GAP (HAP) - Chapter 32: Knots and Quandles

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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 (nUNKNOWNEntity(le)12) if it exists; fail otherwise.
`GaussCodeKnot(n,k)` Returns a Gauss Code (with orientations) for the k-th knot with n crossings (nUNKNOWNEntity(le)12) 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 (nUNKNOWNEntity(le)12) 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, nUNKNOWNEntity(le)6. If nUNKNOWNEntity(ge)7, it returns fail.
`Quandle(n,k)` Returns the k-th quandle of size n (nUNKNOWNEntity(le)6) 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 nUNKNOWNEntity(ge)7, 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 e UNKNOWNEntity(isin) Q and an element stigma UNKNOWNEntity(isin) G; 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 e UNKNOWNEntity(isin) Q and an element stigma UNKNOWNEntity(isin) 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)` 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 nUNKNOWNEntity(le)iUNKNOWNEntity(le)m.
`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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