# Arcs on N(k,1): mathematical contract Research and implementation: 7 September 2026. ## Scope The integrated Laboratory (`index.html`, `assets/general_lab.js`) represents **individual simple marked arcs on a compact nonorientable surface with one boundary component and no punctures**, for k ≥ 2. Endpoints may coincide. Marked points remain fixed throughout isotopies. As in the laboratory, a contractible based loop or a curve homotopic relative endpoints to a single boundary segment is excluded. An arc cutting off a disk containing other boundary marks is not automatically excluded. This is a complete *representation language*, not a finite list of classes, not a canonical normal form for isotopy, and not an implementation of the whole quasi-cluster algebra. It is deliberately separate from the older fixed-spine preview in `marked_surface.js`. Its generators are dual to the cuts below; do not interchange its words with the older schema's words. ## Sources checked before implementation 1. Grégoire Dupont and Frédéric Palesi, [Quasi-cluster algebras from non-orientable surfaces](https://arxiv.org/abs/1105.1560), [full paper](https://arxiv.org/pdf/1105.1560), Journal of Algebraic Combinatorics 42 (2015). §2.2 defines marked arcs and one-sided closed curves; Proposition 3.7 gives the rank; Theorem 7.2 distinguishes the finite disk/Möbius cases from the infinite cases. In our notation χ = 1 − k and the quasi-triangulation rank is 3k + n − 3. 2. D. B. A. Epstein, [Curves on 2-manifolds and isotopies](https://wrap.warwick.ac.uk/id/eprint/34905/), Acta Mathematica 115 (1966), 83–107, DOI 10.1007/BF02392203. This is the relevant background for the distinction between homotopy and isotopy of embedded curves. A word by itself does not certify an embedding. The implementation does not rely on an unrestricted homotopy-to-isotopy assertion for possibly self-intersecting input. 3. Marcus Schaefer, Eric Sedgwick and Daniel Štefankovič, [Algorithms for Normal Curves and Surfaces](https://www.cs.rochester.edu/~stefanko/Publications-new/C4.pdf), §2. This explains encoding embedded curves by ordered intersections with edges. **The paper assumes orientability in its initial setup.** We do not cite it as a theorem proving our nonorientable algorithm; the elementary cut-disk argument below accounts explicitly for the gluing orientation. Our chord diagrams are not the paper's compressed normal coordinates. 4. Michael S. Floater, [One-to-one piecewise linear mappings over triangulations](https://doi.org/10.1090/S0025-5718-02-01466-7), Mathematics of Computation 72 (2003), 685–696. The convex-combination injectivity result under a convex boundary condition supplies the background for the disk chart used in the crosscap drawing. The implementation also checks convergence and strictly positive triangle areas at runtime. ## Quotient and conventions Take a closed disk whose boundary is divided, in positive cyclic order, into B, a1+, a1−, a2+, a2−, …, ak+, ak−. Glue `ai+` to `ai−` with equal boundary parameters t ↦ t. Thus both copies have the **same direction as the perimeter**. These are twisted identifications, not the opposite-perimeter-direction identifications of an orientable handle. Leave B unpaired. The quotient is N(k,1). The polygon's vertices form one unmarked boundary vertex, and its link is an interval. There is one face, k+1 edges and one vertex, hence χ = 1 − k; the gluing is nonorientable. Place P1,…,Pn in this order in the **interior of B**. This avoids putting an endpoint on a cut vertex or having to choose a corner sector at a marked point. The cut-disk views draw either a polygon or curved paired sides. The quarter-circle assigned to B in the cover changes layout, not topology. A crossing `(a,r)` goes from the r-th port on `a+` to the r-th port on `a−`. A crossing `(-a,r)` goes in the reverse direction. The r-th ports in a pair are identified. For each a, the ranks must be a permutation of 1,…,m, where m is the number of crossings of that cut in the path. A port cannot be reused. There is no restriction on how often a cut occurs and no 512-letter cutoff. ## Integrated views The existing crosscap-count control at the bottom of the Laboratory selects the surface without an access code. k=0 and k=1 retain their previous backends; k=2,…,24 use the complete cut-disk representation above. The existing arc register, undo, clear, language buttons, PNG exports and 3D camera controls are shared. Saved diagrams live in the Laboratory's common `state.arcs`. Only admissible drafts are rendered, in every view including 3D and PNG exports. Rejected drafts remain editable, with an explanation, but show an empty surface rather than a red self-intersecting curve. For an intersecting single-crossing draft, the UI can offer the opposite crossing if it passes the full admissibility check. This is an explicit alternative chosen by the user, not an isotopy simplification or an automatic change of the entered arc. The base view is a large circular disk with k smaller crossed circles inside, as in the crosscap convention of Dupont–Palesi §2.2. Opposite points of each small circle are identified; these circles are not additional free boundaries. The marked points are