Article
Kyungpook Mathematical Journal 2021; 61(1): 205-212
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, 5-40-10 Otsuka, Bunkyo-ku, Tokyo 112-0012, Japan
e-mail : s6115m15@math.chs.nihon-u.ac.jp
College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan
e-mail : ichihara.kazuhiro@nihon-u.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 3-space, Kauffman introduced
-
Figure 1. Forbidden moves
F and forbidden detour moveFd
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.
F2 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
Fd -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 (
Fd -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.
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