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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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    <9>
        <about hap: knots and quandles></about hap: knots and quandles>
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    <10>
        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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    <12>
        <about hap: knots and quandles></about hap: knots and quandles>
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    <13>
        <about hap: knots and quandles></about hap: knots and quandles>
    </13>
    <14>
        <about hap: knots and quandles></about hap: knots and quandles>
    </14>
    <15>
        <about hap: knots and quandles>Knots and Quandles </about hap: knots and quandles>
    </15>
    <16>
        <about hap: knots and quandles>             Sub-package by Cédric FRAGNAUD and Graham ELLIS </about hap: knots and quandles>
    </16>
    <17>
        <about hap: knots and quandles>A quandle (Q</about hap: knots and quandles>
    </17>
    <18>
        <about hap: knots and quandles>            set Q equipped with a binary operation ▹ : Q × Q → Q satisfying the</about hap: knots and quandles>
    </18>
    <19>
        <about hap: knots and quandles>            following axioms:</about hap: knots and quandles>
    </19>
    <20>
        <about hap: knots and quandles></about hap: knots and quandles>
    </20>
    <21>
        <about hap: knots and quandles>            1/ ∀∃∈For all a ∈ Q</about hap: knots and quandles>
    </21>
    <22>
        <about hap: knots and quandles>            2/ ∀ a</about hap: knots and quandles>
    </22>
    <23>
        <about hap: knots and quandles>            3/ ∀ a</about hap: knots and quandles>
    </23>
    <24>
        <about hap: knots and quandles></about hap: knots and quandles>
    </24>
    <25>
        <about hap: knots and quandles>            One can check that for any group G and n ∈ ℤ</about hap: knots and quandles>
    </25>
    <26>
        <about hap: knots and quandles>            a quandle with the operation x ▹ y = y-nxyn </about hap: knots and quandles>
    </26>
    <27>
        <about hap: knots and quandles>            ∀ x</about hap: knots and quandles>
    </27>
    <28>
        <about hap: knots and quandles></about hap: knots and quandles>
    </28>
    <29>
        <about hap: knots and quandles>            A quandle Q is said to be connected if the inner</about hap: knots and quandles>
    </29>
    <30>
        <about hap: knots and quandles>            automorphism group Inn Q acts transitively on Q.</about hap: knots and quandles>
    </30>
    <31>
        <about hap: knots and quandles>            In other words</about hap: knots and quandles>
    </31>
    <32>
        <about hap: knots and quandles>            pair a</about hap: knots and quandles>
    </32>
    <33>
        <about hap: knots and quandles>            . </about hap: knots and quandles>
    </33>
    <34>
        <about hap: knots and quandles>            ▹ an = b.</about hap: knots and quandles>
    </34>
    <35>
        <about hap: knots and quandles></about hap: knots and quandles>
    </35>
    <36>
        <about hap: knots and quandles>            A quandle Q is said to be latin if ∀ a</about hap: knots and quandles>
    </36>
    <37>
        <about hap: knots and quandles>            c ∈ Q such that a = b ▹ c.</about hap: knots and quandles>
    </37>
    <38>
        <about hap: knots and quandles>gap> Q:=Quandle(5</about hap: knots and quandles>
    </38>
    <39>
        <about hap: knots and quandles>            &amp;lt;magma with 5 generators></about hap: knots and quandles>
    </39>
    <40>
        <about hap: knots and quandles>            gap> Display(MultiplicationTable(Q));</about hap: knots and quandles>
    </40>
    <41>
        <about hap: knots and quandles>            [ [  1</about hap: knots and quandles>
    </41>
    <42>
        <about hap: knots and quandles>              [  3</about hap: knots and quandles>
    </42>
    <43>
        <about hap: knots and quandles>              [  4</about hap: knots and quandles>
    </43>
    <44>
        <about hap: knots and quandles>              [  5</about hap: knots and quandles>
    </44>
    <45>
        <about hap: knots and quandles>              [  2</about hap: knots and quandles>
    </45>
    <46>
        <about hap: knots and quandles>            gap> IsConnectedQuandle(Q);</about hap: knots and quandles>
