Tohoku Mathematical Journal
2026
March
SECOND SERIES VOL. 78, NO. 1

Tohoku Math. J.
78 (2026), 1-30

Title PROPERLY IMMERSED CURVES IN ARBITRARY SURFACES VIA APPARENT CONTOURS ON SPINES OF TRAVERSING FLOWS

Author Carlo Petronio

(Received November 14, 2023)
Abstract. Let $\Sigma$ be a compact surface with boundary and $F$ be the set of the orbits of a traversing flow on $\Sigma$. If the flow is generic, its orbit space is a spine$G$ of $\Sigma$, namely $G$ is a graph embedded in $\Sigma$ and $\Sigma$ is a regular neighbourhood of $G$. Moreover an extra structure on $G$ turns it into a flow-spine, from which one can reconstruct $\Sigma$ and $F$. In this paper we study properly immersed curves $C$ in $\Sigma$. We do this by considering generic $C$'s and their apparent contour relative to $F$, namely the set of points of $G$ corresponding to orbits that either are tangent to $C$, or go through a self-intersection of $C$, or meet the boundary of $C$. We translate this apparent contour into a decoration of $G$ that allows one to reconstruct $C$, and then we allow $C$ to vary up to homotopy within a fixed generic $F$, and next also $F$ to vary up to homotopy, and we identify a finite set of local moves on decorated graphs that translate these homotopies.

Mathematics Subject Classification. Primary 57R42; Secondary 37E35, 57R40, 58D10.
Key words and phrases. Properly immersed curves, surface topology and spines, traversing flows, apparent contours.

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