Starting surface
Disk + crosscap
Möbius strip M2
Regular endpoints lie on the unique boundary. Select the crosscap twice to draw the one-sided quasi-arc.
HPM
Interactive mathematics
Möbius Laboratory
No installation
Marked Möbius strip Mn
Regular arcs and the one-sided quasi-arc
Choose endpoints among the green marked points to add a regular arc, or select the crosscap twice to add the one-sided closed curve. The laboratory classifies curves by isotopy and automatically rejects duplicates.
Marked points
Topological type
Select two green points for a regular arc. Select the same point twice for a loop, or the crosscap twice for the quasi-arc.
Double cover
2n points
Orientable annulus
outer boundary
inner boundary
τ half-turn + boundary exchange
Wilson · construction in the double cover
Snake graph or band graph
Select an additional regular arc for its snake graph, or select the closed quasi-arc for its band graph.
Isotopy classes
Accepted arcs
0 internal curves
0 lifts
0 duplicates rejected
0 interior intersections
Not yet a triangulation
A triangulation of Mn must be a maximal compatible family of size n.
Interior intersections among lifts in the double cover will be listed here.
No arcs yet. Choose two points, or select the crosscap twice to add the quasi-arc.
How does the laboratory recognize equivalent arcs?
It does not compare pixels. A canonical signature records the unordered endpoints, whether the arc passes through the crosscap, and, when relevant, the chosen boundary side. Reversing the direction therefore never creates a new arc.
A regular arc has two lifts exchanged by the involution τ. A lift’s endpoints lie on the same boundary when the arc avoids the crosscap, and on opposite boundaries when it passes through it.
The quasi-arc is a one-sided closed curve with no marked endpoint. Unlike a regular arc, it lifts to a single closed curve in the annulus.
Different classes may intersect without being duplicates. The laboratory keeps them all because it visualizes families of arcs and does not require a triangulation.
A regular arc produces a snake graph. The one-sided closed curve produces a band graph: the tile sequence is first shown cut open, and the blue edges b and b′ indicate the gluing that closes the band. Each tile corresponds to a quadrilateral around an intersection with the lifted triangulation.