    </46>
    <47>
        <about hap: knots and quandles>            true</about hap: knots and quandles>
    </47>
    <48>
        <about hap: knots and quandles>            gap> IsLatin(Q);</about hap: knots and quandles>
    </48>
    <49>
        <about hap: knots and quandles>            true</about hap: knots and quandles>
    </49>
    <50>
        <about hap: knots and quandles></about hap: knots and quandles>
    </50>
    <51>
        <about hap: knots and quandles>Let Q be a set</about hap: knots and quandles>
    </51>
    <52>
        <about hap: knots and quandles>            Q</about hap: knots and quandles>
    </52>
    <53>
        <about hap: knots and quandles>            Then (Q</about hap: knots and quandles>
    </53>
    <54>
        <about hap: knots and quandles></about hap: knots and quandles>
    </54>
    <55>
        <about hap: knots and quandles>              G is a transitive group on Q.</about hap: knots and quandles>
    </55>
    <56>
        <about hap: knots and quandles></about hap: knots and quandles>
    </56>
    <57>
        <about hap: knots and quandles></about hap: knots and quandles>
    </57>
    <58>
        <about hap: knots and quandles>              stigma ∈ Z(Ge)</about hap: knots and quandles>
    </58>
    <59>
        <about hap: knots and quandles></about hap: knots and quandles>
    </59>
    <60>
        <about hap: knots and quandles></about hap: knots and quandles>
    </60>
    <61>
        <about hap: knots and quandles>              ⟨stigmaG⟩ = G (that is</about hap: knots and quandles>
    </61>
    <62>
        <about hap: knots and quandles>                subgroup of G containing stigma is all of G).</about hap: knots and quandles>
    </62>
    <63>
        <about hap: knots and quandles></about hap: knots and quandles>
    </63>
    <64>
        <about hap: knots and quandles>            From a Quandle Envelope (Q</about hap: knots and quandles>
    </64>
    <65>
        <about hap: knots and quandles>              can construct a Quandle (Q</about hap: knots and quandles>
    </65>
    <66>
        <about hap: knots and quandles></about hap: knots and quandles>
    </66>
    <67>
        <about hap: knots and quandles>                  for all x</about hap: knots and quandles>
    </67>
    <68>
        <about hap: knots and quandles>                  x ▹ y=(ŷ(stigma))(x)   </about hap: knots and quandles>
    </68>
    <69>
        <about hap: knots and quandles></about hap: knots and quandles>
    </69>
    <70>
        <about hap: knots and quandles>            Such a quandle is</about hap: knots and quandles>
    </70>
    <71>
        <about hap: knots and quandles>              connected. This property is used to construct all the connected</about hap: knots and quandles>
    </71>
    <72>
        <about hap: knots and quandles>              quandles of size n.</about hap: knots and quandles>
    </72>
    <73>
        <about hap: knots and quandles></about hap: knots and quandles>
    </73>
    <74>
        <about hap: knots and quandles>gap> Q:=[1..9];; e:=2;;</about hap: knots and quandles>
    </74>
    <75>
        <about hap: knots and quandles>            G:=TransitiveGroup(9</about hap: knots and quandles>
    </75>
    <76>
        <about hap: knots and quandles>            gap> IsQuandleEnvelope(Q</about hap: knots and quandles>
    </76>
    <77>
        <about hap: knots and quandles>            QE:=QuandleQuandleEnvelope(Q</about hap: knots and quandles>
    </77>
    <78>
        <about hap: knots and quandles>            true</about hap: knots and quandles>
    </78>
    <79>
        <about hap: knots and quandles>            &amp;lt;magma with 9 generators></about hap: knots and quandles>
    </79>
    <80>
        <about hap: knots and quandles>            gap> IsQuandle(QE); IsConnectedQuandle(QE);</about hap: knots and quandles>
    </80>
    <81>
        <about hap: knots and quandles>            true</about hap: knots and quandles>
    </81>
    <82>
        <about hap: knots and quandles>            true</about hap: knots and quandles>
    </82>
    <83>
        <about hap: knots and quandles>            gap> ConnectedQuandles(20); time;</about hap: knots and quandles>
    </83>
    <84>
        <about hap: knots and quandles>            [ &amp;lt;magma with 20 generators></about hap: knots and quandles>
    </84>
    <85>
        <about hap: knots and quandles>            generators></about hap: knots and quandles>
    </85>
    <86>
