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Class math.topol.Braid.polys 

java.lang.Object
   |
   +----math.topol.Braid.polys
public class polys
extends Object
Computes general 2-variable (HOMFLYPT) polynomial from braid presentation of a knot. In 1991, Hugh Morton and Hamish Short gave an algorithm for computing the so-called HOMFLY polynomial (from which it is easy to obtain the Jones Polynomial, the classical Alexander polynomial, and the ``2-variable'' polynomial) from a braid presentation of a link. The given method has worst-case running time proportional to the square of the number of crossings of a braid, once a fixed-cost (depending only on the braid-index) initialization is made. This implementation is based on pascal code of Hugh Morton and Hamish Short.

Version:
$Id: polys.java,v 1.4 1999/02/12 20:35:15 djun Exp djun $

Variable Index

o zerovect

Constructor Index

o polys()

Method Index

o binomial(int, int)
Get small binomial coefficient ($n$ choose $k$) as an int, for small values of $k$.
o init()
Initializes the variables required for the polynomial computations.
o pointer(int, int)
Get factorial expansion of $g$ down to $(r1 - 1)$ position.
o triang(int, int)
Returns large binomial coefficient $(-1)^k1 \times k!/k1! (k-k1)!$
o two_variable(PrintWriter, StringBuffer)
Computes the two-variable polynomial from a braid.

Variables

o zerovect 
 public static final int zerovect[]

Constructors

o polys 
 public polys()

Methods

o init 
 public void init()
Initializes the variables required for the polynomial computations. This method must be called before any polynomial may be calculated. Since it initializes an structures which are factorially large in the size of the input, it may consume significant computational resources. Once the global tables are initiallized, however, subsequent polynomial computations are fast.

o two_variable 
 public void two_variable(PrintWriter out,
                          StringBuffer braidstr)
Computes the two-variable polynomial from a braid.

o triang 
 public static float triang(int k,
                            int k1)
Returns large binomial coefficient $(-1)^k1 \times k!/k1! (k-k1)!$

o pointer 
 public static int[] pointer(int g,
                             int r1)
Get factorial expansion of $g$ down to $(r1 - 1)$ position.

o binomial 
 public static int binomial(int v,
                            int v1)
Get small binomial coefficient ($n$ choose $k$) as an int, for small values of $k$. 

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