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Kauffman-Harary conjecture for virtual knots

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Kauffman-Harary conjecture for virtual knots
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Williamson, Mathew
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Fox colorings
Diagrams with distinct colors
Determinants
Links
Jones polynomial
Dissertations, Academic -- Mathematics -- Masters -- USF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

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Abstract:
ABSTRACT: In this paper, we examine Fox colorings of virtual knots, and moves called k-swap moves defined for virtual knot diagrams. The k-swap moves induce a one-to-one correspondence between colorings before and after the move, and can be used to reduce the number of virtual crossings. For the study of colorings, we characterize families of alternating virtual knots to generalize (2, n)-torus knots, alternating pretzel knots, and alternating 2-bridge knots. The k-swap moves are then applied to prove a "virtualization" of the Kauffman-Harary conjecture, originally stated for classical knot diagrams, for the above families of virtual pretzel knot diagrams.
Thesis:
Thesis (M.A.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
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by Mathew Williamson.
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Title from PDF of title page.
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Document formatted into pages; contains 40 pages.

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Kauffman-Harary conjecture for virtual knots
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ABSTRACT: In this paper, we examine Fox colorings of virtual knots, and moves called k-swap moves defined for virtual knot diagrams. The k-swap moves induce a one-to-one correspondence between colorings before and after the move, and can be used to reduce the number of virtual crossings. For the study of colorings, we characterize families of alternating virtual knots to generalize (2, n)-torus knots, alternating pretzel knots, and alternating 2-bridge knots. The k-swap moves are then applied to prove a "virtualization" of the Kauffman-Harary conjecture, originally stated for classical knot diagrams, for the above families of virtual pretzel knot diagrams.
502
Thesis (M.A.)--University of South Florida, 2006.
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Includes bibliographical references.
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System requirements: World Wide Web browser and PDF reader.
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Advisor: Masahiko Saito, Ph.D.
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Fox colorings.
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Determinants.
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Jones polynomial.
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Kauman-HararyConjectureforVirtualKnots by MathewWilliamson Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofArts DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:MasahikoSaito,Ph.D. MohamedElhamdadi,Ph.D. MileKrajcevski,Ph.D. DateofApproval: April2,2007 Keywords:Foxcolorings,diagramswithdistinctcolors,de terminants,links,Jones Polynomial c r Copyright2006,MathewWilliamson

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TableofContents ListofFigures iii Abstract v 1BackgroundandMotivation 1 1.1Introduction.................................... 1 1.2Preliminaries................................... 2 1.2.1ClassicalKnots..............................21.2.2VirtualKnots...............................51.2.3Quandles.................................71.2.4ColoringsofKnots............................81.2.5TheFundamentalQuandle........................91.2.6Kauman-HararyConjecture...................... 11 1.2.7BracketPolynomialandtheJonesPolynomial........ .....12 2SwapMovesandSomeFamiliesofVirtualKnotDiagrams14 2.1The k -SwapMoves................................14 2.1.1Denitionof k -SwapMoves.......................14 2.1.2ColoringsandtheDeterminantunder k -SwapMoves.........14 2.1.3QuandlesandtheJonesPolynomialunder k -SwapMoves.......17 2.2SomeFamiliesofAlternatingVirtualKnotDiagrams.... .........19 2.2.1Closed2-StringVirtualBraidDiagrams............ ....19 2.2.2VirtualPretzelKnotDiagrams.................... .26 2.2.3Virtual2-BridgeKnotDiagrams................... ..31 2.3Kauman-HararyConjectureforAlternatingVirtualKno tsandSwapMoves36 3Conclusion 38 i