        <about hap: knots and quandles>              &amp;lt;magma with 20 generators></about hap: knots and quandles>
    </86>
    <87>
        <about hap: knots and quandles>            generators></about hap: knots and quandles>
    </87>
    <88>
        <about hap: knots and quandles>              &amp;lt;magma with 20 generators></about hap: knots and quandles>
    </88>
    <89>
        <about hap: knots and quandles>            generators></about hap: knots and quandles>
    </89>
    <90>
        <about hap: knots and quandles>              &amp;lt;magma with 20 generators> ]</about hap: knots and quandles>
    </90>
    <91>
        <about hap: knots and quandles>            3364296</about hap: knots and quandles>
    </91>
    <92>
        <about hap: knots and quandles>Let's denote Rx the</about hap: knots and quandles>
    </92>
    <93>
        <about hap: knots and quandles>            mapping defined by Rx : Q→Q</about hap: knots and quandles>
    </93>
    <94>
        <about hap: knots and quandles>            We define the right multiplication group G of a quandle Q by G=〈Rx</about hap: knots and quandles>
    </94>
    <95>
        <about hap: knots and quandles>            x ∈ Q〉.</about hap: knots and quandles>
    </95>
    <96>
        <about hap: knots and quandles>            We also define the automorphism group Aut(Q)={f:Q→Q}.</about hap: knots and quandles>
    </96>
    <97>
        <about hap: knots and quandles>            It can be proven that Rx is a subgroup of Aut(Q).</about hap: knots and quandles>
    </97>
    <98>
        <about hap: knots and quandles>gap></about hap: knots and quandles>
    </98>
    <99>
        <about hap: knots and quandles>            Q:=ConnectedQuandle(8</about hap: knots and quandles>
    </99>
    <100>
        <about hap: knots and quandles>            m6</about hap: knots and quandles>
    </100>
    <101>
        <about hap: knots and quandles>            gap> A:=AutomorphismGroupQuandle(Q);; a:=Random(A);;</about hap: knots and quandles>
    </101>
    <102>
        <about hap: knots and quandles>            gap> q^a;</about hap: knots and quandles>
    </102>
    <103>
        <about hap: knots and quandles>            m4</about hap: knots and quandles>
    </103>
    <104>
        <about hap: knots and quandles>            gap> R:=RightMultiplicationGroupOfQuandle(Q);; r:=Random(R);;</about hap: knots and quandles>
    </104>
    <105>
        <about hap: knots and quandles>            gap> q^r;</about hap: knots and quandles>
    </105>
    <106>
        <about hap: knots and quandles>            m3</about hap: knots and quandles>
    </106>
    <107>
        <about hap: knots and quandles>A knot is an</about hap: knots and quandles>
    </107>
    <108>
        <about hap: knots and quandles>            embedding of the circle S1 in ℝ3.</about hap: knots and quandles>
    </108>
    <109>
        <about hap: knots and quandles></about hap: knots and quandles>
    </109>
    <110>
        <about hap: knots and quandles>            To study these structures</about hap: knots and quandles>
    </110>
    <111>
        <about hap: knots and quandles>            projections of these knots into ℝ2</about hap: knots and quandles>
    </111>
    <112>
        <about hap: knots and quandles>            instance</about hap: knots and quandles>
    </112>
    <113>
        <about hap: knots and quandles>            subject to the constraint that the preimage of any (x</about hap: knots and quandles>
    </113>
    <114>
        <about hap: knots and quandles>            contains at most two points.</about hap: knots and quandles>
    </114>
    <115>
        <about hap: knots and quandles></about hap: knots and quandles>
    </115>
    <116>
        <about hap: knots and quandles>            Crossing points occur when the preimage of a point in ℝ2</about hap: knots and quandles>
    </116>
    <117>
        <about hap: knots and quandles>            contains more than one point.</about hap: knots and quandles>
    </117>
    <118>
        <about hap: knots and quandles></about hap: knots and quandles>
    </118>
    <119>
        <about hap: knots and quandles>            At these crossing points</about hap: knots and quandles>
    </119>
    <120>
        <about hap: knots and quandles>            is nearer to the ℝ2 plane as the under-crossing point and</about hap: knots and quandles>
    </120>
    <121>
        <about hap: knots and quandles>            the point farther away as the over-crossing point. An arc is a line</about hap: knots and quandles>
