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27 Knots and Links

### 27 Knots and Links

`PureCubicalKnot(L)` `PureCubicalKnot(n,i)`

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.

`ViewPureCubicalKnot(L)` Inputs a pure cubical link L and displays it.
`KnotSum(K,L)` 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.
`KnotGroup(K)` Inputs a pure cubical link K and returns the fundamental group of its complement. The group is returned as a finitely presented group.
`AlexanderMatrix(G)` Inputs a finitely presented group G whose abelianization is infinite cyclic. It returns the Alexander matrix of the presentation.
`AlexanderPolynomial(K)` `AlexanderPolynomial(G)` Inputs either a pure cubical knot K or a finitely presented group G whose abelianization is infinite cyclic. The Alexander Polynomial is returned.
`ProjectionOfPureCubicalComplex(K)` 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)` `ReadPDBfileAsPureCubicalComplex(file,m ,c)`

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.

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.

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