Jones polynomials of periodic links
For an oriented periodic link L with period \(r^ q\) (the power of a prime integer) and its factor link \(L_*\), the author proves several relationships between the Jones polynomials of L and \(L_*\), for example, \[ V_ L(t)\equiv [V_{L_*}(t)]^{r_ q}\quad mod(r,\xi_ r(t)), \] where \(\xi_ r(t)=\sum^{r-1}_{j=0}(-t)^ j-t^{(r- 1)/2}\). For the case that L is not split, he proved more precise formula. As an applications, he proved that \(10_{105}\) cannot have period 7. This fact together with the previous work of \textit{G. Burde} and \textit{H. Zieschang} [Knots (1985; Zbl 0568.57001)] imply that \(10_{105}\) has no period.
scientific article; zbMATH DE number 4080418