Knots and Links

PureCubicalKnot
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
ViewPureCubicalKnot(L)

Inputs a pure cubical link L and displays it.

KnotSum
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
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
AlexanderMatrix(G)

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

AlexanderPolynomial
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
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
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.