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References 39 ii

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ListofFigures 1.1Twodiagramsofanequivalentknot:atrefoil.......... .......3 1.2TheReidemeistermoves............................ .3 1.3TheHopflink...................................41.4Overandunderarcsofacrossing..................... ...4 1.5Anugatorycrossing............................... .5 1.6Analternatingandanon-alternatingdiagram......... ........5 1.7Positiveandnegativecrossings.................... ......6 1.8Typesofcrossings................................ .6 1.9ClassicalandvirtualReidemeistermoves............ ........6 1.10RelationofquandlestotheReidemeistermoves....... .........7 1.11Coloringrelations.............................. ...8 1.12Atrefoilcoloredby R 3 ..............................9 1.13Trefoilanditsrelations......................... .....10 1.14Figure8knot...................................1 1 1.15Bracketcrossingcomputations.................... ......12 2.1A k -swapmove..................................15 2.2Arcsundera k -swap...............................15 2.3Dihedralcoloringproof........................... ...16 2.4A k -swaponthefundamentalinvolutoryquandle............. ..18 2.5A k -swap'seectonthebracketpolynomial................. .18 2.6Examplesofvirtualtwists:thetopis[ v; 3 ; 2 ;v ]andthebottomis[2 ; 3 ; 1]19 2.7Thecase n =1oftheReducingLemma.....................21 2.8Thecasewhen n isodd, x 1 = v and x 2 = ; ...................22 2.9Thecasewhen n isevenand x i = v .......................23 2.10ThetwocasesfromLemma2.2.5: D 1 haseven n and x 2 = v ,while D 2 has odd n and x 2 = ; ..................................24 iii

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2.11The2-stringvirtualbraiddiagram D = T (2 ; [ v; 2 ; 1 ; 2 ;v ])..........25 2.12Thegeneralsetupofapretzelknot.................. ......26 2.13Thealternatingvirtualpretzelknot P ([ v; 1 ; 1 ;v ] ; [ 1 ; 3] ; [ v; 1 ; 1 ;v ]) : ...27 2.14Onenon-alternatingpretzelknotdiagramconstructio n,andthetwoalternatingpretzelknotdiagrams............................2 8 2.15Thevirtualevenpretzelknotdiagram D = P ([ v; 2 ; 2] ; [ v; 1 ; 3] ; [ 6])...30 2.16Theoddvirtualpretzelknotdiagram D = P ([ v; 5] ; [ v; 1 ; 1 ; 1] ; [ v; 5])....31 2.17Thegeneral2-bridgevirtualknotdiagram........... ........32 2.18The2-bridgevirtualknotdiagram B ([ v; 2 ; 2 ;v ] ; [ v; 1 ;v ] ; [ 1 ;v ]).....32 2.19Thethreecasestocheck........................... ..33 2.20Thecaseswhen r = n ...............................34 iv

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Kauman-HararyConjectureforVirtualKnots MathewWilliamson ABSTRACT Inthispaper,weexamineFoxcoloringsofvirtualknots,and movescalled k -swapmoves denedforvirtualknotdiagrams.The k -swapmovesinduceaone-to-onecorrespondence betweencoloringsbeforeandafterthemove,andcanbeusedt oreducethenumberofvirtual crossings.Forthestudyofcolorings,wecharacterizefami liesofalternatingvirtualknots togeneralize(2 ;n )-torusknots,alternatingpretzelknots,andalternating 2-bridgeknots. The k -swapmovesarethenappliedtoprovea"virtualization"oft heKauman-Harary conjecture,originallystatedforclassicalknotdiagrams ,fortheabovefamiliesofvirtual pretzelknotdiagrams. v

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1BackgroundandMotivation 1.1Introduction In[KVKT],Kaumandenesanextensionofclassicalknotthe ory,calledvirtualknottheory,motivatedbyGausscodesandthickenedsurfaces.Manyc lassicalknotinvariantscan begeneralizedtovirtualknottheory,includingquandlekn otcolorings.Fox([F])described coloringsofknotdiagramsby Z n andrelatedthemtoAlexanderpolynomialsandhomomorphismsfromtheknotgrouptodihedralgroups.Sincethen,co loringsofknotdiagramsby quandleshavebeenextensivelystudied.Thispaperconside rsvirtualknotcolorings,along withavirtualizationofaclassicalknotconjecturebyKau manandHarary([KH]). Conjecture1.1.1(Kauman-HararyConjecture1) Let D beareducedalternating knotdiagramwithaprimedeterminant p .TheneverynontrivialFox's p -coloringof D assignsdierentcolorstodierentarcsof D Theconjecturewasrstposedin[KH],andprovedfortoruskn ots T (2 ;n )inthesame paper.Inalaterpaper[KL],itwasshownthattheKauman-Ha raryconjectureholdsfor any2-bridgeknot(rationalknots)withoutrestrictionson theknotdeterminant.Later,in [AM],theconjecturewasprovedforallMontesinoslinksand amoregeneralconjecturewas madeinvolvingthehomologyofthedoublecoverofthe3-sphe re S 3 branchedalongalink. Finally,in[AS],aconjectureassociatedwiththeAlexande rquandlewasintroduced,which generalizestheoriginalconjectureforFoxcolorings. ThispaperdealswithananalogueoftheoriginalKauman-Ha raryconjectureforvirtual knots.Toaccomplishthis,weintroduceamovecalledthe k -swapmove.Itisinstructiveto notethatKaumanstudied1-swapmoves(heonlydescribedwh attheydid,andhedidnot haveanameforthem)in[KVKT].Heshowedthatthe1-swapdoes notchangetheJones polynomialandtheinvolutoryquandlebutmaychangethefun damentalquandle.This facthadacorollarywhichwasusedtoeasilyconstructavirt ualknotthathadanontrivial fundamentalgroupbutatrivialJonespolynomial.Toputthi sresultintoperspective,the 1

