A partial order on prime knots can be defined by declaring J ≥ K if there exists an epimorphism from the knot group of J onto the knot group of K. Suppose that J is a 2-bridge knot that is strictly greater than m distinct, nontrivial knots. In this paper we determine a lower bound on the crossing number of J in terms of m. Using this bound we answer a question of Suzuki regarding the 2-bridge epimorphism number EK(n) which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number n. We establish our results using techniques associated to parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot. arXiv:1810.05146v1 [math.GT]

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