Journal of the
Korean Mathematical Society JKMS

In this paper, we give a recursive formula for the Jones polynomial of a $2$-bridge knot or link with Conway normal form $C(-2n_1,$ $2n_2,$ $-2n_3, \ldots, (-1)^r2n_r)$ in terms of $n_1, n_2, \ldots, n_r$. As applications, we also give a recursive formula for the Jones polynomial of a $3$-periodic link $L^{(3)}$ with rational quotient $L = C(2, n_1, -2, n_2, \ldots,$ $n_r, (-1)^r 2)$ for any nonzero integers $n_1, n_2, \ldots, n_r$ and give a formula for the span of the Jones polynomial of $L^{(3)}$ in terms of $n_1, n_2, \ldots, n_r$ with $n_i \neq \pm 1$ for all $i = 1, 2, \ldots, r$.

Keywords: Jones polynomial, $2$-bridge knot, span, periodic link with rational quotient

MSC numbers: Primary 57M25