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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <!-- saved from url=(0082)http://hamilton.nuigalway.ie/Hap/www/SideLinks/About/About <html> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"> <title>AboutHap</title> </head> <body style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);" link="#0 vlink="#000066" alink="#000066"> <br> <table style="text-align: left; margin-left: auto; margin-right: auto; color: #000 border="0" cellpadding="20" cellspacing="10"> <tbody style="height: 3869.87px;"> <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;"><br> </td> <td style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span style="font-weight: bold;">About HAP: Knots and Quandles<br> </span></big></td> <td style="text-align: right; vertical-align: top;"><br> </td> </tr> </tbody> </table> <big><span style="font-weight: bold;"></span></big><br> </th> </tr> <tr align="center"> <td style="vertical-align: top; background-color: rgb(255, 255, 255);"><big style="font-weight: bold;">Knots and Quandles <br> </big> Sub-package by Cédric FRAGNAUD and Graham ELLIS </td> </tr> <tr> <td style="background-color: white;">A quandle (Q, ▹) is a non-empty set Q equipped with a binary operation ▹ : Q × Q → Q satisfying the following axioms:<br> <br> 1/ ∀∃∈For all a ∈ Q, a ▹ a = a.<br> 2/ ∀ a, b ∈ Q, ∃! c ∈ Q such that a = c ▹ b.<br> 3/ ∀ a, b, c ∈ Q, (a ▹ b) ▹ c = (a ▹ c) ▹ (b ▹ c).<br> <br> One can check that for any group G and n ∈ ℤ, the magma (G, ▹) forms a quandle with the operation x ▹ y = y<sup>-n</sup>xy<sup>n</sup> , ∀ x, y ∈ G. Such a quandle is called the n-Fold Conjugation Quandle.<br> <br> A quandle <em>Q</em> is said to be connected if the inner automorphism group <em>Inn Q</em> acts transitively on <em>Q</em>. In other words, <em>Q</em> is connected if and only if for each pair a, b in <em>Q</em> there are a<sub>1</sub>, a<sub>2</sub>, . . . , a<sub>n</sub> in <em>Q</em> such that a ▹ a<sub>1</sub> ▹· · · ▹ a<sub>n</sub> = b.<br> <br> A quandle <em>Q</em> is said to be latin if ∀ a, b ∈ <em>Q</em>, ∃ c ∈<em> Q</em> such that a = b ▹ c.</td> </tr> <tr> <td style="background-color: #ffffcc;">gap> Q:=Quandle(5,21);<br> <magma with 5 generators><br> gap> Display(MultiplicationTable(Q));<br> [ [ 1, 3, 4, 5, 2 ],<br> [ 3, 2, 5, 1, 4 ],<br> [ 4, 5, 3, 2, 1 ],<br> [ 5, 1, 2, 4, 3 ],<br> [ 2, 4, 1, 3, 5 ] ]<br> gap> IsConnectedQuandle(Q);<br> true<br> gap> IsLatin(Q);<br> true<br> </td> </tr> <tr> <td style="background-color: white;">Let Q be a set, e an element in Q, G a permutation group, and stigma an element in G.<br> Then (Q,G,e,stigma) describes a Quandle Envelope if :<br> <ul> <li>G is a transitive group on Q.</li> </ul> <ul> <li>stigma ∈ Z(G<sub>e</sub>), the center of the stabilizer of e.</li> </ul> <ul> <li>⟨stigma<sup>G</sup>⟩ = G (that is, the smallest normal subgroup of G containing stigma is all of G).</li> </ul> <p style="height: 9px;">From a Quandle Envelope (Q,G,e,stigma), we can construct a Quandle (Q, ▹):</p> <p style="margin-top: -1px; height: 18px;"> for all x,y in Q, x ▹ y=(ŷ(stigma))(x) , where ŷ ∈ G satisfies ŷ(e)=y.</p> <p style="margin-top: -1px; height: 18px;">Such a quandle is connected. This property is used to construct all the connected quandles of size n.</p> </td> </tr> <tr> <td style="background-color: #ffffcc;">gap> Q:=[1..9];; e:=2;; G:=TransitiveGroup(9,15);; st:=(1,8,7,4,9,5,3,6);;<br> gap> IsQuandleEnvelope(Q,G,e,st); QE:=QuandleQuandleEnvelope(Q,G,e,st);<br> true<br> <magma with 9 generators><br> gap> IsQuandle(QE); IsConnectedQuandle(QE);<br> true<br> true<br> gap> ConnectedQuandles(20); time;<br> [ <magma with 20 generators>, <magma with 20 generators>, <magma with 20 generators>, <br> <magma with 20 generators>, <magma with 20 generators>, <magma with 20 generators>, <br> <magma with 20 generators>, <magma with 20 generators>, <magma with 20 generators>, <br> <magma with 20 generators> ]<br> 3364296</td> </tr> <tr> <td style="background-color: white;">Let's denote Rx the mapping defined by R<sub>x</sub> : Q→Q, y ↦y▹x.