.. _sphx_glr_api_gallery_topology_curves:

Curves
------

Topological invariants of closed curves: linking and turning numbers.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Gauss&#x27;s double integral \frac{1}{4\pi}\oint\oint \frac{(r_1 - r_2)\cdot(dr_1 \times dr_2)}{|r_1 - r_2|^3} counts how many times one closed curve winds around another. It is an integer that no deformation can change without passing one curve through the other. Below: the unlink (0), the Hopf link (\pm 1, the sign set by the orientations), and torus links whose components wind n times around each other.">

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  .. image:: /api/gallery/topology/curves/images/thumb/sphx_glr_plot_01_gauss_linking_number_thumb.png
    :alt:

  :doc:`/api/gallery/topology/curves/plot_01_gauss_linking_number`

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      <div class="sphx-glr-thumbnail-title">Gauss's linking number (1833)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="As you walk once around a closed plane curve, your heading turns by a whole number of full turns. Hopf proved that for every simple closed curve that number is \pm 1. Curves that cross themselves can have any turning number: 0 for a figure eight, 2 for a limaçon with an inner loop, 4 for a circle carrying three small inner loops.">

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  .. image:: /api/gallery/topology/curves/images/thumb/sphx_glr_plot_02_hopf_umlaufsatz_thumb.png
    :alt:

  :doc:`/api/gallery/topology/curves/plot_02_hopf_umlaufsatz`

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      <div class="sphx-glr-thumbnail-title">Hopf's Umlaufsatz: the turning number of a closed curve (1935)</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/topology/curves/plot_01_gauss_linking_number
   /api/gallery/topology/curves/plot_02_hopf_umlaufsatz