## 1.7  LinKnot functions and KL notation

The program Knot 2000 (K2K) was created by Mitsuyuki Ochiai and Noriko Imafuji from the Graduate School of human culture, Nara Women's University, Nara, Japan. In 2003 the Mathematica-based program LinKnot created by Slavik Jablan and Radmila Sazdanovic from the Mathematical Institute, Belgrade, Serbia, was added to it. LinKnot is completely compatible with Knot2000 and provides tools for working with KLs given in Conway notation without any restriction on the number of crossings.

An input for the original version of the program K2K is a KL diagram traced by mouse on the mouse tracking window, by using the function GetPdatabyTracking (webMathematica GetPdatabyTracking). The LinKnot function fCreatePData (webMathematica fCreatePData) gives the possibility to use a Conway symbol of a KL as an input. For example, from the Conway symbol of the non-alternating link K="111*2.2.2.-2 0.2 0.-2 0" it calculates the corresponding P-data

 {{14,3},{-16,-20,-12,23,29,-6,-30,32,-2,-24,-4,7,-18,14,-28,26,9}}.
(2)

P-data is a list having two entries. The first is a list of the numbers of crossings in each component, and the second is a list of numbers derived from the KL, called P-word. P-data are the basic input for all other K2K and LinKnot functions. The first of them is the function ShowKnotfromPdata (webMathematica ShowKnotfromPdata) for drawing KL diagrams with rendering, as a Mathematica graphics. For example, the given animation shows the drawing of the non-alternating 2-component link given by its P-data. Its 3D-image that can be rotated and transformed you can get with rendering by OpenGL, by using the function ShowKnotbyOpengl (webMathematica ShowKnotbyOpengl).

The LinKnot function fCreateGraphics (webMathematica fCreateGraphics) enables a communication with the program KnotPlot written by Robert Sharein. As an input for this function you can use the Conway notation of KLs as a Mathematica string. The function fCreateGraphics creates the file graphics.txt that can be loaded in KnotPlot by writing "load graphics.txt" in the KnotPlot command line. After that, you can work with it in KnotPlot in the same way as with any other KnotPlot file. Next figure shows the drawing of the alternating 6-component link

 K = "101*2 1.2 1 0.2 1.2 1 0.2 1.2 1 0.2 1.2 1 0.2 1.2 1 0"

created by the function fCreateGraphics and processed in KnotPlot, before and after relaxation and 3D-rotation.  Most of illustrations in this book are creatad by exporting LinKnot graphics of KLs to KnotPlot.