evenly spaced on the large circle, starting at the top. The planar view uses a radial homeomorphism to a regular polygon; curved segments are sampled for display. The double cover uses two disk copies, with the second drawn reflected. Each crossing switches sheets; both lifts are drawn. Thus a+ in one sheet is glued to a− in the other sheet at the same rank. Reflecting one disk displays opposite orientations on the two sheets. The glued cover is connected and orientable, has χ=2−2k and two boundary components, hence genus k−1. For k=2, the 3D view shows a classical Klein bottle with a disk removed and the selected arc on it (construction below). For k>2, it bends the *cut disk* into a bowl, explicitly labeled as a gluing diagram. The camera controls are shared. In every view, the geometry is derived from the same certified sequence of disk segments. The views show the active arc or draft; they do not claim simultaneous compatibility of saved arcs. Starting at the specified point on B, draw a chord to the first entry port, jump to its matched exit port, draw the next chord, and end at the specified point on B. The order of the supplied crossings makes this a single connected path; no disconnected extra closed component can be hidden in the input. ## Simplicity certificate Two chords of a convex disk with four distinct endpoints intersect in their interiors if and only if their endpoints alternate in cyclic boundary order. The engine uses only integer side indices and integer port ranks for this predicate. It never uses pixel distances or a floating-point tolerance to certify simplicity. Shared terminal marked endpoints are permitted; degenerate or duplicate chord segments are not. The ranks guarantee distinct crossings on the glued seams, each with exactly two incident half-segments, one from each side. Consequently disjoint chord interiors glue to an arc with embedded interior; a repeated terminal point is the allowed based-loop case. If endpoints alternate, no deformation of the chords with those endpoints fixed can repair that diagram. The UI displays the conflicting chords in red and does not save it as an admissible arc. **This does not prove that the word has no simple representative with a different ordering of ports.** Change the ranks to specify a different drawing when appropriate. ## Why every simple marked arc has a diagram 1. Start with a representative of any such isotopy class. Perturb its interior away from the unmarked cut vertex and transverse to the k cut edges, with finitely many crossings. This fixes its marked endpoints, also when equal. 2. Cut the surface open. The arc becomes a finite collection of disjoint disk arcs, meeting the paired edges in corresponding ordered points. Record the cuts in path order and each point's rank on its side. The two copies of each cut have the same rank order by the chosen gluing. 3. A disjoint family of disk arcs has nonalternating endpoints and can be straightened to the associated chords while preserving those endpoints and their cyclic order. For a repeated terminal endpoint the two terminal germs keep their order as well. The disk isotopy can be chosen to fix its boundary, so it descends through the gluing. 4. The resulting data are accepted by the simplicity check and reconstruct a representative of the original isotopy class. The essentiality convention below only removes classes already excluded in the laboratory definition. Thus the absence of a fixed passage bound matters: every individual class has a finite diagram, although the collection of classes is infinite. This argument establishes coverage, **not uniqueness** of diagrams. Rank insertion, manual editing, path reversal and JSON import/export are available in the UI. The boundary-twist example also supplies an explicit infinite family at fixed k. Starting with the based arc a1 at the only marked point, each collar twist prepends W and appends W inverse, where W = a1²…ak². On each paired side, the four new ports have ranks 1, m+4, m+3, 2 in traversal order and the m old ports shift by 2. The two new outer strands at each end run alongside the disk boundary, so they introduce no crossing. After t twists the word is W^t a1 W^(-t). For k≥2 these are distinct reduced based words (even after reversing arc orientation), hence distinct relative-endpoint isotopy classes. The UI allows any nonnegative number of twists, subject to finite resources. ## Trivial and boundary-segment exclusions The dual cut spine is a rank-k rose. Relative to connectors inside the disk, crossings give a word in its free generators. Adjacent inverse letters cancel. We use that word only to reject the following elementary classes: * empty reduced word and identical endpoints: contractible; * empty reduced word between Pi and P(i+1), in either direction: the straight boundary segment inside B; * from Pn to P1, the reduced word `1 1 2 2 … k k`: the boundary segment passing the unmarked vertex; in reverse, the inverse word. To check the last convention, trace the link of the unmarked vertex starting at the end of B: it crosses `a1+` near its initial end, then `a1+` near its terminal end, then similarly each next pair. This gives two positive crossings