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 ≤ 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 ≤ 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 ≤ 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 ≤ 6. If n ≥ 7, it returns fail.
Examples: 1 , 2 , 3 , 4 , 5 , 6
‣ 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
‣ 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:
‣ 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 ∈ Q and an element stigma ∈ 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 ∈ 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.
‣ 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:
‣ 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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