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questionofwhetheranontrivialclassicallinkcouldhavea trivialJonespolynomialwasonly answeredin2000([ES]),butforknotsitisstillanopenques tion.Invirtualknottheory, however,thequestionwasansweredquickly.SoKauman'sap proachtowhatwecalla1swapmovewastoshowthatwecanbuildnontrivialvirtualkno tsfromtrivialclassicalknots, keepingsomeinvariantsunchanged.Ourapproachistosimpl ifyvirtualknotdiagramsand applyresultsfromclassicalknottheorytovirtualknotthe ory. Thepaperisorganizedinthefollowingway.Section1organi zesthebackgroundneeded forthemainresults.Itexplainsvirtualknots,quandles,k notcolorings,andtheKaumanHararyconjecture.InSection2.1,the k -swapmoveforvirtualknotsisdenedandshown tobeacoloringinvariantovervirtualknots.AlsoinSectio n2.1istheresultthatthe k swapmovedoesnotchangetheKaumanbracket,andtherefore theJonespolynomialis unaectedaswell.Section2.2and2.3showthatalternating closed2-stringvirtualbraid diagrams,alternatingvirtualpretzelknotdiagrams,anda lternatingvirtual2-bridgeknot diagramseachsatisfytheKauman-Hararyconjecture,usin gthe k -swapmovedenedin Section2.1. Finally,wewouldliketothankDr.GregoryMcColmforhisnum erouscontributionsto thisthesis.Withouthishelp,itwouldnothavebeenaccompl ished. 1.2Preliminaries 1.2.1ClassicalKnotsThemostfundamentalofallquestionsinknottheoryisexami ningwhethertwoknotsare thesameknot.Thisisasubtleproblem,andmanyinvariantsa ndtechniqueshavebeen foundtohelp.Toprocedefurther,weneedtoknowwhataknoti s.A knot istheimage ofadierentiableembedding f : S 1 R 3 fromacircle S 1 tothe3-dimensionalEuclidiean space R 3 .Twoknotsareequivalentifthereisadieomorphismof R 3 thattakesoneknot totheother.Thisdenitionexcludesknotswithlimitpoint s,andpolygonalknots,bothof whichwedonotneed.Ifinterested,check[CP]formoreinfor mation. Sinceknotsaresubsetsof R 3 ,wecanmovethestrandsofaknotin R 3 aslongastwo strandsdonotpassthrougheachother.Projectingaknotdow nintothestandard xy -plane givesusa knotdiagram .Sincewecanmovestrandsin R 3 withoutcuttingthem,andsince wecanrotatetheknotatwill,thereareinnitelymanyknotd iagramsofanygivenknot. Thismakesrecognizingthattwoknotsareequivalentdicul t.Foranexampleoftwoknot 2

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diagramswhoseknotisequivalent,seeFigure1.1.Toseetha ttheseknotdiagramsare Figure1.1:Twodiagramsofanequivalentknot:atrefoil equivalent,wecanemploytheReidemeistermovestotransfo rmonediagramintotheother. ThesemovesarepicturedinFigure1.2. III) I) II) Figure1.2:TheReidemeistermoves Itisknownthattwoknotdiagramsareequivalentif,andonly if,onediagramcan betransformedintotheotherusinganitesequenceofReide meistermoves.Itisusually impracticaltotryandndasequenceofReidemeistermovest oprovethattwoknotdiagrams arediagramsofanequivalentknot.Thus,inthisthesis,wes tudyequivalenceclassesofknots bydiagramsandtheirmoves,insteadofbydierentialmappi ngs,andweturnourattention toknotinvariantswhichtellusiftwoknotdiagramsarediag ramsofdierentknots.A fewknotinvariantsarediscussedlaterinthispapersowedo nottalkaboutanyspecic invariantsyet. Whenwetalkofknots,wemeanthatwehaveonlyonecomponent( orstrand).Cutting theknotoncewouldleaveuswithonelinesegment.Thereisno reasontorestrictourselvesto 3

