<Chapter><Heading> Knots and Quandles </Heading>

<Section><Heading> &nbsp;</Heading>

Knots 

<ManSection>

<Func Name="PresentationKnotQuandle" Arg="gaussCode"/>
<Description>
<P/>

Inputs a Gauss Code of a knot (with the orientations; see <M>GaussCodeOfPureCubicalKnot</M> in HAP package) and outputs the generators and relators of the knot quandle associated (in the form of a record).
<P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>4</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="PD2GC" Arg="PD"/>
<Description>
<P/>

Inputs a Planar Diagram of a knot; outputs the Gauss Code associated (with the orientations).
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="PlanarDiagramKnot" Arg="n,k"/>
<Description>
<P/>

Returns a Planar Diagram for the <M>k</M>-th knot with <M>n</M> crossings (<M>n \leq 12</M>) if it exists; fail otherwise.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="GaussCodeKnot" Arg="n,k"/>
<Description>
<P/>

Returns a Gauss Code (with orientations) for the <M>k</M>-th knot with <M>n</M> crossings (<M>n \leq 12</M>) if it exists; fail otherwise.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Func Name="PresentationKnotQuandleKnot" Arg="n,k"/>
<Description>
<P/>

Returns generators and relators (in the form of a record) for the <M>k</M>-th knot with <M>n</M> crossings (<M>n \leq 12</M>) if it exists; fail otherwise.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="NumberOfHomomorphisms" Arg="genRelQ,finiteQ"/>
<Description>
<P/>

Inputs generators and relators <M>genRelQ</M> of a knot quandle (in the form of a record, see above) and a finite quandle <M>finiteQ</M>; outputs the number of homomorphisms from the former to the latter.
<P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="PartitionedNumberOfHomomorphisms" Arg="genRelQ,finiteQ"/>
<Description>
<P/>

Inputs generators and relators <M>genRelQ</M> of a knot quandle (in the form of a record, see above) and a finite connected quandle <M>finiteQ</M>; outputs a partition of the number of homomorphisms from the former to the latter.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>1</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

Quandles 

<ManSection>

<Func Name="ConjugationQuandle" Arg="G,n"/>
<Description>
<P/>

Inputs a finite group <M>G</M> and an integer <M>n</M>; outputs the associated <M>n</M>-fold conjugation quandle.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="FirstQuandleAxiomIsSatisfied" Arg="M"/>

<Func Name="SecondQuandleAxiomIsSatisfied" Arg="M"/>

<Func Name="ThirdQuandleAxiomIsSatisfied" Arg="M"/>
<Description>
<P/>

Inputs a finite magma <M>M</M>; returns true if <M>M</M> satisfy the first/second/third axiom of a quandle, false otherwise.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Func Name="IsQuandle" Arg="M"/>
<Description>
<P/>

Inputs a finite magma <M>M</M>; returns true if <M>M</M> is a quandle, false otherwise.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="Quandles" Arg="n"/>
<Description>
<P/>

Returns a list of all quandles of size <M>n</M>, <M>n \leq 6</M>. If <M>n \geq 7</M>, it returns fail.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutPersistent.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCoveringSpaces.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>4</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>5</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>6</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="Quandle" Arg="n,k"/>
<Description>
<P/>

Returns the <M>k</M>-th quandle of size <M>n</M> (<M>n \leq 6</M>) if such a quandle exists, fail otherwise.
<P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutPersistent.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCoveringSpaces.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutCoverinSpaces.html</Link><LinkText>4</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>5</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>6</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>7</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="IdQuandle" Arg="Q"/>
<Description>
<P/>

Inputs a quandle <M>Q</M>; and outputs a list of integers [<M>n</M>,<M>k</M>] such that <M>Q</M> is isomorphic to <M>Quandle(n,k)</M>. If <M>n \geq 7</M>, then it returns [<M>n</M>,fail] (where <M>n</M> is the size of <M>Q</M>).
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Var Name="IsLatin"/>
<Description>
<P/>

Inputs a finite quandle <M>Q</M>; returns true if <M>Q</M> is latin, false otherwise.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Var Name="IsConnectedQuandle"/>
<Description>
<P/>

Inputs a finite quandle <M>Q</M>; returns true if <M>Q</M> is connected, false otherwise.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Func Name="ConnectedQuandles" Arg="n"/>
<Description>
<P/>

Returns a list of all connected quandles of size <M>n</M>.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="ConnectedQuandle" Arg="n,k"/>
<Description>
<P/>

Returns the <M>k</M>-th quandle of size <M>n</M> if such a quandle exists, fail otherwise.
<P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>4</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="IdConnectedQuandle" Arg="Q"/>
<Description>
<P/>

Inputs a connected quandle <M>Q</M>; and outputs a list of integers [<M>n</M>,<M>k</M>] such that <M>Q</M> is isomorphic to <M>ConnectedQuandle(n,k)</M>.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>1</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="IsQuandleEnvelope" Arg="Q,G,e,stigma"/>
<Description>
<P/>

Inputs a set <M>Q</M>, a permutation group <M>G</M>, an element <M>e \in Q</M> and an element <M>stigma \in G</M>; returns true if this structure describes a quandle envelope, false otherwise.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="QuandleQuandleEnvelope" Arg="Q,G,e,stigma"/>
<Description>
<P/>

Inputs a set <M>Q</M>, a permutation group <M>G</M>, an element <M>e \in Q</M> and an element <M>stigma \in G</M>.
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.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

<ManSection>

<Func Name="KnotInvariantCedric" Arg="genRelQ,n,m"/>
<Description>
<P/>

Inputs generators and relators of a knot quandle (in the form of a record, see above) and two integers <M>n</M> and <M>m</M>; outputs a list [<M>n</M>1,<M>n</M>2,...,<M>n</M>k] where <M>n</M>j is a partition of the number of homomorphisms from the considered knot quandle to the <M>j</M>-th connected quandle of size <M>n \leq i \leq m</M>.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Var Name="RightMultiplicationGroupAsPerm"/>
<Description>
<P/>

Inputs a connected quandle <M>Q</M>; output its right multiplication group whose elements are permutations.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Var Name="RightMultiplicationGroup"/>
<Description>
<P/>

Inputs a connected quandle <M>Q</M>; output its right multiplication group whose elements are mappings from <M>Q</M> to <M>Q</M>.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Func Name="AutomorphismGroupQuandleAsPerm" Arg="Q"/>
<Description>
<P/>

Inputs a connected quandle <M>Q</M>; outputs its automorphism group whose elements are permutations.
<P/><B>Examples:</B> 
</Description>

























































































































</ManSection>

<ManSection>

<Func Name="AutomorphismGroupQuandle" Arg="Q"/>
<Description>
<P/>

Inputs a connected quandle <M>Q</M>; outputs its automorphism group whose elements are mappings from <M>Q</M> to <M>Q</M>.
<P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>1</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>2</LinkText></URL>&nbsp;,
<URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>3</LinkText></URL>&nbsp; 

</Description>

























































































































</ManSection>

</Section>
</Chapter>
