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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Knots and Quandles 
 </big></td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: center;"><big
style="font-weight: bold;">Knots and Quandles 
 </big> Sub-package by Cédric FRAGNAUD and Graham ELLIS </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
quandle (Q, ▹) is a non-empty set Q equipped with a binary operation ▹
: Q × Q → Q satisfying the following axioms: 
 
1/ ∀ a ∈ Q, a ▹ a = a. 
2/ ∀ a, b ∈ Q, ∃! c ∈ Q such that a = c ▹ b. 
3/ ∀ a, b, c ∈ Q, (a ▹ b) ▹ c = (a ▹ c) ▹ (b ▹ c). 
 
One can check that for any group G and n ∈ ℤ, the magma (G, ▹) forms a
quandle with the operation x ▹ y = y-nxyn , ∀ x,
y ∈ G. Such a quandle is called the n-Fold Conjugation Quandle. 
 
A quandle Q is said to be connected if the inner automorphism
group Inn Q acts transitively on Q. In other words,
 Q is connected if and only if for each pair a, b in Q
there are a1, a2, . . . , an in Q
such that a ▹ a1 ▹· · · ▹ an
= b. 
 
A quandle Q is said to be latin if ∀ a, b ∈ Q, ∃ c ∈
Q such that a = b ▹ c. </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
Q:=Quandle(5,21); 
<magma with 5 generators> 
gap> Display(MultiplicationTable(Q)); 
[ [ 1, 3, 4, 5, 2 ], 
 [ 3, 2, 5, 1, 4 ], 
 [ 4, 5, 3, 2, 1 ], 
 [ 5, 1, 2, 4, 3 ], 
 [ 2, 4, 1, 3, 5 ] ] 
gap> IsConnected(Q); 
true 
gap> IsLatin(Q); 
true </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=DihedralGroup(64);; 
gap> Q:=ConjugationQuandle(G,1); 
<magma with 19 generators> 
gap> Size(Q); 
64 
gap> IsConnected(Q); 
false 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Let
Q be a set, e an element in Q, G a permutation group, and stigma an
element in G. 
Then (Q,G,e,stigma) describes a Quandle Envelope if : 
 <ul>
 <li>G is a transitive group on Q.</li>
 </ul>
 <ul>
 <li>stigma ∈ Z(Ge), the center of the stabilizer of
e.</li>
 </ul>
 <ul>
 <li>⟨stigmaG⟩ = G (that is, the smallest normal
subgroup of G containing stigma is all of G).</li>
 </ul>
 From a Quandle Envelope (Q,G,e,stigma),
we can construct a Quandle (Q, ▹):
 
 for all x,y in Q,
 x ▹ y=(ŷ(stigma))(x)
 , where ŷ ∈ G satisfies ŷ(e)=y.
 Such a quandle is
connected. This property is used to construct all the connected
quandles of size n.
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Q:=[1..9];;
e:=2;;
G:=TransitiveGroup(9,15);;
st:=(1,8,7,4,9,5,3,6);; 
gap> IsQuandleEnvelope(Q,G,e,st);
QE:=QuandleQuandleEnvelope(Q,G,e,st); 
true 
<magma with 9 generators> 
gap> IsQuandle(QE); IsConnected(QE); 
true 
true 
gap> ConnectedQuandles(20); time; 
[ <magma with 20 generators>, <magma with 20 generators>,
<magma with 20 generators>, 
 <magma with 20 generators>, <magma with 20
generators>, <magma with 20 generators>, 
 <magma with 20 generators>, <magma with 20
generators>, <magma with 20 generators>, 
 <magma with 20 generators> ] 
3364296</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Let's
denote
Rx the mapping defined by Rx : Q→Q, y ↦y▹x. 
We define the right multiplication group G of a quandle Q by G=〈Rx,
x
∈
Q〉. 
We also define the automorphism group Aut(Q)={f:Q→Q}. 
It can be proven that Rx is a subgroup of Aut(Q).</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Q:=ConnectedQuandle(8,2);;
q:=Random(Q); 
m6 
gap> A:=AutomorphismGroupQuandle(Q);; a:=Random(A);; 
gap> q^a; 
m4 
gap> R:=RightMultiplicationGroupOfQuandle(Q);; r:=Random(R);; 
gap> q^r; 
m3</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
knot is an embedding of the circle S1 in ℝ3. 
 
To study these structures, we use knot diagrams, which are projections
of these knots into ℝ2, defined, for instance, by f : ℝ3
→ ℝ2; (x,y,z) → (x,y) subject to the constraint that
the preimage of any (x, y) ∈ ℝ2 contains at most two points. 
 
Crossing points occur when the preimage of a point in ℝ2
contains more than one point. 
 
At these crossing points, we denote the point in the preimage that is
nearer to the ℝ2 plane as the under-crossing point and the
point farther away as the over-crossing point. An arc is a line that
connects two crossing points in the knot diagram, with a line break
occurring when an undercrossing point is mapped to the arc. 
 
