[Paper Review] A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody

John Berge|ArXiv.org|Oct 16, 2009

Advanced Numerical Analysis Techniques3 references4 citations

TL;DR

This paper classifies pairs of disjoint, nonparallel, primitive simple closed curves in the boundary of a genus two handlebody. It shows such pairs either lie on opposite ends of a standard or twisted product of a once-punctured torus with an interval, or, when one curve is a proper power of a primitive, they are either separated by a separating disk or bound a nonseparating essential annulus in the handlebody.

ABSTRACT

Embeddings of pairs of disjoint nonparallel primitive simple closed curves in the boundary of a genus two handlebody are classified. Briefly, two disjoint primitives either lie on opposite ends of a product $F \boldsymbol{ imes} I$, or they lie on opposite ends of a kind of "twisted" product $F \widetilde{\boldsymbol{ imes}} I$, where $F$ is a once-punctured torus. If one of the curves is a proper power of a primitive, the situation is simpler. Either the curves lie on opposite sides of a separating disk in the handlebody, or they bound a nonseparating essential annulus in the handlebody.

Motivation & Objective

• To classify embeddings of pairs of disjoint, nonparallel, primitive simple closed curves in the boundary of a genus two handlebody.
• To determine the topological configurations that arise when such curves are disjoint and nonparallel, particularly in relation to the handlebody's cutting disks.
• To extend the classification to cases where one curve is a proper power of a primitive, simplifying the topological structure.
• To characterize when such pairs are separated by a separating disk or bound a nonseparating essential annulus in the handlebody.

Proposed method

• Uses Heegaard diagrams and their underlying graphs to analyze the intersection patterns of curves with a complete set of cutting disks.
• Employs R-R diagrams (relative-representation diagrams) to model curves on the boundary of the handlebody, particularly focusing on connections in once-punctured tori.
• Applies graph-theoretic techniques to the graphs $G_{eta}$ and $G_{eta}$ derived from cutting the handlebody along a complete set of disks.
• Uses isotopy and bandsum operations to minimize intersection numbers and deduce topological configurations.
• Distinguishes cases based on whether edges in the graph $G_{eta}$ are parallel or nonparallel, leading to different topological types.
• Applies the concept of proper powers in the fundamental group to reduce the complexity of the classification when one curve is not primitive.

Experimental results

Research questions

• RQ1 What are the possible topological configurations for a pair of disjoint, nonparallel, primitive curves in the boundary of a genus two handlebody?
• RQ2 How do the intersection patterns with a complete set of cutting disks determine whether the pair lies in a standard or twisted product structure?
• RQ3 What happens when one curve in such a pair is a proper power of a primitive curve?
• RQ4 Under what conditions are such pairs separated by a separating disk or bound a nonseparating essential annulus in the handlebody?
• RQ5 Can the classification be fully described using R-R diagrams and graph-theoretic invariants of the induced graphs on the 2-sphere boundary?

Key findings

• A pair of disjoint nonparallel primitives in the boundary of a genus two handlebody lies on opposite ends of either a standard product $F imes I$ or a twisted product $F ilde{ imes}I$, where $F$ is a once-punctured torus.
• When one curve is a proper power of a primitive, the pair is either separated by a separating disk or bounds a nonseparating essential annulus in the handlebody.
• The classification depends on the structure of the graph $G_{eta}$ formed by the intersection of the curves with the cutting disks; non-parallel edges lead to twisted product structures.
• If the graph $G_{eta}$ has only parallel edges, the curves lie in a once-punctured torus neighborhood, leading to annular or separating configurations.
• The condition for a pair to be both Type I and Type II is that the curves are separated or have an R-R diagram with wrapping numbers satisfying $q = ps \pm 1$.
• The minimal intersection number with the cutting disks' boundaries determines the canonical configuration, and bandsum operations are used to reduce intersections and verify minimality.