GAP (HAP commands) - Chapter 32: Knots and Quandles

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32 Knots and Quandles
32.1
32.1-1 PresentationKnotQuandle
32.1-2 PD2GC
32.1-3 PlanarDiagramKnot
32.1-4 GaussCodeKnot
32.1-5 PresentationKnotQuandleKnot
32.1-6 NumberOfHomomorphisms
32.1-7 PartitionedNumberOfHomomorphisms
32.1-8 ConjugationQuandle
32.1-9 FirstQuandleAxiomIsSatisfied
32.1-10 IsQuandle
32.1-11 Quandles
32.1-12 Quandle
32.1-13 IdQuandle
32.1-14 IsLatin
32.1-15 IsConnectedQuandle
32.1-16 ConnectedQuandles
32.1-17 ConnectedQuandle
32.1-18 IdConnectedQuandle
32.1-19 IsQuandleEnvelope
32.1-20 QuandleQuandleEnvelope
32.1-21 KnotInvariantCedric
32.1-22 RightMultiplicationGroupAsPerm
32.1-23 RightMultiplicationGroup
32.1-24 AutomorphismGroupQuandleAsPerm
32.1-25 AutomorphismGroupQuandle

### 32 Knots and Quandles

#### 32.1

Knots

##### 32.1-1 PresentationKnotQuandle

`‣ PresentationKnotQuandle`( gaussCode ) | ( function )

Inputs a Gauss Code of a knot (with the orientations; see \(GaussCodeOfPureCubicalKnot\) in HAP package) and outputs the generators and relators of the knot quandle associated (in the form of a record).

Examples: 1 , 2 , 3 , 4

##### 32.1-2 PD2GC

`‣ PD2GC`( PD ) | ( function )

Inputs a Planar Diagram of a knot; outputs the Gauss Code associated (with the orientations).

Examples: 1 , 2 , 3

##### 32.1-3 PlanarDiagramKnot

`‣ PlanarDiagramKnot`( n, k ) | ( function )

Returns a Planar Diagram for the \(k\)-th knot with \(n\) crossings (\(n \leq 12\)) if it exists; fail otherwise.

Examples: 1 , 2 , 3

##### 32.1-4 GaussCodeKnot

`‣ GaussCodeKnot`( n, k ) | ( function )

Returns a Gauss Code (with orientations) for the \(k\)-th knot with \(n\) crossings (\(n \leq 12\)) if it exists; fail otherwise.

Examples:

##### 32.1-5 PresentationKnotQuandleKnot

`‣ PresentationKnotQuandleKnot`( n, k ) | ( function )

Returns generators and relators (in the form of a record) for the \(k\)-th knot with \(n\) crossings (\(n \leq 12\)) if it exists; fail otherwise.

Examples: 1 , 2 , 3

##### 32.1-6 NumberOfHomomorphisms

`‣ NumberOfHomomorphisms`( genRelQ, finiteQ ) | ( function )

Inputs generators and relators \(genRelQ\) of a knot quandle (in the form of a record, see above) and a finite quandle \(finiteQ\); outputs the number of homomorphisms from the former to the latter.

Examples: 1 , 2 , 3

##### 32.1-7 PartitionedNumberOfHomomorphisms

`‣ PartitionedNumberOfHomomorphisms`( genRelQ, finiteQ ) | ( function )

Inputs generators and relators \(genRelQ\) of a knot quandle (in the form of a record, see above) and a finite connected quandle \(finiteQ\); outputs a partition of the number of homomorphisms from the former to the latter.

Examples: 1

Quandles

##### 32.1-8 ConjugationQuandle

`‣ ConjugationQuandle`( G, n ) | ( function )

Inputs a finite group \(G\) and an integer \(n\); outputs the associated \(n\)-fold conjugation quandle.

Examples: 1 , 2

##### 32.1-9 FirstQuandleAxiomIsSatisfied

`‣ FirstQuandleAxiomIsSatisfied`( M ) | ( function )

`‣ SecondQuandleAxiomIsSatisfied`( M ) | ( function )

`‣ ThirdQuandleAxiomIsSatisfied`( M ) | ( function )

Inputs a finite magma \(M\); returns true if \(M\) satisfy the first/second/third axiom of a quandle, false otherwise.

Examples:

##### 32.1-10 IsQuandle

`‣ IsQuandle`( M ) | ( function )

Inputs a finite magma \(M\); returns true if \(M\) is a quandle, false otherwise.

