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arXiv · 1810.05146

Remarks on Suzuki's Knot Epimorphism Number

Jim Hoste·Joshua Ocana Mercado·Patrick D. Shanahan
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Abstract

A partial order on prime knots can be defined by declaring $J\ge K$ 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 $\mbox{EK}(n)$ 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.

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Jim Hoste, Joshua Ocana Mercado, Patrick D. Shanahan. 2018-10-11. Remarks on Suzuki's Knot Epimorphism Number. https://arxiv.org/abs/1810.05146

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