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.

References ↓
Marked points 2
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.
Starting surface

Möbius strip M2

Disk + crosscap
Möbius strip with marked points A disk with a central crosscap. The green points and crosscap are interactive.

Regular endpoints lie on the unique boundary. Select the crosscap twice to draw the one-sided quasi-arc.

Planar representation

Fundamental rectangle

Twisted sides
Planar representation of the Möbius strip A fundamental rectangle whose vertical sides are identified in opposite directions. Every selected arc is shown in the rectangle.

The left and right sides are identified in opposite directions; every selected arc is also visible in this planar model.

Double cover

Orientable annulus

2n points
Orientable double cover of the Möbius strip An annulus with two lifts of every marked point, two lifts of each regular arc, and one closed lift of the quasi-arc.
outer boundary inner boundary half-turn + boundary exchange

Construction in the double cover

Snake graph or band graph

0 tiles

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.

Demo

Exhaustive mutation exploration

Exchange Graph Laboratory

Ready to generate Open demo Reduce demo

Generate every quasi-triangulation reachable by valid mutations, then explore how one flip connects it to the next.

Starting triangulation
The graph keeps its own T₀ snapshot and never overwrites the locked Mutation Lab seed.
Locked seed: — Graph start: — Currently viewing: —
Discovered 0 Processed 0 Queue 0 Generation has not started.

Drag empty space to pan, use the wheel or zoom buttons, and drag a node to reposition it. Select a node or edge to inspect its mutation data.

Exchange graph of quasi-triangulations Nodes are quasi-triangulations and edges are single valid mutations.

Keyboard: use Tab to reach graph items, Enter to select, arrow keys to pan, plus or minus to zoom, and 0 to fit the graph.

Find mutation path

Choose a start and target after generating the graph.

Graph statistics

Pending generation
Quasi-triangulations
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Mutation edges
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Connected components
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Maximum distance
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Graph diameter
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Average degree
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Research Debug Mode

Inspect the exact queue, canonicalization, mutation, and validation state without opening the browser console.

Current queue size
0
Visited triangulations
0
Current triangulation
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Mutation being attempted
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Canonical key
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Existing or new state
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Validation result
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No mutation diagnostics yet.

Generation Log
  1. Generation has not started.

Sources

References

The examples and constructions used in the laboratory are based on the following articles.

  1. Grégoire Dupont & Frédéric Palesi (2011), Quasi-cluster algebras from non-orientable surfaces arXiv:1105.1560. Primary source for quasi-arcs, ranks, quasi-mutations, orientable double covers, and finite type.
    Open the article ↗
  2. Sergey Fomin, Michael Shapiro & Dylan Thurston (2008), Cluster algebras and triangulated surfaces. Part I Acta Mathematica 201, arXiv:math/0608367. Primary source for polygon arcs, triangulations, flips, and exchange relations.
    Open the article ↗
  3. Gregg Musiker, Ralf Schiffler & Lauren Williams (2011), Positivity for cluster algebras from surfaces Advances in Mathematics 227, arXiv:0906.0748. Primary source for the ordered-crossing construction of polygon snake graphs.
    Open the article ↗
  4. İlke Çanakçı & Ralf Schiffler (2013), Snake graph calculus and cluster algebras from surfaces Journal of Algebra 382, arXiv:1209.4617. Source for abstract snake graphs, crossings, and resolutions.
    Open the article ↗
  5. Véronique Bazier-Matte (2025), Quasi-cluster algebras: An overview Representations of Algebras and Related Topics, pp. 85–119, EMS Press. DOI: 10.4171/ECR/21/3. Source for Definition 3.2 and the Figure 12 example.
    Open the article ↗
  6. Jon Wilson (2019), Positivity for quasi-cluster algebras arXiv:1912.12789. Source for the snake- and band-graph construction in the orientable double cover.
    Open the article ↗