• Kyungpook Mathematical Journal
• /
• v.58 no.1
• /
• pp.183-202
• /
• 2018

${\Delta}-moves$ 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 ${\Delta}-move$. 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 ${\Delta}-moves$ transforming $K_1$ to $K_2$ if two virtual knots $K_1$ and $K_2$ are related by a finite sequence of ${\Delta}-moves$.

• Communications of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.591-604
• /
• 2018

We consider a family of ribbon 2-knots with trivial Alexander polynomial. We give nonabelian SL(2, C)-representations from the groups of these knots, and then calculate the twisted Alexander polynomials associated to these representations, which allows us to classify this family of knots.

• Kyungpook Mathematical Journal
• /
• v.55 no.3
• /
• pp.741-752
• /
• 2015

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two independent canonical orientations.

• Kyungpook Mathematical Journal
• /
• v.46 no.4
• /
• pp.509-525
• /
• 2006

We give formulas for the first four coefficient polynomials of the Kauffman's link polynomial involving linking numbers and the coefficient polynomials of the Kauffman polynomials of the one- and two-component sublinks. We use mainly the Dubrovnik polynomial, a version of the Kauffman polynomial.

• Kyungpook Mathematical Journal
• /
• v.61 no.1
• /
• pp.205-212
• /
• 2021

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.

• Kyungpook Mathematical Journal
• /
• v.56 no.3
• /
• pp.1017-1045
• /
• 2016

Using a combination of calculational and theoretical approaches, we establish results that relate two knot invariants, the Alexander polynomial, and the number of quandle colourings using any finite linear Alexander quandle. Given such a quandle, specified by two coprime integers n and m, the number of colourings of a knot diagram is given by counting the solutions of a matrix equation of the form AX = 0 mod n, where A is the m-dependent colouring matrix. We devised an algorithm to reduce A to echelon form, and applied this to the colouring matrices for all prime knots with up to 10 crossings, finding just three distinct reduced types. For two of these types, both upper triangular, we found general formulae for the number of colourings. This enables us to prove that in some cases the number of such quandle colourings cannot distinguish knots with the same Alexander polynomial, whilst in other cases knots with the same Alexander polynomial can be distinguished by colourings with a specific quandle. When two knots have different Alexander polynomials, and their reduced colouring matrices are upper triangular, we find a specific quandle for which we prove that it distinguishes them by colourings.

• Communications of the Korean Mathematical Society
• /
• v.35 no.3
• /
• pp.1037-1043
• /
• 2020

We show that if M and V are Seifert forms such that M is metabolic and has nontrivial Alexander polynomial, then there exists a knot K having Seifert form M ⊕ V that is not concordant to any knot with Seifert form V.

• Kyungpook Mathematical Journal
• /
• v.56 no.2
• /
• pp.639-645
• /
• 2016

In [3] the main theorem was erroneously stated. We needed to assume that the irrationality measure of 1/r is finite to prove the theorem.

• Journal of the Korean Mathematical Society
• /
• v.52 no.2
• /
• pp.389-402
• /
• 2015

A symmetric union is a diagram of a knot, obtained from diagrams of a knot in the 3-space and its mirror image, which are symmetric with respect to an axis in the 2-plane, by connecting them with 2-tangles with twists along the axis and 2-tangles with no twists. This paper presents an invariant of knots with symmetric union presentations, which is called the minimal twisting number, and the minimal twisting number of $10_{42}$ is shown to be two. This paper also presents a sufficient condition for non-amphicheirality of a knot with a certain symmetric union presentation.

• Kyungpook Mathematical Journal
• /
• v.48 no.2
• /
• pp.255-280
• /
• 2008

We derive a formula for the colored Jones polynomial of 2-bridge knots. For a twist knot, a more explicit formula is given and it leads to a relation between the degree of the colored Jones polynomial and the crossing number.