.. _Dictionary:

***********
Dictionary
***********
Here are explained some usefull terms.

.. _codes:

PD code and EM code
====================

EM code (Ewing-Millett code) and PD code (planar diagram code) are methods of 
descirbing knots in detail. To find EM code of your knot diagram:
    
* Give each crossing a number, 
* Give each crossing a sign according to,
* For each crossing:

  * Name "a" direction of outgoing overpassing arc, 
  * Name every other direction in a clockwise order "b", "c" and "d",
  * In this way, every arc -- by which we mean here continuous piece 
    of chain between two neighbour crossings -- consists of two ends, 
    each described by a number and a letter.


* A code for a crossing consists of its number, its sign, and four two-character 
  descriptions of opposite ends of four arcs coming out of the crossing (in order
  "a", "b", "c", "d").  
* A code for a structure consists of a list of codes for crossings.

.. figure:: _static/emcode.png
    :scale: 25%
    :alt: EM code

Finding PD code is easier:

* Go along the structure according to its orientation and after each 
  crossing asign next number to it (starting from 1).
* Each crossing is described by "X" symbol with numbers of its neighbouring
  arcs: counter-clockwise starting from ingoing underpassing.
* Structure code is described by a list of its crossings.

In case of spatial graphs (theta-curves, handcuffs) PD code can be extended. 
In such a case every vertex connected to three arcs is described by "V" 
symbol with numbers of its neighbouring arcs in any order.

.. figure:: _static/pdcode.png
    :scale: 25%
    :alt: PD code


.. _reidemeister_moves:

Reidemeister moves
===================
Set of basic moves that change knot diagram but doesn't alter knot topology.

.. figure:: _static/ReidemeisterMoves.gif
    :scale: 80%
    :alt: Reidemeister moves
    
    Three types of Reidemeister moves


.. _KMT_algorithm:

KMT algorithm
==============
Based on `Koniaris's and Muthukamar's method <https://doi.org/10.1063/1.460889>`_.  
This algorithm analyzes all triangles in a chain made by three consecutive 
points, and removes the middle point in case a given triangle is not 
intersected by any other segment of the chain. In effect, after a number of 
iterations, the initial chain is replaced by (much) shorter and simpler chain 
of the same topological type. 
 
.. figure:: _static/kmt.png
    :scale: 40%
    :alt: KMT algorithm
    
    Representation of KMT algorithm