower Bound of Forbidden Detour Number
In this section, we consider the lower bound for the forbidden detour number of a virtual knot. Our argument bases on the following result of Sakurai given in [9] for the forbidden move.
Let
Then
holds for some integers
Theorem 3.1.
Let
for some integers ℓ and
To prove this, we recall some definitions about the affine index polynomial used in [9].
First, we define virtual knot invariants by indexes of arrows for a Gauss diagram. Let
For an arrow
The
Then the
and, we define the
We remark that this is different from the original definition by Kauffman in [5]. However Sakurai showed in [9, Proposition
Let
and
Also note that this moves preserves the indexes and the signs of all the other arrows.
Therefore, we obtain the following.
It concludes that
holds for some integers ℓ and
Then, for each
It follows that the coefficients
Thus, for
It concludes that
which completes the proof.
References
 A. S. Crans, B. Mellor, and S. Ganzell. The forbidden number of a knot, Kyungpook Math. J. 55(2) (2015), 485506.
 M. Goussarov, M. Polyak, and O. Viro. Finitetype invariants of classical and virtual knots, Topology 39(5) (2000), 10451068.
 T. Kanenobu. Forbidden moves unknot a virtual knot, J. Knot Theory Ramifications 10(1) (2001), 8996.
 L. H. Kauﬀman. Virtual knot theory, European J. Combin. 20(7) (1999), 663690.
 L. H. Kauﬀman. An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), 1340007. 30 pp.
 S. V. Matveev. Generalized surgeries of threedimensional manifolds and representations of homology spheres, Mat. Zametki 42(2) (1987), 268278.
 H. Murakami and Y. Nakanishi. On a certain move generating linkhomology, Math. Ann. 284(1) (1989), 7589.
 S. Nelson. Unknotting virtual knots with Gauss diagram forbidden moves, J. Knot Theory Ramifications 10(6) (2001), 931935.
 M. Sakurai. An affine index polynomial invariant and the forbidden move of virtual knots, J. Knot Theory Ramifications 25(7)(2016), 1650040. 13 pp.
 S. Satoh and K. Taniguchi. The writhes of a virtual knot, Fund. Math. 225(1) (2014), 327342.