Knots
‣ 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).
‣ PD2GC( PD ) | ( function ) |
Inputs a Planar Diagram of a knot; outputs the Gauss Code associated (with the orientations).
‣ PlanarDiagramKnot( n, k ) | ( function ) |
Returns a Planar Diagram for the \(k\)-th knot with \(n\) crossings (\(n \leq 12\)) if it exists; fail otherwise.
‣ 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:
‣ 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.
‣ 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.
‣ 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
‣ ConjugationQuandle( G, n ) | ( function ) |
Inputs a finite group \(G\) and an integer \(n\); outputs the associated \(n\)-fold conjugation quandle.
‣ 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:
‣ IsQuandle( M ) | ( function ) |
Inputs a finite magma \(M\); returns true if \(M\) is a quandle, false otherwise.
‣ 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
‣ 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
‣ 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:
‣ IsLatin | ( global variable ) |
Inputs a finite quandle \(Q\); returns true if \(Q\) is latin, false otherwise.
Examples:
‣ IsConnectedQuandle | ( global variable ) |
Inputs a finite quandle \(Q\); returns true if \(Q\) is connected, false otherwise.
Examples:
‣ ConnectedQuandles( n ) | ( function ) |
Returns a list of all connected quandles of size \(n\).
‣ ConnectedQuandle( n, k ) | ( function ) |
Returns the \(k\)-th quandle of size \(n\) if such a quandle exists, fail otherwise.
‣ 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
‣ 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.
‣ 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.
‣ 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:
‣ RightMultiplicationGroupAsPerm | ( global variable ) |
Inputs a connected quandle \(Q\); output its right multiplication group whose elements are permutations.
Examples:
‣ RightMultiplicationGroup | ( global variable ) |
Inputs a connected quandle \(Q\); output its right multiplication group whose elements are mappings from \(Q\) to \(Q\).
Examples:
‣ AutomorphismGroupQuandleAsPerm( Q ) | ( function ) |
Inputs a connected quandle \(Q\); outputs its automorphism group whose elements are permutations.
Examples:
‣ AutomorphismGroupQuandle( Q ) | ( function ) |
Inputs a connected quandle \(Q\); outputs its automorphism group whose elements are mappings from \(Q\) to \(Q\).
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