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A [Formula: see text]-move is a local change for knots and links which changes 4 half twists to 0 half twists or vice versa. In 1979, Yasutaka Nakanishi conjectured that every knot can be changed by [Formula: see text]-moves to the unknot. This is still an open problem. In this paper, we consider the [Formula: see text]-move distance of knots, which is the minimal number of [Formula: see text]-moves needed to deform one into the other. In particular, the [Formula: see text]-move unknotting number of a knot is the [Formula: see text]-move distance to the unknot. We give a table of the [Formula: see text]-move unknotting number of knots with up to nine crossings.

We recall a skein relation of the first coefficient Kauffman polynomial for knots. By using the skein relation, we show that there exist infinitely many knots whose Whitehead doubles have the trivial first coefficient Kauffman polynomial.

Abe and Tange constructed a sequence of slice disks with the same exterior. Moreover, they showed that these slice disks are ribbon disks. We call the boundaries of the ribbon disks Abe-Tange's ribbon knots. In this paper, we classify Abe-Tange's ribbon knots completely by the Γ-polynomial. Therefore, we see that all ribbon disks in the sequence are mutually inequivalent.

For coprime integers p ( > 0 ) and q, the ( p , q ) -cable Γ-polynomial of a knot K is the Γ-polynomial of the ( p , q ) -cable knot of K, where the Γ-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we give some results on Vassiliev knot invariants derived from the cable Γ-polynomials. In particular, we show that all Vassiliev knot invariants of order ≤4 are determined by the cable Γ-polynomials.

The Γ-polynomial is an invariant of an oriented link in the 3-sphere, which is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. It is known that the HOMFLYPT and Kauffman polynomials, their 2-cable versions, and the satellite versions of the Alexander and Jones polynomials are invariant under mutation. On the other hand, there exists a mutant knot pair which is distinguished by the 3-cable version of the HOMFLYPT polynomial. In this paper, we show that the 3-cable version of the Γ-polynomial is invariant under mutation.

We study the braid indices of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. The MFW inequality is known for giving a lower bound of the braid index of a link by applying the HOMFLYPT polynomial. Therefore, it is not easy to determine the braid indices of the Kanenobu knots. In our previous paper, we gave upper bounds and sharper lower bounds of the braid indices of the Kanenobu knots by applying the 2-cable version of the zeroth coefficient HOMFLYPT polynomial. In this paper, we give sharper upper bounds of the braid indices of the Kanenobu knots.

We call smoothing a self-crossing point of an oriented link diagram self-smoothing. By self-smoothing repeatedly, we obtain an oriented link diagram without self-crossing points. In this paper, we show that every knot has an oriented diagram which becomes a two-component oriented link diagram without self-crossing points by a single self-smoothing.

For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the [Formula: see text]-polynomial is computable in polynomial time, the [Formula: see text]-cable [Formula: see text]-polynomial is also computable in polynomial time. In this paper, we show that the [Formula: see text]-cable [Formula: see text]-polynomial completely classifies the unoriented knots up to ten crossings including the chirality information.

For coprime integers [Formula: see text] and [Formula: see text], the [Formula: see text]-cable [Formula: see text]-polynomial of a knot is the [Formula: see text]-polynomial of the [Formula: see text]-cable knot of the knot, where the [Formula: see text]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial [Formula: see text]-cable [Formula: see text]-polynomial, that is, the [Formula: see text]-cable [Formula: see text]-polynomial of the trivial knot. Moreover, we see that the knots have the trivial [Formula: see text]-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials.

It is known that every knot bounds a singular disk with only clasp singularities, which is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. It is known that the Conway polynomials of knots with clasp number at most two are characterized. In this paper, we focus on the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials, which is called the [Formula: see text]-polynomial. As a result, we characterize the [Formula: see text]-polynomials of knots with clasp number at most two. Moreover, it is known that there exist two homeomorphic classes of clasp disks with two clasp singularities, which are called types [Formula: see text] and [Formula: see text]. We show that there exist infinitely many knots which bound clasp disks of not doubled knots but both types [Formula: see text] and [Formula: see text], not type [Formula: see text] but type [Formula: see text], not type [Formula: see text] but type [Formula: see text], respectively.

In this paper, we calculate the Kauffman polynomials [Formula: see text] of Kanenobu knots [Formula: see text] with [Formula: see text] half twists and determine their spans on the variable [Formula: see text] completely. As an application, we determine the arc index of infinitely many Kanenobu knots. In particular, we give sharper lower bounds of the arc index of [Formula: see text] by using canonical cabling algorithm and the 2-cable [Formula: see text]-polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations.

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of [Formula: see text]-cable links and Whitehead doubles of all prime knots with up to eight crossings.

In our previous paper, we studied the braid index of the Kanenobu knot k(n) for n ≥ 0. In this paper, we study the braid index of the Kanenobu knot K(a, b) for a, b ∈ ℤ. In particular, k(n) is K(2n, -2n). The MFW inequality is known for giving a lower bound of the braid index of an oriented link by applying the HOMFLYPT polynomial. The HOMFLYPT polynomial of K(a, b) is given by Professor Taizo Kanenobu. Therefore, we have a lower bound of the braid index of K(a, b). The purpose of this paper is to give an upper bound of the braid index of K(a, b). As a result, we determine the braid indices of infinitely many Kanenobu knots.

The zeroth coefficient polynomial is a one variable polynomial contained in the HOMFLYPT polynomial. In this paper, we give a basic computation of the zeroth coefficient polynomial of a 2-cable knot. In particular, we compute the zeroth coefficient polynomials of the 2-cable knots of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov–Rozansky homology. As a result, we distinguish the Kanenobu knots completely. Moreover, we estimate the braid indices of the Kanenobu knots.