  
  [1X32 [33X[0;0YKnots and Quandles[133X[101X
  
  
  [1X32.1 [33X[0;0Y [133X[101X
  
  [33X[0;0YKnots[133X
  
  [1X32.1-1 PresentationKnotQuandle[101X
  
  [33X[1;0Y[29X[2XPresentationKnotQuandle[102X( [3XgaussCode[103X ) [32X function[133X
  
  [33X[0;0YInputs   a   Gauss   Code   of   a   knot   (with   the   orientations;  see
  [22XGaussCodeOfPureCubicalKnot[122X  in  HAP  package) and outputs the generators and
  relators of the knot quandle associated (in the form of a record).[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap5.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,                             3
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              4
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-2 PD2GC[101X
  
  [33X[1;0Y[29X[2XPD2GC[102X( [3XPD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  Planar Diagram of a knot; outputs the Gauss Code associated (with
  the orientations).[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-3 PlanarDiagramKnot[101X
  
  [33X[1;0Y[29X[2XPlanarDiagramKnot[102X( [3Xn[103X, [3Xk[103X ) [32X function[133X
  
  [33X[0;0YReturns  a  Planar Diagram for the [22Xk[122X-th knot with [22Xn[122X crossings ([22Xn ≤ 12[122X) if it
  exists; fail otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-4 GaussCodeKnot[101X
  
  [33X[1;0Y[29X[2XGaussCodeKnot[102X( [3Xn[103X, [3Xk[103X ) [32X function[133X
  
  [33X[0;0YReturns  a Gauss Code (with orientations) for the [22Xk[122X-th knot with [22Xn[122X crossings
  ([22Xn ≤ 12[122X) if it exists; fail otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-5 PresentationKnotQuandleKnot[101X
  
  [33X[1;0Y[29X[2XPresentationKnotQuandleKnot[102X( [3Xn[103X, [3Xk[103X ) [32X function[133X
  
  [33X[0;0YReturns  generators and relators (in the form of a record) for the [22Xk[122X-th knot
  with [22Xn[122X crossings ([22Xn ≤ 12[122X) if it exists; fail otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-6 NumberOfHomomorphisms[101X
  
  [33X[1;0Y[29X[2XNumberOfHomomorphisms[102X( [3XgenRelQ[103X, [3XfiniteQ[103X ) [32X function[133X
  
  [33X[0;0YInputs  generators  and relators [22XgenRelQ[122X of a knot quandle (in the form of a
  record,  see  above)  and  a  finite  quandle [22XfiniteQ[122X; outputs the number of
  homomorphisms from the former to the latter.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap5.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,                             3
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) [133X
  
  [1X32.1-7 PartitionedNumberOfHomomorphisms[101X
  
  [33X[1;0Y[29X[2XPartitionedNumberOfHomomorphisms[102X( [3XgenRelQ[103X, [3XfiniteQ[103X ) [32X function[133X
  
  [33X[0;0YInputs  generators  and relators [22XgenRelQ[122X of a knot quandle (in the form of a
  record,  see  above)  and  a  finite  connected  quandle  [22XfiniteQ[122X; outputs a
  partition of the number of homomorphisms from the former to the latter.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutQuandles.html[107X) [133X
  
  [33X[0;0YQuandles[133X
  
  [1X32.1-8 ConjugationQuandle[101X
  
  [33X[1;0Y[29X[2XConjugationQuandle[102X( [3XG[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  finite  group  [22XG[122X  and an integer [22Xn[122X; outputs the associated [22Xn[122X-fold
  conjugation quandle.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) [133X
  
  [1X32.1-9 FirstQuandleAxiomIsSatisfied[101X
  
  [33X[1;0Y[29X[2XFirstQuandleAxiomIsSatisfied[102X( [3XM[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSecondQuandleAxiomIsSatisfied[102X( [3XM[103X ) [32X function[133X
  [33X[1;0Y[29X[2XThirdQuandleAxiomIsSatisfied[102X( [3XM[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  finite  magma [22XM[122X; returns true if [22XM[122X satisfy the first/second/third
  axiom of a quandle, false otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-10 IsQuandle[101X
  
  [33X[1;0Y[29X[2XIsQuandle[102X( [3XM[103X ) [32X function[133X
  
  [33X[0;0YInputs a finite magma [22XM[122X; returns true if [22XM[122X is a quandle, false otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-11 Quandles[101X
  
  [33X[1;0Y[29X[2XQuandles[102X( [3Xn[103X ) [32X function[133X
  
  [33X[0;0YReturns a list of all quandles of size [22Xn[122X, [22Xn ≤ 6[122X. If [22Xn ≥ 7[122X, it returns fail.[133X
  
  [33X[0;0Y[12XExamples:[112X      1      ([7X../www/SideLinks/About/aboutPersistent.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutCoveringSpaces.html[107X) ,                        3
  ([7X../www/SideLinks/About/aboutCoverinSpaces.html[107X) ,                         4
  ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,                             5
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              6
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-12 Quandle[101X
  