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onecomponent,sowecallanydierentiablyembeddedcircle swithmorethanonecomponent a link .Weadopttheconventionthatwhenwemeanalinkwithonecomp onent,wewill specicallysayknot.Thisishelpfulbecausemanyinvarian tsofknotsarealsoinvariantsof links.Foranexampleofalink,seeFigure1.3.NotethatReid emeistermovesapplytolinks aswellasknots. Figure1.3:TheHopflink Nowwepresentsomebasicdenitionsoftheanatomyofalink. A crossing isa4-valent vertexinalinkdiagramwhichhasoverandunderinformation retained.Thearcthatisan unbrokenlineisanoverarcandthetwobrokenarcsaretheund erarcs.SeeFigure1.4. Over Under Under Figure1.4:Overandunderarcsofacrossing Alinkdiagramissaidtohavea nugatorycrossing ,orremovablecrossing,ifthereexists acircleintheplaneofthediagramthatintersectsthediagr amatasinglecrossing.This situationisshowninFigure1.5.Thiscrossingisnugatoryb ecauseitcanberemoved simplybyrotatingaportionofthelink.InthecaseofFigure 1.5,the F portionwasripped horizontallyfrombottomtotop.Adiagramwithnonugatoryc rossingsissaidtobe reduced Denition1.2.1 Adiagram D is alternating ifwecantravelalongthelinkandpassover andunderalternately.Otherwise,itiscalled non-alternating .Alink K is alternating if thereexistsatleastonelinkdiagram D of K suchthatthelinkdiagram D isalternating. SeeFigure1.6foranalternatingandanon-alternatinglink diagram,respectively.The 4

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F J JF Figure1.5:Anugatorycrossing linkcorrespondingtothelinkdiagramontherightofFigure 1.6isthesameasthelinkon theleft.WecanuseaReidemeisterIImoveontheright-handl inktotransformitintothe linkontheleft. Figure1.6:Analternatingandanon-alternatingdiagram Sometimesitisusefultoassignonorientationtolinkdiagr ams.Thenitmakessenseto talkaboutpositiveornegativecrossingsinadiagram.Fort hisconvention,seeFigure1.7. 1.2.2VirtualKnotsInclassicalknottheory,weconsiderprojectionsoflinkst osomeplanetodenethelink diagrams.Forthemotivationsofvirtualknottheory,weenc ouragethereadertoconsult [KVKT].Virtualknottheoryissimilarinusingdiagramsexc eptthatthereisanextra typeofcrossingcalledavirtualcrossing.Thus,therearet hreetypesofcrossingsforan orientedvirtualknotdiagram:positiveornegativeclassi calcrossingsandvirtualcrossings (seeFigure1.8).Furthermore,avirtuallinkdiagramisalt ernatingifwecantravelalong 5

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PositiveNegative Figure1.7:Positiveandnegativecrossings thelinkpassingoverandunderalternately,justasinthecl assicalcase.Notethatvirtual crossingsarenotconsidered"over"or"under",sowejusttr aveltothenextclassicalcrossing. positivevirtual negative Figure1.8:Typesofcrossings Twovirtualdiagramsareequivalentifonecanbetransforme dintotheotherbyanite sequenceofextendedReidemeistermoves(showninFigure1. 9)combinedwithorientation preservinghomeomorphismsoftheplanetoitself,asinthec lassicalcase.MovesI,II,and IIIarejusttheclassicalReidemeistermoves.Avariationo fthetypeIIImovewithasingle virtualcrossingandtwoclassicalcrossingsisprohibited III) I) III') II') I') III") II) Figure1.9:ClassicalandvirtualReidemeistermoves 6