per generator. This also handles n=1 and the two different segments for n=2. If n>2 a longer boundary interval containing other marked points is allowed. The word is not used to claim that arbitrary input is simple. Exclusions are combined with the independent geometric embedding certificate above. ## Software checks and limits ### Readable itinerary The main editor lists the traversed crosscaps in order. A denotes the original `a+` semicircle and B denotes `a−`; `A → B` encodes a positive crossing, and `B → A` a negative one. A position selector appears only for repeated visits to the same crosscap. Exchanging positions swaps the two ranks, preserving their permutation. Removing or changing a crosscap renumbers only that crosscap's remaining positions. Moving a step changes traversal order while preserving its position. All of the original encoding is still expressible; the compact textarea remains available under the advanced disclosure. ### The punctured Klein model Reference: Gregorio Franzoni, [The Klein Bottle: Variations on a Theme](https://www.math-art.eu/Documents/pdfs/rtx120801076p.pdf), Notices of the AMS 59 (2012), DOI 10.1090/noti880. We use Dickson's classical piecewise parametrization (equation 4), including its nonsmooth joins, as a surface visualization. It is not a spatial embedding. Intersecting sheets in 3D are distinct surface points unless the parameter gluing identifies them. After shifting the transverse parameter by pi/2, its twist is v ↦ −v. Cutting the two one-sided core curves leaves an annulus parameterized by 0 ≤ t ≤ 4pi, 0 ≤ v ≤ pi. Equal points on each boundary are t and t+2pi; t=0 and t=4pi are also glued. A small disk centered at (0,pi/2) is removed, and the remaining t=0 seam is opened. The resulting domain is a rectangle with a semicircular indentation on each side, hence a disk. Its boundary order gives the original two pairs of crosscap sides. A convex harmonic chart transfers each certified chord into this domain. A monotone angular map near the removed disk distributes all marked points around its boundary. The same surface parametrization maps the mesh, the marks and every arc segment. In particular, matched crosscap ports coincide in 3D, rather than being connected by an invented straight line through space. The Klein tests check both parameter seam identities, the quotient mesh's Euler characteristic −1 and its single boundary component, positive chart triangles, marked-point positions, and continuity of winding arcs at all recollements. Meshing and curve sampling remain numerical approximations. ### Transferring arcs to the crossed-circle drawing `crosscap_view.js` triangulates the planar disk with k holes, opening a slit from each hole to the outer boundary. Slit banks are distinct vertices, even where they share screen coordinates. A collar homeomorphism places all marked points around the free boundary while keeping all slit tips in its unmarked gap. In the cut boundary order, each pair ai+, ai− becomes the two consecutive halves of the corresponding inner circle, traversed in the same direction. Equal ranks therefore become antipodal points. The added slit banks and unmarked outer intervals contain no arc endpoints. Contracting these auxiliary trees recovers the original B,a1+,a1−,… cut convention; this contraction can be performed away from the arc. Thus it does not discard winding classes. The opened mesh is a topological disk. Its boundary vertices are mapped in order to a strictly convex polygon and its interior vertices to positive averages of their neighbors. Straight noncrossing chords in that convex chart are transferred triangle by triangle. Disjoint radial collars round the small polygonal holes to exact circles; the outer angular collar is monotone on every concentric circle. These homeomorphisms preserve the curves' topology. For readability, the display shortens strands across triangles containing neither a crosscap nor another strand. Rounded corners stay inside similarly checked empty triangles. Each such move is an isotopy relative to all ports and the other strands. This is display simplification, not a canonical geodesic or an algorithm for identifying isotopy classes. The chart and curves use floating-point arithmetic. Nonlinear pieces are adaptively sampled at a local coordinate error of 1e-6 (well below a screen pixel at the default scale). This is a numerical drawing, not a formal proof of arbitrary-precision rendering. The exact integer admissibility certificate is independent of these display coordinates. `crosscap_view.test.js` checks 69 charts covering every k from 2 through 24 with 1, 2 and 12 marks: boundary order, orientation of every mesh triangle, antipodal ports and separated holes. It additionally checks winding examples, round-hole clearance and absence of spurious crossings after simplification. `arc_diagrams.test.js` compares integer chord alternation with an independent Euclidean segment-intersection calculation on 2,040 small diagrams; tests reversal, repeated cuts, common endpoints, boundary wrapping, malformed ranks, an accepted simple 600-crossing diagram, and boundary-twist examples at k=2 up to 801 crossings. These tests supplement the