The primitive curve complex for a handlebody

Cho, Sangbum; Lee, Jung Hoon

doi:10.2140/agt.2026.26.973

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The primitive curve complex for a handlebody

Sangbum Cho and Jung Hoon Lee

Algebraic & Geometric Topology 26 (2026) 973–988

DOI: 10.2140/agt.2026.26.973

Abstract

A simple closed curve in the boundary surface of a handlebody is called primitive if there exists an essential disk in the handlebody whose boundary circle intersects the curve transversely in a single point. The primitive curve complex is then defined to be the full subcomplex of the curve complex for the boundary surface, spanned by the vertices of primitive curves. Given any two primitive curves, we construct a sequence of primitive curves from one to the other one satisfying a certain property. As a consequence, we prove that the primitive curve complex for the handlebody is connected.

Keywords

handlebody, curve complex, primitive curve, dual disk

Mathematical Subject Classification

Primary: 57K30

References

Publication

Received: 1 February 2024

Revised: 15 December 2024

Accepted: 28 March 2025

Published: 1 April 2026

Authors

Sangbum Cho
Department of Mathematics Education Hanyang University Seoul South Korea

Jung Hoon Lee
Department of Mathematics and Institute of Pure and Applied Mathematics Jeonbuk National University Jeonju South Korea

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Volume 26, issue 3 (2026)