In this article we study a partial ordering on knots in where if there is an epimorphism from the knot group of onto the knot group of which preserves peripheral structure. If is a –bridge knot and , then it is known that must also be –bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given –bridge knot , produces infinitely many –bridge knots with . After characterizing all –bridge knots with or less distinct boundary slopes, we use this to prove that in any such pair, is either a torus knot or has 5 or more distinct boundary slopes. We also prove that –bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering. This result provides some evidence for the conjecture that all pairs of –bridge knots with arise from the Ohtsuki–Riley–Sakuma construction.
"Epimorphisms and boundary slopes of $2$–bridge knots." Algebr. Geom. Topol. 10 (2) 1221 - 1244, 2010. https://doi.org/10.2140/agt.2010.10.1221