    </121>
    <122>
        <about hap: knots and quandles>            that connects two crossing points in the knot diagram</about hap: knots and quandles>
    </122>
    <123>
        <about hap: knots and quandles>            break occurring when an undercrossing point is mapped to the arc.</about hap: knots and quandles>
    </123>
    <124>
        <about hap: knots and quandles></about hap: knots and quandles>
    </124>
    <125>
        <about hap: knots and quandles>            We may give a knot diagram an orientation</about hap: knots and quandles>
    </125>
    <126>
        <about hap: knots and quandles>            travelling around the knot. This allows us to categorize crossings</about hap: knots and quandles>
    </126>
    <127>
        <about hap: knots and quandles>            as either positive or negative:</about hap: knots and quandles>
    </127>
    <128>
        <about hap: knots and quandles></about hap: knots and quandles>
    </128>
    <129>
        <about hap: knots and quandles></about hap: knots and quandles>
    </129>
    <130>
        <about hap: knots and quandles>                There exists different ways to describe a knot diagram: Planar</about hap: knots and quandles>
    </130>
    <131>
        <about hap: knots and quandles>                Diagram</about hap: knots and quandles>
    </131>
    <132>
        <about hap: knots and quandles></about hap: knots and quandles>
    </132>
    <133>
        <about hap: knots and quandles></about hap: knots and quandles>
    </133>
    <134>
        <about hap: knots and quandles></about hap: knots and quandles>
    </134>
    <135>
        <about hap: knots and quandles>            An other way to describe a knot is to use quandles. From a knot K</about hap: knots and quandles>
    </135>
    <136>
        <about hap: knots and quandles>            we can construct the knot quandle Q(K)</about hap: knots and quandles>
    </136>
    <137>
        <about hap: knots and quandles>            arcs of K</about hap: knots and quandles>
    </137>
    <138>
        <about hap: knots and quandles></about hap: knots and quandles>
    </138>
    <139>
        <about hap: knots and quandles></about hap: knots and quandles>
    </139>
    <140>
        <about hap: knots and quandles>              This figure gives us </about hap: knots and quandles>
    </140>
    <141>
        <about hap: knots and quandles>              b = c</about hap: knots and quandles>
    </141>
    <142>
        <about hap: knots and quandles></about hap: knots and quandles>
    </142>
    <143>
        <about hap: knots and quandles>gap></about hap: knots and quandles>
    </143>
    <144>
        <about hap: knots and quandles>            K:=PureCubicalKnot(3</about hap: knots and quandles>
    </144>
    <145>
        <about hap: knots and quandles>            gap> G:=GaussCodeOfPureCubicalKnot(K);;</about hap: knots and quandles>
    </145>
    <146>
        <about hap: knots and quandles>            gap> P:=PresentationKnotQuandle(G);</about hap: knots and quandles>
    </146>
    <147>
        <about hap: knots and quandles>            rec( generators := [ 1 .. 3 ]</about hap: knots and quandles>
    </147>
    <148>
        <about hap: knots and quandles>            3 ]</about hap: knots and quandles>
    </148>
    <149>
        <about hap: knots and quandles>From this example</about hap: knots and quandles>
    </149>
    <150>
        <about hap: knots and quandles>            the generators of the Trefoil Knot Quandle are the arcs 1</about hap: knots and quandles>
    </150>
    <151>
        <about hap: knots and quandles>            these generators satisfy the relations above.</about hap: knots and quandles>
    </151>
    <152>
        <about hap: knots and quandles></about hap: knots and quandles>
    </152>
    <153>
        <about hap: knots and quandles>            Nb: [[a1 </about hap: knots and quandles>
    </153>
    <154>
        <about hap: knots and quandles>            ▹ a2 = a3</about hap: knots and quandles>
    </154>
    <155>
        <about hap: knots and quandles>            or negative crossing.</about hap: knots and quandles>
    </155>
    <156>
        <about hap: knots and quandles>We</about hap: knots and quandles>
    </156>
    <157>
        <about hap: knots and quandles>            can also easily go from a Planar Diagram representation of a knot to</about hap: knots and quandles>
    </157>
    <158>
