Quelle aboutQuandles2.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/aboutQuandles2.html

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
 <meta http-equiv="content-type"
content="text/html; charset=ISO-8859-1">
 <title>AboutHap</title>
</head>
<body
style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);"
link="#000066" vlink="#000066" alink="#000066">
 
<table
style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);"
border="0" cellpadding="20" cellspacing="10">
 <tbody>
 <tr align="center">
 <th style="vertical-align: top;">
 <table style="width: 100%; text-align: left;" cellpadding="2"
cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a
href="aboutPeripheral.html">Previous</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Knots and Quandles 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutPersistent.html">next</a> 
 </td>
 </tr>
 </tbody>
 </table>
 <big></big> 
 </th>
 </tr>
 <tr>
 <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 ] 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">
 <table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutPeripheral.html">Previous
Page</a> 
 </td>
 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
 </td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutPersistent.html">Next
page 
 </a> </td>
 </tr>
 </tbody>
 </table>
 <a href="aboutTopology.html"> 
 </a> </td>
 </tr>
 </tbody>
</table>
 
 
</body>
</html>

Messung V0.5

¤ Dauer der Verarbeitung: 0.11 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.