Examples: 1 , 2 , 3

##### 32.1-11 Quandles

`‣ Quandles`( n ) | ( function )

Returns a list of all quandles of size \(n\), \(n \leq 6\). If \(n \geq 7\), it returns fail.

Examples: 1 , 2 , 3 , 4 , 5 , 6

##### 32.1-12 Quandle

`‣ Quandle`( n, k ) | ( function )

Returns the \(k\)-th quandle of size \(n\) (\(n \leq 6\)) if such a quandle exists, fail otherwise.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7

##### 32.1-13 IdQuandle

`‣ IdQuandle`( Q ) | ( function )

Inputs a quandle \(Q\); and outputs a list of integers [\(n\),\(k\)] such that \(Q\) is isomorphic to \(Quandle(n,k)\). If \(n \geq 7\), then it returns [\(n\),fail] (where \(n\) is the size of \(Q\)).

Examples:

##### 32.1-14 IsLatin

`‣ IsLatin` | ( global variable )

Inputs a finite quandle \(Q\); returns true if \(Q\) is latin, false otherwise.

Examples:

##### 32.1-15 IsConnectedQuandle

`‣ IsConnectedQuandle` | ( global variable )

Inputs a finite quandle \(Q\); returns true if \(Q\) is connected, false otherwise.

Examples:

##### 32.1-16 ConnectedQuandles

`‣ ConnectedQuandles`( n ) | ( function )

Returns a list of all connected quandles of size \(n\).

Examples: 1 , 2 , 3

##### 32.1-17 ConnectedQuandle

`‣ ConnectedQuandle`( n, k ) | ( function )

Returns the \(k\)-th quandle of size \(n\) if such a quandle exists, fail otherwise.

Examples: 1 , 2 , 3 , 4

##### 32.1-18 IdConnectedQuandle

`‣ IdConnectedQuandle`( Q ) | ( function )

Inputs a connected quandle \(Q\); and outputs a list of integers [\(n\),\(k\)] such that \(Q\) is isomorphic to \(ConnectedQuandle(n,k)\).

Examples: 1

##### 32.1-19 IsQuandleEnvelope

`‣ IsQuandleEnvelope`( Q, G, e, stigma ) | ( function )

Inputs a set \(Q\), a permutation group \(G\), an element \(e \in Q\) and an element \(stigma \in G\); returns true if this structure describes a quandle envelope, false otherwise.

Examples: 1 , 2 , 3

##### 32.1-20 QuandleQuandleEnvelope

`‣ QuandleQuandleEnvelope`( Q, G, e, stigma ) | ( function )

Inputs a set \(Q\), a permutation group \(G\), an element \(e \in Q\) and an element \(stigma \in G\). If this structure describes a quandle envelope, the function returns the quandle from this quandle envelope; and fail otherwise. Nb: this quandle is a connected quandle.

Examples: 1 , 2 , 3

##### 32.1-21 KnotInvariantCedric

`‣ KnotInvariantCedric`( genRelQ, n, m ) | ( function )

Inputs generators and relators of a knot quandle (in the form of a record, see above) and two integers \(n\) and \(m\); outputs a list [\(n\)1,\(n\)2,...,\(n\)k] where \(n\)j is a partition of the number of homomorphisms from the considered knot quandle to the \(j\)-th connected quandle of size \(n \leq i \leq m\).

Examples:

##### 32.1-22 RightMultiplicationGroupAsPerm

`‣ RightMultiplicationGroupAsPerm` | ( global variable )

Inputs a connected quandle \(Q\); output its right multiplication group whose elements are permutations.

Examples:

##### 32.1-23 RightMultiplicationGroup

`‣ RightMultiplicationGroup` | ( global variable )

Inputs a connected quandle \(Q\); output its right multiplication group whose elements are mappings from \(Q\) to \(Q\).

Examples:

##### 32.1-24 AutomorphismGroupQuandleAsPerm

`‣ AutomorphismGroupQuandleAsPerm`( Q ) | ( function )

Inputs a connected quandle \(Q\); outputs its automorphism group whose elements are permutations.

Examples:

##### 32.1-25 AutomorphismGroupQuandle

`‣ AutomorphismGroupQuandle`( Q ) | ( function )

Inputs a connected quandle \(Q\); outputs its automorphism group whose elements are mappings from \(Q\) to \(Q\).

Examples: 1 , 2 , 3

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