2-Bridge knot boundary slopes: diameter and genus

Abstract

We prove that for $2$-bridge knots, the diameter, $D$, of the set of boundary slopes is twice the crossing number, $c$. This constitutes partial verification of a conjecture that, for all knots in $S^{3}$, $D \leq 2 c$. In addition, we characterize the $2$-bridge knots with four or fewer boundary slopes and show that they each have a boundary slope of genus two or less.