We may give a knot diagram an orientation, i.e. a direction of
travelling around the knot. This allows us to categorize crossings as
either positive or negative:
 <div style="text-align: center;"><img
title="A positive and neative crossing"
style="width: 278px; height: 181px;"
src="data:image/png;base64,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"
alt="A positive and neative crossing"> 
 <div style="text-align: left;"> 
There exists different ways to describe a knot diagram: Planar Diagram,
Gauss Code, Dowker Notation, Conway Notation.</div>
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">An
other way to describe a knot is to use quandles. From a knot K, we can
construct the knot quandle Q(K), whose generators are the arcs of K,
and relations are associated to the crossings: 
 
 <div style="text-align: center;"><img alt="Relators knot quandles"
src="Images/a_b_c_neg.png"> 
This figure gives us "a ▹ b = c" at a negative crossing, and "a ▹^-1
b = c" (or "c ▹ b = a") at a positive one.

</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
K:=PureCubicalKnot(3,1);; 
gap> G:=GaussCodeOfPureCubicalKnot(K);; 
gap> P:=PresentationKnotQuandle(G); 
rec( generators := [ 1 .. 3 ], relators := [ [ [ 3, 2 ], 1 ], [ [ 1, 3
], 2 ], [ [ 2, 1 ], 3 ] ] )</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">From
this
example,
we
see
that the generators of the Trefoil Knot Quandle
are the arcs 1, 2 and 3; these generators satisfy the relations above. 
 
 Nb: [[a1 ,a2 ],a3 ] means
a1 ▹ a2 = a3, no matter if we consider
a positive or negative crossing.</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">We
can also easily go from a Planar Diagram representation of a knot to a
its Gauss Code (with orientations of crossings).</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
PD:=PlanarDiagramKnot(3,1); 
[ [ 1, 4, 2, 5 ], [ 3, 6, 4, 1 ], [ 5, 2, 6, 3 ] ] 
gap> G:=PD2GC(PD); 
[ [ [ -1, 3, -2, 1, -3, 2 ] ], [ -1, -1, -1 ] ] </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Using

quandles,
we
can
construct
an knot invariant: a list made of the number
of Homomorphisms beetween the knot (in the form of a record) and a
connected quandle.</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
K:=PresentationKnotQuandleKnot(8,2); 
rec( generators := [ 1 .. 8 ], relators := [ [ [ 8, 2 ], 1 ], [ [ 2, 5
], 1 ], [ [ 2, 6 ], 3 ], [ [ 3, 7 ], 4 ], [ [ 4, 8 ], 5 ], 
[ [ 6, 1 ], 5 ], [ [ 6, 3 ], 7 ], [ [ 7, 4 ], 8 ] ] ) 
gap> Q:=ConnectedQuandle(9,2);; 
gap> NumberOfHomomorphisms(K,Q); time; 
9 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following code shows how the humber of quandle homomorphisms K--->Q
from a knot quandle K to a finite quandle Q can be used to distinguish
between knots. The code establishes that by using only connected finite
quandles Q of order <=13 one can distinguish between all prime knots
on at most eight crossings. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
L:=[];;
#List of prime knots on <= 8 crossings 
gap> for n in [1..8] do 
> for i in [1..NumberOfPrimeKnots(n)] do 
> Add(L,PresentationKnotQuandleKnot(n,i)); 
> od;od; 
 
gap> inv:=function(K,n); #A knot invariant 
> return List(ConnectedQuandles(n),x->NumberOfHomomorphisms(K,x)); 
> end;; 
gap> C:=Classify(L,K->inv(K,3));; 
gap> List(C,Size); 
[ 11, 23, 1 ] 
gap> C4:=RefineClassification(C,K->inv(K,4));; 
gap> List(C4,Size); 
[ 8, 3, 6, 17, 1 ] 
gap> 
gap> C5:=RefineClassification(C4,K->inv(K,5));; 
gap> List(C5,Size); 
[ 5, 2, 1, 1, 1, 1, 1, 1, 4, 3, 12, 1, 1, 1 ] 
gap> C6:=RefineClassification(C5,K->inv(K,6));; 
gap> List(C6,Size); 
[ 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 12, 1, 1, 1 ] 
gap> C7:=RefineClassification(C6,K->inv(K,7));; 
gap> List(C7,Size); 
[ 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 8, 2, 1, 1, 1 ] 
gap> C8:=RefineClassification(C7,K->inv(K,8));; 
gap> List(C8,Size); 
[ 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 6, 2, 2, 1, 1, 1 ] 
gap> C9:=RefineClassification(C8,K->inv(K,9));; 
gap> List(C9,Size); 
[ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1,
1, 1, 1, 1, 1, 1 ] 
gap> C10:=RefineClassification(C9,K->inv(K,10));; 
gap> List(C10,Size); 
[ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1,
1, 1, 1, 1, 1, 1 ] 
gap> C11:=RefineClassification(C10,K->inv(K,11));; 
gap> List(C11,Size); 
[ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1 ] 
gap> List(C12,Size); 
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] 
gap> C13:=RefineClassification(C12,K->inv(K,13));; 
gap> List(C13,Size); 
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] 
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