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 ≤ 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 ≤ 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 ≤ 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 ≤ 6. If n ≥ 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 ≤ 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 ≥ 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 ∈ Q and an element stigma ∈ 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 ∈ Q and an element stigma ∈ 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 [n1,n2,...,nk] where nj is a partition of the number of homomorphisms from the considered knot quandle to the j-th connected quandle of size n ≤ i ≤ 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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