Narrow Results

Some evaluations of the Gordian distance from a knot to another knot are given by using three polynomial invariants called the HOMFLY polynomial, the Jones polynomial and the Q-polynomial. Furthermore, Gordian distances between a lot of pairs of knots are determined.

Qazaqzeh and Chbili showed that for any quasi-alternating link, the degree of the Q-polynomial is less than its determinant. We give a refinement of their evaluation.

We study the relation between the maximum z-degree of Kauffman polynomial and the determinant for a quasi-alternating link. This gives a stronger criterion for deciding whether a non-alternating link is quasi-alternating or not than the known criterion in terms of Q-polynomials. Also, we determine all quasi-alternating links with determinant 5.

We give necessary and sufficient conditions for a given polynomial to be a plucking polynomial of a rooted tree. We discuss the fact that different rooted trees can have the same polynomial.

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.

We call a link (knot) $L$ to be strongly Jones (respectively, Homfly) undetectable, if there are infinitely many links which are not isotopic to $L$ but share the same Jones (respectively, Homfly) polynomial as $L$. We reconstruct Kanenobu’s knot [Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97(1) (1986), 158–162] and give two new constructions. Using these constructions, we give some examples of strongly Jones undetectable: $8_8$, $8_9$, $10_{22}$, $10_{35}$, $10_{155}$, $4_1\#4_1$, $5_2\#5_2!$ ($5_2!$ is the mirror image of $5_2$) and etc. For some special cases, these constructions will be shown to be strongly Jones undetectable and strongly Homfly undetectable.