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1.2.3QuandlesWepresentabriefreviewofquandlesandquandlecolorings.Denition1.2.2 ([JD])A quandle X ,isasetwithabinaryoperation : X X X satisfyingthefollowingconditions:(Q1)Forany x in X x x = x (Q2)Forany x;y in X ,thereexistsaunique z in X suchthat x = z y (Q3)Forany x;y;z in X ,( x y ) z =( x z ) ( y z ). Thecondition(Q2)isalsoequivalenttothefollowingcondi tion: (Q2')Thereisabinaryoperation 1 : X X X suchthat( x y ) 1 y = x =( x 1 y ) y forany x;y in X TherelationofquandlestolinksisshowninFigure1.10.Eac hquandleaxiomcorresponds tooneoftheReidemeistermoves.See[BE,JD,MS]formoredet ailedaspectsofquandles. Thefollowingisacoupleofsomecommonlyusedexamplesofqu andles. x x x x x I) II) z y x z x y x y y z ( x z ) ( y z ) y z ( x y ) z x z y yxyx x x y z III) Figure1.10:RelationofquandlestotheReidemeistermoves Example1.2.3 Let= Z [ t;t 1 ]beaLaurentPolynomialringover Z ,andlet J be anidealof.Thenthequotientring =J withthebinaryoperationdenedby x y = tx +(1 t ) y forany x;y 2 =J isaquandle,calledan Alexanderquandle .Theoperation 1 isgivenby x 1 y = t 1 x +(1 t 1 ) y 7

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Example1.2.4 Let m beapositiveinteger.Forelements i;j 2 Z m ,dene i j =2 j i (mod m ).Then denesaquandlecalleda dihedralquandle R m oforder m .Alsonotethat thedihedralquandle R m isisomorphicto = ( m;t +1).Furthermore, 1 isthesameas Occassionally,wewillneedanothertypeofquandlewhichdo esnotdependuponthe orientationsoftheknotitself.Denition1.2.5 [TM]An involutoryquandle X ,isaquandlesuchthat( x y ) y = x for all x and y 2 X Thenotionofinvolutoryquandlesrstappearedasearlyas1 942([TM]).Also,notice thatthedihedralquandleisaninvolutoryquandle,butthat ,ingeneral,theAlexander quandleisnot.1.2.4ColoringsofKnotsLet X beaquandle, D beanorientedvirtualknotdiagram,and A bethesetof(over)-arcs. Thenormalvectorsaregiveninsuchawaythattheorderedpai r(tangent,normal)agrees withtheorientationofthestandardorientationofthe xy -plane.SeeFigure1.11. Denition1.2.6 ([FR])A coloring C isamap C : A! X suchthatateveryclassical crossing,therelation C ( ) C ( )= C ( r )holds,wherethenormaltotheoverarc points fromthearc tothearc r (seeFigure1.11).Ateveryvirtualcrossing,thecoloring C ( )= a holdsforonearc ,and C ( )= b holdsfortheotherarc .Theimage C ( )iscalleda color ofthearc .Thecolorsintheorderedpair h a;b i arecalledthe sourcecolors C ( )= b C ( )= a C ( r )= c = a b C ( )= b C ( )= a C ( )= b C ( )= a Figure1.11:Coloringrelations LetCol X ( D )denotethesetofcoloringsofaknotdiagram D ofaknot K byaquandle X .Thenthereisaone-to-onecorrespondencebetweenthesets ofcoloringsoftwodiagrams ofthesameknot.Inparticular,thenumberofelements j Col X ( D ) j ofCol X ( D )foranite quandle X isaknotinvariant.Anyknotdiagram D hasatleastonecoloringforagiven 8

