On nonhyperbolicity of P/P and P/SF knots in $S^3$

Sungmo Kang

Published 2020-03-31Version 1

In [B90] or an available version [B18], Berge constructed twelve families of primitive/primitve(or simply P/P) knots, which are referred to as the Berge knots. It is proved in [B08] or independently in [G13] that all P/P knots are the Berge knots, which gives the complete classification of P/P knots. However, the hyperbolicity of the Berge knots was undetermined in [B90]. As a natural generalization of P/P knots, Dean introduced primitive/Seifert(or simply P/SF) knots in $S^3$. The classification of hyperbolic P/SF knots has been performed and the complete list of hyperbolic P/SF knots is given in [BK20]. In this paper, we provide necessary, sufficient, or equivalent conditions for P/P or P/SF knots being nonhyperbolic, and their applications on some P/SF knots. The results of this paper will be used in proving the hyperbolicity of P/P and P/SF knots in [K20a] and [K20b] completing the classification of hyperbolic P/P and P/SF knots in [B90] and [BK20] respectively.