Crosscap number and epimorphisms of two-bridge knot groups

TL;DR

This paper explores the relationship between the crosscap number of 2-bridge knots and a partial order defined by epimorphisms of their fundamental groups, establishing inequalities and classifying cases of equality.

Contribution

It establishes a lower bound on the crosscap number for 2-bridge knots related by epimorphisms and classifies all pairs where this bound is tight.

Findings

If K > J, then γ(K) ≥ 3γ(J) - 4.

The inequality is sharp for certain pairs of knots.

A similar inequality relates the genus of knots, g(K) ≥ 3g(J) - 1.

Abstract

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism $f:\pi_1(S^3-K) \longrightarrow \pi_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K> J$, then $\gamma(K) \geq 3\gamma(J) -4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K >J$, then $g(K) \geq 3 g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.