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<Chapter><Heading> Knots and Links</Heading> <Section><Heading> </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> , <URL><Link>../tutorial/chap2.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../tutorial/chap3.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../tutorial/chap4.html</Link><LinkText>4</LinkText></URL> , <URL><Link>../tutorial/chap6.html</Link><LinkText>5</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCoveringSpaces.html</Link><LinkText>6</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>7</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>8</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>9</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>10</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>11</LinkText></URL>
</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> , <URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>2</LinkText></URL>
</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> , <URL><Link>../tutorial/chap3.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../tutorial/chap6.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>4</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>5</LinkText></URL>
</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>
</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> , <URL><Link>../tutorial/chap2.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../tutorial/chap5.html</Link><LinkText>3</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>4</LinkText></URL>
</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>../tutorial/chap5.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutPersistent.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutKnots.html</Link><LinkText>3</LinkText></URL>
</Description> </ManSection> </Section> </Chapter>
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