libisotopy

A C++ shared library that computes the real scheme (ambient isotopy type in $\mathbb{RP}^2$) of a T-curve given by Viro's combinatorial patchworking: a unimodular triangulation of $d\cdot\Delta_2$ together with a sign distribution on its lattice points.

The algorithm runs in near-quadratic time in the degree $d$ via union-find on the reflected signed triangulation, region-adjacency-graph traversal, and rooted-tree canonicalization. It is described in:

Geiselmann, Joswig, Kastner, Mundinger, Pokutta, Spiegel, Wack, Zimmer. Fast Isotopy Computation for T-Curves. ICMS 2026. arXiv:2604.09221

and was used to verify the realizability of all 121 degree-seven real schemes (121 Patchworked Curves of Degree Seven; data at dmg-lab/CombinatorialPatchworking).

Status. The core algorithm (Isotopy::Graph) is tested and stable. The surrounding build/packaging is geared toward our own use; if you embed the library elsewhere, please report any issues.

Features

• Isotopy::Graph — computes the real scheme of a patchwork and emits its Viro (Rohlin–Viro) notation.
• Queries: viro_notation(), even_regions(), odd_regions(), is_mcurve().

Input format

Isotopy::Graph(delta, sign, edges) takes:

• delta — the degree $d$.
• sign — the sign distribution on the lattice points of $d\cdot\Delta_2$, in the standard lattice-point order (a std::vector<bool> of length $\tfrac12(d{+}1)(d{+}2)$).
• edges — the triangulation as an edge list over lattice-point indices (std::vector<std::pair<int,int>>).

Documentation

Man pages are generated by Doxygen from the header:

make docs                            # regenerate from include/isotopy_graph.h
man docs/man/man3/Isotopy_Graph.3   # Isotopy::Graph class reference

Worked examples are the test cases in tests/isotopy_graph.cpp.

Building

A Makefile drives the build and tests:

make        # build the shared library
make test   # run tests
make clean  # remove build artifacts

Usage

Include the header and link against the library:

#include "isotopy_graph.h"

g++ myfile.cpp -L/path/to/lib -lisotopy

Example (degree 2):

#include "isotopy_graph.h"
int main() {
    int delta = 2;
    std::vector<bool> sign = {true, true, true, true, true, true};
    std::vector<std::pair<int, int>> edges =
        {{0,1},{1,2},{0,3},{3,4},{2,4},{1,3},{1,4},{4,5},{3,5}};
    Isotopy::Graph g(delta, sign, edges);
    g.isotopy_type();
    std::string notation = g.viro_notation();
    int even = g.even_regions();
    int odd  = g.odd_regions();
    bool mcurve = g.is_mcurve();
}