argument; they are not a formal verification or an external mathematical review. The engine takes O(L²) time in the explicit number L of segments. Memory, safe machine integers and browser performance remain practical limits. The integrated surface selector currently supports up to 24 crosscaps. Large diagrams can be slow or visually dense; PNG and JSON export are available. The drawn order and the certificate come from the same segment objects. Saved identifiers only deduplicate identical diagrams up to reversal; they are explicitly not isotopy identifiers. This editor does not compute minimal intersection numbers, quasi-triangulations, flips, snake graphs, or simultaneous compatibility of independently saved diagrams. It does not claim that a planar or 3D sketch is an embedding of the surface in space. The quotient is specified by antipodal identification in the base view and by the paired-side arrows in the cut-disk views. ## Animated gluing explanation The question-mark button next to the planar title opens an explanatory dialog. It includes the user-supplied five-stage reference diagram and a continuous parametric reconstruction, with play/pause, a scrubber and stage controls. Nothing runs automatically. Closing the dialog or hiding the page pauses it. For one crosscap the model is an annulus whose inner circle is identified antipodally. Cutting along two opposite radii gives two half-annuli. Their material coordinates are `u,r ∈ [0,1]`, with `r=0` on the inner semicircle. The initial antipodal pairing is `upper(u,0) ~ lower(1-u,0)`; the two copies of each radial cut pair at equal `r`. These material coordinates persist through the animation. Both half-annuli unfold to rectangles. Rotating the upper rectangle by π around its vertical axis in **three dimensions** aligns the inner-edge pairing pointwise. Translating the pieces together then joins these edges. The remaining red/blue pairings combine into the oppositely oriented vertical sides of the usual Möbius rectangle. The unfolding is a linear interpolation from the half-annulus coordinates to rectangle coordinates. Writing `s` for its eased progress, `ρ=30+110r`, and `θ=πu`, the absolute Jacobian divided by 110 is `(1-s)²ρπ + s(1-s)(ρπ+220)sin(θ) + 220s² > 0`. Thus each material patch stays regular during unfolding. The rotation is rigid in space; its planar **projection** can become edge-on. The cut and glue operations are changes of presentation of the quotient surface, not a claim of an ambient planar isotopy between the original and final pictures. The red arrows point toward the inner circle on both copies of the left radial cut. The blue arrows point outward. Reversing both arrows of a pair does not change its gluing. After turning and joining, this convention gives upward arrows on the whole left side and downward arrows on the whole right side, without reversing either colour at an intermediate stage. ### Whole-surface construction for k >= 2 `global_gluing.js` starts with the same disk and arrangement of k crosscaps as the lab's base view. It has no single-collar selector. All k circles are opened in the same animation, ending at the actual lab's `(2k+1)`-gon with cyclic word `B C1A C1B ... CkA CkB`. B is free; Ai and Bi for each crosscap are paired in the same perimeter direction and with equal parameters. 1. Draw disjoint radial slits from each crosscap to the outer boundary. 2. Open the slits, deforming each circular hole into a notch within its own angular sector. Keep both antipodal half-circle labels throughout. The paired auxiliary slit edges shrink to vertices at the end of this step. 3. Increase the radial coordinate of the notched boundary to obtain a circle. 4. Contract the k-1 green boundary intervals between successive crosscaps, retaining the interval preceding C1A as B. Redistribute polar angles and straighten the remaining sides into the lab's regular polygon. These are **explicit contractions of auxiliary trees**, not merely a linear interpolation of two arbitrary pictures. On the quotient surface each slit is an embedded arc from the boundary to an interior point. Contracting it preserves the surface type: a regular neighbourhood is a disk meeting the surface boundary in an interval. The later contracted boundary intervals form a proper boundary arc, not the entire boundary component. They likewise have half-disk neighbourhoods. No crosscap edge is contracted. The animation explains the unmarked surface; boundary marks can be transported into the retained free interval before these contractions. It does not animate arcs. For the geometric opening, write `d` and `r` for a hole's radial position and radius and `q` for its sector angle. On its clockwise circumference, use `u in [0,1]`. The initial polar radius is `R0(u)=sqrt(d*d+r*r+2*d*r*cos(2*pi*u))` and the initial relative angle is `theta0(u)=atan2(-r*sin(2*pi*u),d+r*cos(2*pi*u))`. Interpolate radius to `R1(u)=d-r+(1-d+r)*abs(2*u-1)` and relative angle to `beta*(2*u-1)`, where `beta=.78*pi/k`. Both radii strictly decrease on the first half and increase on the second. The relative angle has strictly opposite signs on the two halves. Thus the two halves cannot intersect. For all supported k, `asin(r/d)