Article
Kyungpook Mathematical Journal 2022; 62(2): 347361
Published online June 30, 2022
Copyright © Kyungpook Mathematical Journal.
The Second Reidemeister Moves and Colorings of Virtual Knot Diagrams
Myeong–Ju Jeong*, Yunjae Kim
Department of Mathematics, Korea Science Academy, 111 Baekyang Gwanmun–Ro, Busanjin–Gu, Busan 47162 Korea
email : mjjeong@ksa.kaist.ac.kr
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566 Korea
email : kimholzi@gmail.com
Received: April 22, 2020; Revised: February 10, 2022; Accepted: February 22, 2022
Abstract
Two virtual knot diagrams are said to be
Keywords: virtual knot, Reidemeister moves, coloring, knot polynomial
1. Introduction
L. H. Kauffman introduced virtual knots, which generalize classical knots, and virtual knot invariants ([7, 8]). Nonclassical virtual knots can be represented in a thickened orientable surface of genus greater than 0. Some invariants of classical knots, such as the bracket polynomial, the fundamental group and the quandle, can be naturally extended to virtual knots. Other invariants have be developed for virtual knots. In [5, 6], we gave polynomials related to virtual knot diagrams to study Reidemeister moves. The polynomials provide us with information about the third Reidemeister moves in the deformation of a virtual knot diagram. In this paper we introduce a polynomial, obtained from a coloring of a virtual knot diagram, and use it to give a lower bound for the number of second Reidemeister moves in a Reidermeister sequence between two virtual knot diagrams. We also give examples in which we fully determine the minimal number of Reidemeister moves by using the new polynomial, the
In this paper all knot diagrams are assumed to be oriented. We define the

Figure 1. The sign of a crossing.
In [5], we introduced a polynomial
In [6], we introduced a bridge diagram and parity polynomials of a virtual knot diagram. If we calculate the parity polynomials of two equivalent virtual knot diagrams, the result in the paper gives a lower bound for the number of third Reidemeister moves needed to deform a diagram to the other one.
All Vassiliev invariants of degree
was introduced, where the sum runs over all crossings
Cheng used a similar idea to define a polynomial invariant of virtual knots. Based on Manturov's parity axioms ([10]), Cheng ([1]) introduced the odd writhe polynomial
where
The
Let
The
A

Figure 4. A virtual knot diagram.

Figure 5. Reidemeister moves.
The moves of diagrams shown in Figure 6 are called

Figure 6. Virtual moves.
As we can for a classical knot diagram, we can define a virtual knot from its Gauss diagram by disregarding virtual crossings. See Figure 7, where

Figure 7. Virtual moves.
For two virtually isotopic knot diagrams
In Section we introduce a coloring of a virtual knot diagram
2. A Polynomial of a Virtual Knot Diagram
For a given virtual knot diagram
See Figure 10 for a coloring of a virtual knot diagram. In Figure 9 and Figure 10, the sign of each crossing is denoted by
Assume that
We define
For example if
For a classical knot diagram
If
Similarly for the other types of the first Reidemeister move we get similar result. Then we have the following
Lemma 2.1. If
There are two types of second Reidemeister moves considering the orientations of the two strands involved with second Reidemeister moves as shown in Figure 13 and Figure 14. The two crossings
Lemma 2.2. If
for some integer
Then
for some integer
We note that
Lemma 2.3. If
Similarly if
From the coloring of
By combining Lemma 2.1, Lemma 2.2 and Lemma 2.3 we get a lower bound for the number of second Reidemeister moves deforming a virtual knot diagram to another one as following.
Theorem 2.4. If
where
The difference of cowrithes of two equivalent knot diagrams gives a lower bound for the number of second Reidemeister moves and third Reidemeister moves needed to deform one to the other ([3]). The result can be naturally extended to virtual knot diagrams, but cowrithe is invariant under second Reidemeister move of the type in Figure 14. We will denote the trivial knot diagram by
Example 2.5. Let
If two given virtual knot diagrams seem to be equivalent, we can try to deform one to the other by using Reidemeister moves and virtual moves. By using grades of crossings we may get hints to check whether the two virtual knot diagrams are equivalent or not. If the grade of a crossing is 0 then it may be cancelled by a first Reidemeister move. If two crossings
In [5], we defined a polynomial
For
Theorem 2.6. ([5]) Let
where
Example 2.7. Let

C(K_{2}) c_{1} c_{2} c_{3} c_{4} c_{5} c_{6} c_{7} c_{8} Sign + +   +  + + Grade −1 −2 1 2 −1 1 0 0
Then
From the table we may guess that
Since
We have studied lower bounds for the minimal numbers of Reidemeister moves for two equivalent virtual knot diagrams.
Question. Find upper bounds for the minimal numbers of Reidemeister moves for two equivalent virtual knot diagrams.
Acknowledgements.
This work is supported by the Ministry of Science, ICT and Future Planning.
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