### 31 Knots and Links

#### 31.1

##### 31.1-1 PureCubicalKnot
 ‣ PureCubicalKnot( L ) ( function )
 ‣ PureCubicalKnot( n, i ) ( function )

Inputs a list L=[[m1,n1], [m2,n2], ..., [mk,nk]] 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.

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

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##### 31.1-2 ViewPureCubicalKnot
 ‣ ViewPureCubicalKnot( L ) ( function )

Inputs a pure cubical link L and displays it.

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##### 31.1-3 KnotSum
 ‣ KnotSum( K, L ) ( function )

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

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##### 31.1-4 KnotGroup
 ‣ KnotGroup( K ) ( function )

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

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##### 31.1-5 AlexanderMatrix
 ‣ AlexanderMatrix( G ) ( function )

Inputs a finitely presented group G whose abelianization is infinite cyclic. It returns the Alexander matrix of the presentation.

##### 31.1-6 AlexanderPolynomial
 ‣ AlexanderPolynomial( K ) ( function )
 ‣ AlexanderPolynomial( G ) ( function )

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

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##### 31.1-7 ProjectionOfPureCubicalComplex
 ‣ ProjectionOfPureCubicalComplex( K ) ( function )

Inputs an $n$-dimensional pure cubical complex K 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.

 ‣ ReadPDBfileAsPureCubicalComplex( file ) ( function )
 ‣ ReadPDBfileAsPureCubicalComplex( file, m, c ) ( function )