counting-components

Overview

Python module written in Rust to count components after resolving intersections via surgery. Consider curves γ and δ on a possibly non-orientable surface S, where γ is a two-sided curve, with intersection number k. We encode the intersection using a tuple of a permutation π in S_k, and a subset f of {0,1,...,k-1}. The permutation encodes how strands of δ come back to γ, and f encodes whether the strand comes back with the local orientation flipped.

We take m parallel copies of δ, and n parallel copies of γ, and resolve all the intersections in a consistent manner: we pick a direction on γ, and always turn left.

The functions in this module compute the number of one-sided and two-sided components in the resulting multicurve after resolving all the intersections.

Documentation

Objects

• SignedPermutation: A signed permutation is constructed via SignedPermutation(<perm>, <flipset>), where <perm> is a list containing the numbers 0 to n-1 representing a permutation, and <flipset> is a list of all strands whose local orientation gets flipped.
• PyStrand: A strand is constructed via PyStrand(<strand-type>, m, n), where <strand-type> is either 't' or 'p' corresponding to strands in the transverse or permutation direction. If it is a transverse strand, only the m argument is required, and indicates the index of the transverse strand, and if it is a permutation direction strand, then m and n indicate the permutation and copy index.

Functions

• get_next_major_strand(perm, m, n, strand): Given a permutation perm, m, n, and a strand strand, this function outputs the next strand the strand gets surgered to.
• has_one_component(perm, m, n): Determines whether the resolved multicurve only has one component.
• count_components_with_orientability(perm, m, n): Returns a tuple (x, y), where x is the number of two-sided curves in the resulting multicurve, and y is the number of one-sided curves in the resulting multicurve.
• count_components_upto_complexity(perm, complexity): Returns a list of all (m,n) such that gcd(m,n) == 1, m+n <= k, and the corresponding components. This function uses rayon to run on all available threads.
• two_sided_multicurves_upto_complexity(perm, complexity): Returns a list of all (m,n) such that gcd(m,n) == 1, m+n <= k, such that the resolved multicurve only has two-sided components. This function uses rayon to run on all available threads.

Build instructions

To build this library, you will need to pip install maturin, and the nightly version of the Rust compiler. In the root of this project, run the following commands.

maturin build --release
pip install target/wheels/<file.whl>

You may need to upgrade to the latest version of pip to install the .whl file.