Remarks on Suzuki's Knot Epimorphism Number

TL;DR

This paper establishes a lower bound on the crossing number of 2-bridge knots based on their epimorphism relationships, answering a question about the maximum number of smaller knots related to a given crossing number.

Contribution

It provides a new lower bound on the crossing number of 2-bridge knots with multiple epimorphic images, advancing understanding of knot group epimorphisms and their relation to crossing numbers.

Findings

Derived a lower bound on crossing numbers for 2-bridge knots with multiple epimorphic images.

Answered Suzuki's question on the 2-bridge epimorphism number $ ext{EK}(n)$.

Used continued fraction parsing techniques to establish the results.

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.