Geometrically incompressible non-orientable closed surfaces in lens spaces

TL;DR

This paper characterizes non-orientable closed surfaces with minimal crosscap number in lens spaces using a graph-theoretic approach and provides a method to compute their boundary slopes via continued fractions.

Contribution

It establishes a correspondence between standard form surfaces in lens spaces and edge-paths in a hyperbolic tree, enabling explicit calculation of minimal crosscap numbers.

Findings

Edge-paths in a hyperbolic tree represent standard form surfaces.

Number of edges equals the minimal crosscap number.

Provides a continued fraction method to compute boundary slopes.

Abstract

We consider non-orientable closed surfaces of minimum crosscap number in the $(p,q)$-lens space $L(p,q) \cong V_1 \cup_{\partial} V_2$, where $V_1$ and $V_2$ are solid tori. Bredon and Wood gave a formula for calculating the minimum crosscap number. Rubinstein showed that $L(p,q)$ with $p$ even has only one isotopy class of such surfaces, and it is represented by a surface in a standard form, which is constructed from a meridian disk in $V_1$ by performing a finite number of band sum operations in $V_1$ and capping off the resulting boundary circle by a meridian disk of $V_2$. We show that the standard form corresponds to an edge-path $\lambda$ in a certain tree graph in the closure of the hyperbolic upper half plane. Let $0=p_0/q_0, p_1/q_1, ..., p_k/q_k = p/q$ be the labels of vertices which $\lambda$ passes. Then the slope of the boundary circle of the surface right after the $i$-th band sum is $(p_i, q_i)$. The number of edges of $\lambda$ is equal to the minimum crosscap number. We give an easy way of calculating $p_i / q_i$ using a certain continued fraction expansion of $p/q$.