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quandle X ,the trivialcoloring obtainedbylettingeveryarchavethesamecolor.Also,ifa knotdiagram D canbenon-triviallycoloredbythedihedralquandle R n ,then K issaidto be n -colorable. Example1.2.7 Let X bethedihedralquandle R 3 ,andlet K bethetrefoilasshownin Figure1.12.Now,letthesourcecolorsbegivenby h 0 ; 1 i atthetopofthetwistintheknot. Thenthetrefoiliscoloredby R 3 .Noticethatthisisjustonepossiblecoloringofthetrefoi l fromthesetCol X ( D ). 01=2(0)-2(mod3) 012=2(1)-0(mod3)20=2(2)-1(mod3) 1 Figure1.12:Atrefoilcoloredby R 3 TheexampleaboveisalsocalledaFox3-coloringofthetrefo il.Ingeneral,aFox n coloringisacoloringbythedihedralquandle R n 1.2.5TheFundamentalQuandleDenition1.2.8 ([SJ])Let f x 1 ;x 2 ;:::;x k g bevariablesassignedtoarcsofavirtualknot diagram K ,let x l = x i x j =2 x j x i ,where x j isthevariableassignedtotheoverarcand x i isthevariableassignedtooneoftheunderarcs.Then x l isassignedtotheotherunderarc. Nowlet A bethematrixofrelationsassociatedwiththesetoftheseeq uationssuchthatthe equationscorrespondtotherowsinthematrix,andthe i thvariablecorrespondstothe i th columnofthematrix.If M ij ( K )isthe( i;j )-minorof A ,thenthedeterminantoftheknot K isdenedby Det( K )=gcd( fj Det( M ij ( K )) j :1 i j k g ) : (1.2.1) Thenaturalquestionnowiswhatwouldbeasuitable n tocoloranygivendiagramwith adihedralquandle R n ?ThemostnaturalchoiceisthedeterminantoftheknotDet( K ). 9

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Theorem1.2.9 [F]Let K beaclassicalknot.Foraprime p K is p -colorableif p j Det( K ) Example1.2.10 Let K bethetrefoilshowninFigure1.13.Thenthematrix A isgivenby 26664 12 1 1 12 2 1 1 37775 ; wheretherowscorrespondtotherelationsandthecolumnsco rrespondtothevariables. Thedeterminantofeachminoris 3,sothedeterminantof K willbe3.Thus,thisknot canbecolorednon-triviallyusingthedihedralquandle R 3 .Notethatintheclassicalcase, anyminorof A willhavethesamedeterminantsothegcdofalltheminorsise quivalentto ndingoneoftheminors.Seethegure. x 3 = x 1 x 2 =2 x 2 x 1 x 1 = x 2 x 3 =2 x 3 x 2 x 2 = x 3 x 1 =2 x 1 x 3 x 2 x 1 Figure1.13:Trefoilanditsrelations Remark1.2.11 Considerthegure-eightknotdiagramshownbelowinFigure 1.14.The determinantofthisknotis5,butithasonly4arcs.Thisimpl iesthattheknotiscolored by R 5 ,butnotallelementsof R 5 areusedforcolors. Denition1.2.12 [JD,KVKT,MS]Let f x 1 ;x 2 ;:::;x k g bevariablesassignedtoarcsofa virtualknotdiagram K ,let x l = x i x j beassignedateachcrossing,where x j isthevariable assignedtotheoverarcand x i isthevariableassignedtooneoftheunderarcsfromwhich theorientationofthenormalvectoroftheover-arcpoints. Then x l isassignedtotheother underarc.Thequandle Q ( K )determinedbythesetofgenerators f x 1 ;x 2 ;:::;x k g andthe setofrelations f x l = x i x j g overallcrossingsiscalledthe fundamentalquandle of K .In 10

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Figure1.14:Figure8knot addition,whentherelations( x y ) y = x areimposedforallelementsof Q ( K ),thenthe quandlethusdeterminediscalledthe fundamentalinvolutoryquandle Aquandledenedbythesetofgenerators f x 1 ;x 2 ;:::;x k g andrelations f r 1 ;r 2 ;:::;r m g as aboveisdenotedby h x 1 ;x 2 ;:::;x k j r 1 ;r 2 ;:::;r m i ,andthisnotationiscalleda presentation ofthequandle.Thefundamentalinvolutoryquandleisdenot edby IQ ( K ). 1.2.6Kauman-HararyConjectureInthispaperweconsiderthefollowingconjecturebyKauma nandHarary([KH]). Conjecture1.2.13(Kauman-HararyConjecture1) Let D beareducedalternating knotdiagramwithaprimedeterminant p .TheneverynontrivialFox's p -coloringof D assignsdierentcolorstodierentarcsof D Insteadofrestrictingtheconjecturetoonlyclassicalkno ts,weconsiderthevirtualization oftheKauman-HararyConjecture.Conjecture1.2.14(Kauman-HararyConjecture2) Let D beareducedalternating virtualknotdiagramwithaprimedeterminant p .TheneverynontrivialFox's p -coloringof D assignsdierentcolorstodierentarcsof D KaumanandHararyrstprovedConjecture1.2.13forthefam ilyofrational(or2bridge)classicalknotsin[KL],and[PL].Asaeda,etal.,[A M],provedtheconjectureto betrueforMontesinoslinks(whichincludepretzelknots). OuraimistoproveKaumanHararyConjecture2tobetrueforcertainfamiliesofvirtua lknots. 11

