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32 Knots and Quandles

32 Knots and Quandles

Knots                        
PresentationKnotQuandle(gaussCode)

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)

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

                       
PlanarDiagramKnot(n,k)

Returns a Planar Diagram for the k-th knot with n crossings (n12) if it exists; fail otherwise.

                       
GaussCodeKnot(n,k)

Returns a Gauss Code (with orientations) for the k-th knot with n crossings (n12) if it exists; fail otherwise.

                       
PresentationKnotQuandleKnot(n,k)

Returns generators and relators (in the form of a record) for the k-th knot with n crossings (n12) if it exists; fail otherwise.

                       
NumberOfHomomorphisms(genRelQ,finiteQ)

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)

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.

                       
Quandles                        
ConjugationQuandle(G,n)

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

                       
FirstQuandleAxiomIsSatisfied(M)
SecondQuandleAxiomIsSatisfied(M)
ThirdQuandleAxiomIsSatisfied(M)

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

                       
IsQuandle(M)

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

                       
Quandles(n)

Returns a list of all quandles of size n, n6. If n7, it returns fail.

                       
Quandle(n,k)

Returns the k-th quandle of size n (n6) if such a quandle exists, fail otherwise.

                       
IdQuandle(Q)

Inputs a quandle Q; and outputs a list of integers [n,k] such that Q is isomorphic to Quandle(n,k). If n7, then it returns [n,fail] (where n is the size of Q).

                       
IsLatin(Q)

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

                       
IsConnectedQuandle(Q)

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

                       
ConnectedQuandles(n)

Returns a list of all connected quandles of size n.

                       
ConnectedQuandle(n,k)

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

                       
IdConnectedQuandle(Q)

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

                       
IsQuandleEnvelope(Q,G,e,stigma)

Inputs a set Q, a permutation group G, an element eQ and an element stigmaG; returns true if this structure describes a quandle envelope, false otherwise.

                       
QuandleQuandleEnveloppe(Q,G,e,stigma)

Inputs a set Q, a permutation group G, an element eQ and an element stigmaG. 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)

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 nim.

                       
RightMultiplicationGroupAsPerm(Q)

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

                       
RightMultiplicationGroup(Q)

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

                       
AutomorphismGroupQuandleAsPerm(Q)

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

                       
AutomorphismGroupQuandle(Q)

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

                       

 


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