### 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).

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##### 32.1-2 PD2GC
 ‣ PD2GC( PD ) ( function )

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

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##### 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.

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##### 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.

##### 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.

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##### 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.

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##### 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.

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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.

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##### 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.

##### 32.1-10 IsQuandle
 ‣ IsQuandle( M ) ( function )

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

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##### 32.1-11 Quandles
 ‣ Quandles( n ) ( function )

Returns a list of all quandles of size n, n ≤ 6. If n ≥ 7, it returns fail.

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##### 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.

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##### 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).

##### 32.1-14 IsLatin
 ‣ IsLatin ( global variable )

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

##### 32.1-15 IsConnectedQuandle
 ‣ IsConnectedQuandle ( global variable )

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

##### 32.1-16 ConnectedQuandles
 ‣ ConnectedQuandles( n ) ( function )

Returns a list of all connected quandles of size n.

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##### 32.1-17 ConnectedQuandle
 ‣ ConnectedQuandle( n, k ) ( function )

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

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##### 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).

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##### 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.

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##### 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.

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##### 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.

##### 32.1-22 RightMultiplicationGroupAsPerm
 ‣ RightMultiplicationGroupAsPerm ( global variable )

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

##### 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.

##### 32.1-24 AutomorphismGroupQuandleAsPerm
 ‣ AutomorphismGroupQuandleAsPerm( Q ) ( function )

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

##### 32.1-25 AutomorphismGroupQuandle
 ‣ AutomorphismGroupQuandle( Q ) ( function )

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

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