Non-orientable fundamental surfaces in lens spaces

TL;DR

This paper constructs an infinite sequence of lens spaces with specific triangulations containing minimal non-orientable surfaces, illustrating their properties and structure explicitly.

Contribution

It provides explicit examples of non-orientable fundamental surfaces in lens spaces with minimal crosscap number and detailed geometric structure.

Findings

Existence of an infinite sequence of lens spaces with fundamental non-orientable surfaces

Construction of triangulations containing these surfaces with specific properties

Explicit recursive formulas for the lens spaces parameters

Abstract

We give a concrete example of an infinite sequence of $(p_n, q_n)$-lens spaces $L(p_n, q_n)$ with natural triangulations $T(p_n, q_n)$ with $p_n$ taterahedra such that $L(p_n, q_n)$ contains a certain non-orientable closed surface which is fundamental with respect to $T(p_n, q_n)$ and of minimal crosscap number among all closed non-orientable surfaces in $L(p_n, q_n)$ and has $n-2$ parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of $L(p_n, q_n)$. Actually, we can set $p_0=0, q_0=1, p_{k+1}=3p_k+2q_k$ and $q_{k+1}=p_k+q_k$.