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A partial order on prime knots can be defined by declaring 𝐽β‰₯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 π‘š distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of 𝐽 in terms of π‘š. 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 with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.

Recommended Citation

Hoste, Jim, et al. β€œRemarks on Suzuki’s Knot Epimorphism Number.” Journal of Knot Theory and Its Ramifications, vol. 28, no. 09, Aug. 2019. doi:10.1142/S0218216519500603.

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