Narrow Results

Vaughan Jones was my mentor and friend. I provide here a few personal recollections including the “knotted antennas” project which brought me to Berkeley.

Personal recollections of meetings with Vaughan Jones.

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.

We compare the invariants for classical knots and links defined using the Juyumaya trace on the Yokonuma–Hecke algebras with the HOMFLYPT polynomial. We show that these invariants do not coincide with the HOMFLYPT except in a few trivial cases.

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.

Polyak showed that any Milnor’s $\overline{\mu }$-invariant of length 3 can be represented as a combination of the Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called $\mu$-invariants. We show that any Milnor’s $\mu$-invariant of length $\le k+2$ can be represented as a combination of the HOMFLYPT polynomials of knots obtained from the string link by some operation, if all $\mu$-invariants of length $\le k$ vanish. Moreover, $\mu$-invariants of length $3$ are given by a combination of the Conway polynomials and linking numbers without any vanishing assumption.

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.

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.

Extended strongly periodic links have been introduced by Przytycki and Sokolov as a symmetric surgery presentation of three-manifolds on which the finite cyclic group acts without fixed points. The purpose of this paper is to prove that the symmetry of these links is reflected by the first coefficients of the HOMFLYPT polynomial.

For any alternating prime knot $K$, it is expected that the canonical genus of its Whitehead double is equal to the crossing number of $K$. We introduce local moves of alternating knots, and prove that these local moves preserve the canonical genus of its Whitehead double under a certain condition. By this result, we give a new family of knots which satisfy this conjecture.

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links.

A $4$-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 $4$-moves to the unknot. This is still an open problem. In this paper, we consider the $4$-move distance of knots, which is the minimal number of $4$-moves needed to deform one into the other. In particular, the $4$-move unknotting number of a knot is the $4$-move distance to the unknot. We give a table of the $4$-move unknotting number of knots with up to nine crossings.