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Infinitely many knots with the trivial $(2, 1)$ $(2, 1)$ -cable $Γ$ $Γ$ -polynomial

Osaka City University Advanced Mathematical Institute (OCAMI), Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail Address: takioka@sci.osaka-cu.ac.jp

https://doi.org/10.1142/S021821651850013XCited by:3 (Source: Crossref)

Abstract

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

Keywords:

AMSC: 57M25, 57M27

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