Narrow Results

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.

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.

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsváth and Szabó assume arbitrarily large positive and negative values. As a consequence, we generalize a result by Greene and Watson by proving, for every odd number $\mathrm{\Delta }\ge 5$, the existence of infinitely many non-quasi-alternating homologically thin knots with determinant ${\mathrm{\Delta }}^2$, and a result by Hoffman and Walsh concerning the existence of hyperbolic weight $1$ manifolds, that are not surgery on a knot in $S^3$.

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.