<br> We define the right multiplication group G of a quandle Q by G=〈R<sub>x</sub>, x ∈ Q〉.<br> We also define the automorphism group Aut(Q)={f:Q→Q}.<br> It can be proven that R<sub>x</sub> is a subgroup of Aut(Q).</td> </tr> <tr> <td style="background-color: #ffffcc;">gap> Q:=ConnectedQuandle(8,2);; q:=Random(Q);<br> m6<br> gap> A:=AutomorphismGroupQuandle(Q);; a:=Random(A);;<br> gap> q^a;<br> m4<br> gap> R:=RightMultiplicationGroupOfQuandle(Q);; r:=Random(R);;<br> gap> q^r;<br> m3</td> </tr> <tr> <td style="background-color: white; height: 192px;">A knot is an embedding of the circle S<sup>1</sup> in ℝ<sup>3</sup>.<br> <br> To study these structures, we use knot diagrams, which are projections of these knots into ℝ<sup>2</sup>, defined, for instance, by f : ℝ<sup>3</sup> → ℝ<sup>2</sup>; (x,y,z) → (x,y)<sup></sup> subject to the constraint that the preimage of any (x, y) ∈ ℝ<sup>2</sup> contains at most two points.<br> <br> Crossing points occur when the preimage of a point in ℝ<sup>2</sup> contains more than one point.<br> <br> At these crossing points, we denote the point in the preimage that is nearer to the ℝ<sup>2</sup> 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.<br> <br> 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,iVBORw0KGgoAAAANS alt="A positive and neative crossing"><br> <div style="text-align: left;"><br> 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: white; text-align: left;"> 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:<br> <br> <div style="text-align: center;"><img alt="Relators knot quandles" src="Images/a_b_c_n 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="background-color: rgb(255, 255, 204); vertical-align: top;">gap> K:=PureCubicalKnot(3,1);;<br> gap> G:=GaussCodeOfPureCubicalKnot(K);;<br> gap> P:=PresentationKnotQuandle(G);<br> rec( generators := [ 1 .. 3 ], relators := [ [ [ 3, 2 ], 1 ], [ [ 1, 3 ], 2 ], [ [ 2, 1 ], 3 ] ] )</td> </tr> <tr> <td style="background-color: white;">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.<br> <br> <u>Nb</u>: [[a<sub>1</sub> ,a<sub>2</sub> ],a<sub>3</sub> ] means a<sub>1</sub> ▹ a<sub>2</sub> = a<sub>3</sub>, no matter if we consider a positive or negative crossing.</td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: 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="background-color: #ffffcc; vertical-align: top;">gap> PD:=PlanarDiagramKnot(3,1);<br> [ [ 1, 4, 2, 5 ], [ 3, 6, 4, 1 ], [ 5, 2, 6, 3 ] ]<br> gap> G:=PD2GC(PD);<br> [ [ [ -1, 3, -2, 1, -3, 2 ] ], [ -1, -1, -1 ] ] <br> </td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: 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="background-color: rgb(255, 255, 204); vertical-align: top;">gap> K:=PresentationKnotQuandleKnot(8,2);<br> rec( generators := [ 1 .. 8 ], relators := [ [ [ 8, 2 ], 1 ], [ [ 2, 5 ], 1 ], [ [ 2, 6 ], 3 ], [ [ 3, 7 ], 4 ], [ [ 4, 8 ], 5 ],<br> [ [ 6, 1 ], 5 ], [ [ 6, 3 ], 7 ], [ [ 7, 4 ], 8 ] ] )<br> gap> Q:=ConnectedQuandle(9,2);;<br> gap> NumberHommorphism(Q,K); time;<br> 9<br> 216228<br> gap> Invariant(K,9); time;<br> [ 1, 3, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 64, 64, 8, 9, 9, 9, 9, 81, 9, 9, 9 ]<br> 1327036</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="http://hamilton.nuigalway.ie/Hap/www/SideLinks/About/aboutLieCovers.html">P Page</a><br> </td> <td style="text-align: center; vertical-align: top;"><a href="http://hamilton.nuigalw style="color: rgb(0, 0, 102);">Contents</span></a><br> </td> <td style="text-align: right; vertical-align: top;"><a href="http://hamilton.nuigalwa style="color: rgb(0, 0, 102);">Next page</span><br> </a> </td> </tr> </tbody> </table> <a href="http://math.uni.lu/%7Erahm/subpackage-documentation/aboutTopology.html"><br> </a> </td> </tr> </tbody> </table> <br> <br> </body> </html>
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2026-04-02