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1.2.7BracketPolynomialandtheJonesPolynomialThefollowingmaterialisstandard,see[KKP,CP]formorein formation.Eventhoughthe letter A wasusedearlierforamatrix,wenowfollowthestandardnota tionanduse A asa variableinthissection.Let D beanunorienteddiagramofaknot K .The bracketpolynomial h D i of D isaLaurentpolynomialofavariable A denedbythefollowingaxioms: 1. h D + i = A h D 0 i + A 1 h D 1 i 2. hr D i = d h D i 3. hri = d where d = A 2 A 2 and D + ;D 0 ; and D 1 areidenticaloutsideofasmallballneighborhood,insidewhichtheylooklikeasdepictedinFigure1.15. Theaxiomsgivearecursive computationof h D i .Byrepeatedapplicationofaxiom1,adiagrambecomesaseto fdiagramswithnocrossings.Thevalueofeachoftheseotherdiag ramsiscalculatedbyusing axioms2and3. D + D 0 D 1 Figure1.15:Bracketcrossingcomputations Theorem1.2.15 [KJ]Thebracketpolynomial h D i isinvariantunderReidemeisterIIand IIImoves. Inorderforthebracketpolynomialtobeinvariantunderthe ReidemeisterImove,we needthewritheofaknot([KKP],forexample).The writhe w ( D )ofaknotisthenumber ofpositivecrossingsminusthenumberofnegativecrossing s. Denition1.2.16 Let D beanorienteddiagram,andlet j D j denotethediagram D without orientation.Thenthe normalizedbracketpolynomial isgivenby ~ V D ( A )=( A 3 ) w ( D ) hj D ji ( A ) : ThispolynomialisinvariantunderalltheReidemeistermov es. 12

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Theorem1.2.17 [KJ]Thenormalizedbracketpolynomialisequivalenttothe Jonespolynomialunderachangeofvariable: V L ( A 4 )= ~ V L ( A ) ; where V L ( t ) istheJonespolynomial. 13

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2SwapMovesandSomeFamiliesofVirtualKnotDiagrams 2.1The k -SwapMoves Inthissection,wedene k -swapmovesandinvestigatetheirproperties. 2.1.1Denitionof k -SwapMoves Let A ( K i )bethesetofarcsofthevirtualknotdiagrams K i ,for i =1 ; 2,andlet C i 2 Col n ( K i ) suchthat C i : A ( K i ) R n ,where n isapositiveinteger. Denition2.1.1 Let K 1 beatwistwith k positivecrossingsandonevirtualcrossingat thebottomasdepictedintheleftofFigure2.1.Thena k-swapmove isamovewhich changes K 1 intothetwist K 2 suchthat K 2 hasavirtualcrossingatthetopfollowedby k negativecrossingsasintherightofFigure2.1.InFigure2. 1, K 1 haspositivecrossings,by convention.Thechangefrom K 2 to K 1 ,conversely,isalsocalleda k-swapmove .Notethat ifthetwoarcshaveparallelorientation,thenthecrossing sof K 1 arepositive,butifthearcs haveanti-parallelorientation,thenthecrossingsareneg ative.Thusourconventionforthe positiveornegativesignofatwistwithavirtualcrossingi sthesameasthesignwhenthe arcshaveparallelorientation.2.1.2ColoringsandtheDeterminantunder k -SwapMoves Denition2.1.2 Let D 2 beadiagramobtainedfrom D 1 byapplyinga k -swapmoveonce. Thendeneamap Sp: A ( D 1 ) !A ( D 2 ) asfollows.If 2A ( D 1 )isnotcontainedinthetwistwherethe k -swapmoveisperformed, thendeneSp( )= 2A ( D 2 ).If 2A ( D 1 )iscontainedinthetwist,then isoneofthe arcs i or j inthetwistsuchthat0 i;j l ,where l = k= 2if k iseven,and l =( k 1) = 2 14

