Bulletin of the
Korean Mathematical Society BKMS

Every $(1,1)$-knot is represented by a $4$-tuple of integers $(a,b$, $c,r),$ where $a>0,$ $b\geq 0,$ $c\geq 0,$ $d=2a+b+c,r\in \mathbb{Z}_{d},$ and it is well known that all $2$-bridge knots and torus knots are $(1,1)$-knots. In this paper, we describe some conditions for 4-tuples which determine $2$ -bridge knots and determine all $4$-tuples representing any given $2$-bridge knot.

Keywords: (1,1)-knot, (1,1)-decomposition, cyclic branched covering, crystallization, Dunwoody manifold, Heegaard splitting, Heegaard diagram, 2-bridge knot, torus knot

MSC numbers: Primary 57M25; Secondary 57M27