GAP (HAP commands) - Chapter 31: Knots and Links

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31 Knots and Links
31.1
31.1-1 PureCubicalKnot
31.1-2 ViewPureCubicalKnot
31.1-3 KnotSum
31.1-4 KnotGroup
31.1-5 AlexanderMatrix
31.1-6 AlexanderPolynomial
31.1-7 ProjectionOfPureCubicalComplex
31.1-8 ReadPDBfileAsPureCubicalComplex

### 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.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11

##### 31.1-2 ViewPureCubicalKnot

`‣ ViewPureCubicalKnot`( L ) | ( function )

Inputs a pure cubical link \(L\) and displays it.

Examples: 1 , 2

##### 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.

Examples: 1 , 2 , 3 , 4 , 5

##### 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.

Examples: 1

##### 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.

Examples:

##### 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.

Examples: 1 , 2 , 3 , 4

##### 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.

Examples:

##### 31.1-8 ReadPDBfileAsPureCubicalComplex

`‣ ReadPDBfileAsPureCubicalComplex`( file ) | ( function )

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

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.

Examples: 1 , 2 , 3

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