Remarks on Suzuki’s knot epimorphism number

Abstract

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]

Generating function on epimorphisms between $2$-bridge knot groups

“…, a m ], then the continued fraction expansion is called even. For a 2-bridge knot K(r), an even continued fraction expansion for r is unique up to symmetry and the length, which is the number of the components, is even (see [9], [14] detail). The genus of K(r) for the even continued fraction r = [a 1 , a 2 , .…”