Narrow Results

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 $\mathrm{\Gamma }$-polynomial. As a result, we characterize the $\mathrm{\Gamma }$-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 $0$ and $1$. We show that there exist infinitely many knots which bound clasp disks of not doubled knots but both types $0$ and $1$, not type $0$ but type $1$, not type $1$ but type $0$, respectively.

In this paper, we calculate the Kauffman polynomials $F(K(p,q);a,z)$ of Kanenobu knots $K(p,q)$ with $p,q$ half twists and determine their spans on the variable $a$ 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 $K(2n,-2n)$ by using canonical cabling algorithm and the 2-cable $\mathrm{\Gamma }$-polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations.

For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\mathrm{\Gamma }$-polynomial of a knot is the $\mathrm{\Gamma }$-polynomial of the $(p,q)$-cable knot of the knot, where the $\mathrm{\Gamma }$-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 $(2,1)$-cable $\mathrm{\Gamma }$-polynomial, that is, the $(2,1)$-cable $\mathrm{\Gamma }$-polynomial of the trivial knot. Moreover, we see that the knots have the trivial $\mathrm{\Gamma }$-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials.

For coprime integers $p(>0)$ and $q$, the $(p,q)$-cable $\mathrm{\Gamma }$-polynomial of a knot is the $\mathrm{\Gamma }$-polynomial of the $(p,q)$-cable knot of the knot, where the $\mathrm{\Gamma }$-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the $\mathrm{\Gamma }$-polynomial is computable in polynomial time, the $(p,q)$-cable $\mathrm{\Gamma }$-polynomial is also computable in polynomial time. In this paper, we show that the $(2,1)$-cable $\mathrm{\Gamma }$-polynomial completely classifies the unoriented knots up to ten crossings including the chirality information.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study the $\mathrm{\Gamma }$-polynomial of knots with a given ascending number. We give a lower bound of the ascending numbers by using $\mathrm{\Gamma }$-polynomials. We estimate the ascending numbers for 65 prime knots up to 10 crossings by using $\mathrm{\Gamma }$-polynomials, Conway polynomials, and the determinants.