.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/complexes/plot_03_mayer_vietoris_sequence.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_topology_complexes_plot_03_mayer_vietoris_sequence.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_topology_complexes_plot_03_mayer_vietoris_sequence.py:


The Mayer-Vietoris sequence (1929-1930)
=======================================

Walther Mayer and Leopold Vietoris computed the homology of a union
from its pieces through the long exact sequence

.. math:: \cdots \to H_k(A\cap B) \to H_k(A)\oplus H_k(B) \to H_k(A\cup B) \to H_{k-1}(A\cap B) \to \cdots

Cut a torus into two annuli :math:`A` and :math:`B`. They meet in two
circles. Exactness, plus the rank of the map induced by the inclusions,
predicts :math:`\beta(T^2) = (1, 2, 1)`. The second loop of the torus
comes from the connecting map :math:`\delta`.

.. GENERATED FROM PYTHON SOURCE LINES 17-35

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.topology import circle, mayer_vietoris, sphere, torus
    from mathematicskit.topology.visualizers import plot_complex

    T = torus(16, 6)
    A = T.induced_subcomplex([v for v in T.vertices if v // 6 in range(9)])
    B = T.induced_subcomplex([v for v in T.vertices if v // 6 in (*range(8, 16), 0)])
    cases = {"torus = annulus ∪ annulus": (A, B)}
    C = circle(8)
    cases["circle = arc ∪ arc"] = (C.induced_subcomplex(range(5)), C.induced_subcomplex([4, 5, 6, 7, 0]))
    S = sphere(2, "cross_polytope")
    cases["sphere = disc ∪ disc"] = (S.induced_subcomplex([0, 1, 2, 3, 4]), S.induced_subcomplex([0, 1, 2, 3, 5]))
    for name, (X, Y) in cases.items():
        r = mayer_vietoris(X, Y)
        print(f"{name}: beta(A) = {r.betti_a}, beta(B) = {r.betti_b}, beta(A∩B) = {r.betti_intersection}, rank i_* = {r.rank_inclusion}")
        print(f"{'':>26} predicted beta(A∪B) = {r.predicted_union}, computed = {r.betti_union}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    torus = annulus ∪ annulus: beta(A) = (1, 1, 0), beta(B) = (1, 1, 0), beta(A∩B) = (2, 2), rank i_* = (1, 1, 0)
                               predicted beta(A∪B) = (1, 2, 1), computed = (1, 2, 1)
    circle = arc ∪ arc: beta(A) = (1, 0), beta(B) = (1, 0), beta(A∩B) = (2,), rank i_* = (1, 0)
                               predicted beta(A∪B) = (1, 1), computed = (1, 1)
    sphere = disc ∪ disc: beta(A) = (1, 0, 0), beta(B) = (1, 0, 0), beta(A∩B) = (1, 1), rank i_* = (1, 0, 0)
                               predicted beta(A∪B) = (1, 0, 1), computed = (1, 0, 1)




.. GENERATED FROM PYTHON SOURCE LINES 36-38

The two annuli and their intersection
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-48

.. code-block:: Python


    fig = plt.figure(figsize=(7, 6))
    ax = fig.add_subplot(projection="3d", computed_zorder=False)
    plot_complex(A, ax=ax, face_color="tab:blue", alpha=0.35, vertex_size=0)
    plot_complex(B, ax=ax, face_color="tab:orange", alpha=0.35, vertex_size=0)
    for edge in (A & B).simplices(1):
        ax.plot(*T.coordinates[list(edge)].T, color="k", lw=3)
    ax.view_init(elev=50, azim=-60)
    ax.set_axis_off()
    ax.set_title("A (blue) ∪ B (orange) = torus; A ∩ B = two circles")



.. image-sg:: /api/gallery/topology/complexes/images/sphx_glr_plot_03_mayer_vietoris_sequence_001.png
   :alt: A (blue) ∪ B (orange) = torus; A ∩ B = two circles
   :srcset: /api/gallery/topology/complexes/images/sphx_glr_plot_03_mayer_vietoris_sequence_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'A (blue) ∪ B (orange) = torus; A ∩ B = two circles')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.032 seconds)


.. _sphx_glr_download_api_gallery_topology_complexes_plot_03_mayer_vietoris_sequence.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/topology/complexes/plot_03_mayer_vietoris_sequence.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_03_mayer_vietoris_sequence.ipynb <plot_03_mayer_vietoris_sequence.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_03_mayer_vietoris_sequence.py <plot_03_mayer_vietoris_sequence.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_03_mayer_vietoris_sequence.zip <plot_03_mayer_vietoris_sequence.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_