  [33X[1;0Y[29X[2XQuandle[102X( [3Xn[103X, [3Xk[103X ) [32X function[133X
  
  [33X[0;0YReturns  the  [22Xk[122X-th  quandle of size [22Xn[122X ([22Xn ≤ 6[122X) if such a quandle exists, fail
  otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap5.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutPersistent.html[107X) ,                            3
  ([7X../www/SideLinks/About/aboutCoveringSpaces.html[107X) ,                        4
  ([7X../www/SideLinks/About/aboutCoverinSpaces.html[107X) ,                         5
  ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,                             6
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              7
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-13 IdQuandle[101X
  
  [33X[1;0Y[29X[2XIdQuandle[102X( [3XQ[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  quandle  [22XQ[122X;  and  outputs a list of integers [[22Xn[122X,[22Xk[122X] such that [22XQ[122X is
  isomorphic  to  [22XQuandle(n,k)[122X. If [22Xn ≥ 7[122X, then it returns [[22Xn[122X,fail] (where [22Xn[122X is
  the size of [22XQ[122X).[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-14 IsLatin[101X
  
  [33X[1;0Y[29X[2XIsLatin[102X [32X global variable[133X
  
  [33X[0;0YInputs a finite quandle [22XQ[122X; returns true if [22XQ[122X is latin, false otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-15 IsConnectedQuandle[101X
  
  [33X[1;0Y[29X[2XIsConnectedQuandle[102X [32X global variable[133X
  
  [33X[0;0YInputs a finite quandle [22XQ[122X; returns true if [22XQ[122X is connected, false otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-16 ConnectedQuandles[101X
  
  [33X[1;0Y[29X[2XConnectedQuandles[102X( [3Xn[103X ) [32X function[133X
  
  [33X[0;0YReturns a list of all connected quandles of size [22Xn[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-17 ConnectedQuandle[101X
  
  [33X[1;0Y[29X[2XConnectedQuandle[102X( [3Xn[103X, [3Xk[103X ) [32X function[133X
  
  [33X[0;0YReturns the [22Xk[122X-th quandle of size [22Xn[122X if such a quandle exists, fail otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap5.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,                             3
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              4
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-18 IdConnectedQuandle[101X
  
  [33X[1;0Y[29X[2XIdConnectedQuandle[102X( [3XQ[103X ) [32X function[133X
  
  [33X[0;0YInputs a connected quandle [22XQ[122X; and outputs a list of integers [[22Xn[122X,[22Xk[122X] such that
  [22XQ[122X is isomorphic to [22XConnectedQuandle(n,k)[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutQuandles.html[107X) [133X
  
  [1X32.1-19 IsQuandleEnvelope[101X
  
  [33X[1;0Y[29X[2XIsQuandleEnvelope[102X( [3XQ[103X, [3XG[103X, [3Xe[103X, [3Xstigma[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  set  [22XQ[122X,  a  permutation  group [22XG[122X, an element [22Xe ∈ Q[122X and an element
  [22Xstigma  ∈  G[122X;  returns  true if this structure describes a quandle envelope,
  false otherwise.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-20 QuandleQuandleEnvelope[101X
  
  [33X[1;0Y[29X[2XQuandleQuandleEnvelope[102X( [3XQ[103X, [3XG[103X, [3Xe[103X, [3Xstigma[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  set  [22XQ[122X,  a  permutation  group [22XG[122X, an element [22Xe ∈ Q[122X and an element
  [22Xstigma  ∈  G[122X.  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.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
  [1X32.1-21 KnotInvariantCedric[101X
  
  [33X[1;0Y[29X[2XKnotInvariantCedric[102X( [3XgenRelQ[103X, [3Xn[103X, [3Xm[103X ) [32X function[133X
  
  [33X[0;0YInputs  generators  and relators of a knot quandle (in the form of a record,
  see  above) and two integers [22Xn[122X and [22Xm[122X; outputs a list [[22Xn[122X1,[22Xn[122X2,...,[22Xn[122Xk] where [22Xn[122Xj
  is  a  partition  of  the  number  of homomorphisms from the considered knot
  quandle to the [22Xj[122X-th connected quandle of size [22Xn ≤ i ≤ m[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-22 RightMultiplicationGroupAsPerm[101X
  
  [33X[1;0Y[29X[2XRightMultiplicationGroupAsPerm[102X [32X global variable[133X
  
  [33X[0;0YInputs  a  connected  quandle [22XQ[122X; output its right multiplication group whose
  elements are permutations.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-23 RightMultiplicationGroup[101X
  
  [33X[1;0Y[29X[2XRightMultiplicationGroup[102X [32X global variable[133X
  
  [33X[0;0YInputs  a  connected  quandle [22XQ[122X; output its right multiplication group whose
  elements are mappings from [22XQ[122X to [22XQ[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-24 AutomorphismGroupQuandleAsPerm[101X
  
  [33X[1;0Y[29X[2XAutomorphismGroupQuandleAsPerm[102X( [3XQ[103X ) [32X function[133X
  
  [33X[0;0YInputs  a connected quandle [22XQ[122X; outputs its automorphism group whose elements
  are permutations.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X32.1-25 AutomorphismGroupQuandle[101X
  
  [33X[1;0Y[29X[2XAutomorphismGroupQuandle[102X( [3XQ[103X ) [32X function[133X
  
  [33X[0;0YInputs  a connected quandle [22XQ[122X; outputs its automorphism group whose elements
  are mappings from [22XQ[122X to [22XQ[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              3
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) [133X
  
