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Puncture loops on a non-orientable surface

Aoi Wakuda

TL;DR

The paper investigates when smoothing the intersection of two closed geodesics on a complete hyperbolic non-orientable surface can produce a puncture loop. It develops a hyperbolic-geometric framework, leveraging isometries (hyperbolic and glide-reflection) and explicit trace/length relations to analyze loops formed by concatenating one-sided geodesics, showing that puncture loops arise only for odd $m$ and at most two such $m$, which are consecutive if two exist; it also characterizes how the forward angle at the intersection governs these cases. A key component is the explicit lemma providing trace formulas for all intersection types, enabling the puncture criterion to be expressed in terms of $l_eta$, $l_ heta$, and $l_ ho$. Additionally, the work establishes lower bounds on self-intersections arising from these configurations and connects the results to potential applications in the Goldman Lie algebra with coefficients in $oxed{\mathbb{Z}/2\mathbb{Z}}$. Overall, the results clarify how puncture-winding phenomena distinguish non-orientable surfaces from orientable ones and provide tools for further algebraic-topological investigations on such surfaces.

Abstract

On a connected surface $N$ with negative Euler characteristic, the free homotopy class of a loop obtained by smoothing an intersection of two closed geodesics may wind around a puncture. Chas and Kabiraj showed that this phenomenon does not occur when the surface $N$ is orientable. In this paper, we prove that it occurs when $N$ is non-orientable and both geodesics involved in the smoothing are actually one-sided. In particular, we study a loop obtained by traversing a one-sided closed geodesic and the $m$-th power of another one-sided closed geodesic for odd $m$. Then we show that its free homotopy class may wind aroud a puncture at most two values of $m$. Furthermore, if two such $m$'s exist, they are consecutive odd integers.

Puncture loops on a non-orientable surface

TL;DR

The paper investigates when smoothing the intersection of two closed geodesics on a complete hyperbolic non-orientable surface can produce a puncture loop. It develops a hyperbolic-geometric framework, leveraging isometries (hyperbolic and glide-reflection) and explicit trace/length relations to analyze loops formed by concatenating one-sided geodesics, showing that puncture loops arise only for odd and at most two such , which are consecutive if two exist; it also characterizes how the forward angle at the intersection governs these cases. A key component is the explicit lemma providing trace formulas for all intersection types, enabling the puncture criterion to be expressed in terms of , , and . Additionally, the work establishes lower bounds on self-intersections arising from these configurations and connects the results to potential applications in the Goldman Lie algebra with coefficients in . Overall, the results clarify how puncture-winding phenomena distinguish non-orientable surfaces from orientable ones and provide tools for further algebraic-topological investigations on such surfaces.

Abstract

On a connected surface with negative Euler characteristic, the free homotopy class of a loop obtained by smoothing an intersection of two closed geodesics may wind around a puncture. Chas and Kabiraj showed that this phenomenon does not occur when the surface is orientable. In this paper, we prove that it occurs when is non-orientable and both geodesics involved in the smoothing are actually one-sided. In particular, we study a loop obtained by traversing a one-sided closed geodesic and the -th power of another one-sided closed geodesic for odd . Then we show that its free homotopy class may wind aroud a puncture at most two values of . Furthermore, if two such 's exist, they are consecutive odd integers.

Paper Structure

This paper contains 6 sections, 11 theorems, 36 equations, 6 figures.

Table of Contents

  1. Introduction
  2. isometries on hyperbolic plane
  3. Loops on a non-orientable surface and hyperbolic geometry
  4. Forward angles and lengths.
  5. Smoothing an intersection of two closed geodesics
  6. Self-intersection number of a loop on a non-orientable surface

Key Result

Theorem 2.1

Beardon1983 Let $g$ and $h$ be hyperbolic transformations of the hyperbolic plane and suppose that $A_{g}$ and $A_{h}$ intersect at a point $P$. Denote by $\theta_P$ the angle at P between forward direction of $A_{g}$ and $A_{h}$. Then the composition $g\circ h$ is hyperbolic and

Figures (6)

  • Figure 1: Two intersecting axes $A_g$ and $A_h$ in the upper half-plane $\mathbb{H}$.
  • Figure 2: The value of $\cosh\left(\dfrac{t_g}{2}\right)\sinh\left(\dfrac{t_h}{2}\right) + \sinh\left(\dfrac{t_g}{2}\right)\cosh\left(\dfrac{t_h}{2}\right)\cos\theta_P$ is zero on the left, positive in the middle, and negative on the right.
  • Figure 3: The forward angle $\theta_{P}(X)$
  • Figure 4: Five cases of the function $f(t)$, determined by the values of $r$ and $s$: (i) $s > r$, (ii) $s = r$, (iii) $-r < s < r$, (iv) $s = -r$, (v) $s < -r$.
  • Figure 5: Puncture loops obtained by smoothing an intersection of two closed geodesics.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof : Proof of Theorem \ref{['thm_hyp_gli']}
  • Theorem 2.6
  • Lemma 3.1
  • Theorem 3.2
  • ...and 7 more