        <about hap: knots and quandles>            a its Gauss Code (with orientations of crossings).</about hap: knots and quandles>
    </158>
    <159>
        <about hap: knots and quandles>gap></about hap: knots and quandles>
    </159>
    <160>
        <about hap: knots and quandles>            PD:=PlanarDiagramKnot(3</about hap: knots and quandles>
    </160>
    <161>
        <about hap: knots and quandles>            [ [ 1</about hap: knots and quandles>
    </161>
    <162>
        <about hap: knots and quandles>            gap> G:=PD2GC(PD);</about hap: knots and quandles>
    </162>
    <163>
        <about hap: knots and quandles>            [ [ [ -1</about hap: knots and quandles>
    </163>
    <164>
        <about hap: knots and quandles></about hap: knots and quandles>
    </164>
    <165>
        <about hap: knots and quandles>Using</about hap: knots and quandles>
    </165>
    <166>
        <about hap: knots and quandles>            quandles</about hap: knots and quandles>
    </166>
    <167>
        <about hap: knots and quandles>            number of Homomorphisms beetween the knot (in the form of a record)</about hap: knots and quandles>
    </167>
    <168>
        <about hap: knots and quandles>            and a connected quandle.</about hap: knots and quandles>
    </168>
    <169>
        <about hap: knots and quandles>gap></about hap: knots and quandles>
    </169>
    <170>
        <about hap: knots and quandles>            K:=PresentationKnotQuandleKnot(8</about hap: knots and quandles>
    </170>
    <171>
        <about hap: knots and quandles>            rec( generators := [ 1 .. 8 ]</about hap: knots and quandles>
    </171>
    <172>
        <about hap: knots and quandles>            5 ]</about hap: knots and quandles>
    </172>
    <173>
        <about hap: knots and quandles>            [ [ 6</about hap: knots and quandles>
    </173>
    <174>
        <about hap: knots and quandles>            gap> Q:=ConnectedQuandle(9</about hap: knots and quandles>
    </174>
    <175>
        <about hap: knots and quandles>            gap> NumberHommorphism(Q</about hap: knots and quandles>
    </175>
    <176>
        <about hap: knots and quandles>            9</about hap: knots and quandles>
    </176>
    <177>
        <about hap: knots and quandles>            216228</about hap: knots and quandles>
    </177>
    <178>
        <about hap: knots and quandles>            gap> Invariant(K</about hap: knots and quandles>
    </178>
    <179>
        <about hap: knots and quandles>            [ 1</about hap: knots and quandles>
    </179>
    <180>
        <about hap: knots and quandles>            9</about hap: knots and quandles>
    </180>
    <181>
        <about hap: knots and quandles>            1327036</about hap: knots and quandles>
    </181>
    <182>
        <about hap: knots and quandles></about hap: knots and quandles>
    </182>
    <183>
        <about hap: knots and quandles></about hap: knots and quandles>
    </183>
    <184>
        <about hap: knots and quandles></about hap: knots and quandles>
    </184>
    <185>
        <about hap: knots and quandles></about hap: knots and quandles>
    </185>
    <186>
        <about hap: knots and quandles>                  Previous</about hap: knots and quandles>
    </186>
    <187>
        <about hap: knots and quandles>                      Page</about hap: knots and quandles>
    </187>
    <188>
        <about hap: knots and quandles></about hap: knots and quandles>
    </188>
    <189>
        <about hap: knots and quandles>                  Contents</about hap: knots and quandles>
    </189>
    <190>
        <about hap: knots and quandles></about hap: knots and quandles>
    </190>
    <191>
        <about hap: knots and quandles>                  Next page</about hap: knots and quandles>
    </191>
    <192>
        <about hap: knots and quandles></about hap: knots and quandles>
    </192>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
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        <about hap: knots and quandles></about hap: knots and quandles>
    </196>
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        <about hap: knots and quandles></about hap: knots and quandles>
    </197>
    <198>
        <about hap: knots and quandles>                      Page</about hap: knots and quandles>
    </198>
    <199>
        <about hap: knots and quandles></about hap: knots and quandles>
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