<Chapter><Heading> Knots and Links</Heading>

<Section><Heading> &nbsp;</Heading>

<ManSection>

<Func Name="PureCubicalKnot" Arg="L"/>

<Func Name="PureCubicalKnot" Arg="n,i"/>
<Description>
<P/>

Inputs a list <M>L=[[m1,n1], [m2,n2], ..., [mk,nk]]</M> of pairs of integers
 describing 
 a cubical arc presentation of a 
link with all vertical lines at the front and all horizontal lines at the back. The bottom horizontal line extends from the m1-th column to the n1-th column. 
The second to bottom horizontal line extends from the m2-th column to the n2-th column. And so on. The link is returned 
as a 3-dimensional pure cubical complex.
<P/>

Alternatively the function inputs two
integers <M>n</M>, <M>i</M> and returns the <M>i</M>-th prime knot on <M>n</M> crossings.

<P/><B>Examples:</B> <URL><Link>../tutorial/chap1.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../tutorial/chap2.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../tutorial/chap3.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../tutorial/chap4.html</Link><LinkText>4</LinkText></URL>&nbsp;,
<URL><Link>../tutorial/chap6.html</Link><LinkText>5</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCoveringSpaces.html</Link><LinkText>6</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>7</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>8</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>9</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>10</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>11</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="ViewPureCubicalKnot" Arg="L"/>
<Description>
<P/>

Inputs a pure cubical link <M>L</M> and displays it.
<P/>

<P/><B>Examples:</B> <URL><Link>../tutorial/chap1.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>2</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="KnotSum" Arg="K,L"/>
<Description>
<P/>

Inputs two pure cubical knots <M>K</M>, <M>L</M> and returns their sum as a pure cubical knot. This function is not defined for links with more than one component.
<P/>

<P/><B>Examples:</B> <URL><Link>../tutorial/chap2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../tutorial/chap3.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../tutorial/chap6.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>4</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>5</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="KnotGroup" Arg="K"/>
<Description>
<P/>

Inputs a pure cubical link <M>K</M> and returns the fundamental group of its complement. The group is returned as a finitely presented group.
<P/>

<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>1</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="AlexanderMatrix" Arg="G"/>
<Description>
<P/>

Inputs a finitely presented group <M>G</M> whose abelianization is infinite cyclic. It returns the Alexander matrix of the presentation. 
<P/>

<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Func Name="AlexanderPolynomial" Arg="K"/>

<Func Name="AlexanderPolynomial" Arg="G"/>
<Description>
<P/>

Inputs either a pure cubical knot <M>K</M> or a finitely presented group <M>G</M> whose abelianization is infinite cyclic. The Alexander Polynomial is returned. 
<P/>

<P/><B>Examples:</B> <URL><Link>../tutorial/chap1.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../tutorial/chap2.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="ProjectionOfPureCubicalComplex" Arg="K"/>
<Description>
<P/>

Inputs  an $n$-dimensional
 pure cubical complex <M>K</M> and returns an n-1-dimensional pure cubical complex K'. The returned complex is obtained by projecting Euclidean n-space onto Euclidean n-1-space. 
<P/>

<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Func Name="ReadPDBfileAsPureCubicalComplex" Arg="file"/>

<Func Name="ReadPDBfileAsPureCubicalComplex" Arg="file,m ,c"/>
<Description>
<P/>

Inputs  a protein database file describing a protein, and optionally
 inputs  a positive integer m and character string c. The default values for the optional inputs are m=5 and c="A".  
 It loads the chain of amino acids labelled by c in the file as a 3-dimensional pure cubical complex of the homotopy type of a circle. 
<P/>

It might happen that the function fails to construct a pure cubical complex of the homotopy type of a circle. In this case  retry with a larger integer m.

<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutPersistent.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>2</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

</Section>
</Chapter>

