Primeness of twisted knots

Abstract:

Let $V$ be a standardly embedded solid torus in ${S^3}$ with a meridian-preferred longitude pair $(\mu ,\lambda )$ and $K$ a knot contained in $V$. We assume that $K$ is unknotted in ${S^3}$. Let ${f_n}$ be an orientation-preserving homeomorphism of $V$ which sends $\lambda$ to $\lambda + n\mu$. Then we get a twisted knot ${K_n} = {f_n}(K)$ in ${S^3}$. Primeness of twisted knots is discussed and we prove: A twisted knot ${K_n}$ is prime if $|n| > 5$. Moreover, ${\{ {K_n}\} _{n \in Z}}$ contains at most five composite knots.