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K 2 K 1 k k Figure2.1:A k -swapmove if k isodd,asdepictedintheleftofFigure2.2.Thearcsafterth emovearelabeledasin therightofFigure2.2.ThendeneSp( i )= 0 i andSp( j )= 0 j forall i and j l l 1 1 0 1 1 2 l l 0 0 l l +1 0 l +1 0 l Forodd k : l l +1 0 l 0 l 0 0 0 0 0 0 0 1 0 1 0 l 1 0 l 0 l 0 2 0 1 0 1 0 0 0 0 0 l Figure2.2:Arcsundera k -swap Lemma2.1.3 Thefunction Sp: A ( D 1 ) !A ( D 2 ) isabijection.Proof. Thisfollowsfromthedenition. 15

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Theorem2.1.4 Again,let D 2 bethediagramobtainedfrom D 1 byapplyinga k -swapmove once.Let Sp: A ( D 1 ) !A ( D 2 ) bethebijectiondenedinDenition2.1.2.Let C 1 2 Col n ( D 1 ) .Thenthemapdenedby C 2 = C 1 Sp 1 isan n -coloringof D 2 : C 2 2 Col n ( D 2 ) 3 b 2 a C ( 0 )= b 2 b a b 2 b a C ( 0 )= a kb ( k 1) a ( k +1) b ka kb ( k 1) a ( k +1) b ka C ( 0 )= a b 2 b a kb ( k 1) a ( k +1) b ka a C ( 0 )= b b k k kb ( k 1) a kb ( k 1) a Figure2.3:Dihedralcoloringproof Proof. Let D 1 bethetwistsuchthat D 1 has k positivecrossingswithavirtualcrossing atthebottom.Nowlet h a;b i bethevectorofsourcecolorsatthetopofthetwist,where a = C 1 ( 0 )and b = C 1 ( 0 ).Then,usingtheequation x y =2 y x ,weendupwith h ( k +1) b ka;kb ( k 1) a i atthebottomof D 1 (seetheleftofFigure2.3).Similarly, D 2 isthetwistwithavirtual crossingatthetopfollowedby k negativecrossings.If h a;b i isthevectorofsourcecolorsat thetopofthetwist,then h ( k +1) b ka;kb ( k 1) a i isagainfoundatthebottom(seetherightofFigure2.3). Thus, C 1 ( i )= C 1 (Sp 1 ( 0 i ))= C 2 ( 0 i ),forsome i 2A ( D 1 ). Corollary2.1.5 UndertheassumptionsofTheorem2.1.4,wehavetheequality C 1 ( A ( D 1 ))= C 2 ( A ( D 2 )) 16

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Proof. Let m 2C 1 ( A ( D 1 )) R n .Thenthereisanarc r 2A ( D 1 )suchthat C 1 ( r )= m Then C 2 (Sp( r ))= C 1 (Sp 1 (Sp( r )))= m .Hence, m 2C 2 ( A ( D 2 )). Corollary2.1.6 AgainusingtheassumptionsofTheorem2.1.4,weseethat Det( D 1 )=Det( D 2 ) : Proof. FromTheorem2.1.4,weknowthatthearcsdonotchangesothem atrixdenedin thedenitionofdeterminantdoesnotchange. Corollary2.1.7 IftheassumptionsofTheorem2.1.4aretrue,then j Col p ( D 1 ) j = j Col p ( D 2 ) j Proof. UsingCorollary2.1.5,eachcoloringof D 1 hasacorrespondingcoloringof D 2 ,and C 1 ( A ( D 1 ))= C 2 ( A ( D 2 )).Theresultfollows. Corollary2.1.8 If D 2 isthediagramobtainedfrom D 1 byapplyinga k -swapmoveonce, andif D 1 isalternating,then D 2 isalternatingaswell. Proof. AgainusingTheorem2.1.4,thearcsdonotchangebetween D 1 and D 2 sotheresult follows. 2.1.3QuandlesandtheJonesPolynomialunder k -SwapMoves Nextweinvestigatethefundamentalinvolutoryquandleand theJonespolynomial. Theorem2.1.9 Let X bethefundamentalinvolutoryquandleofavirtualknot K 1 with adiagram D 1 .If D 2 isobtainedfrom D 1 bya k -swapmove,then X isthefundamental involutoryquandleofthevirtualknot K 2 ,where D 2 isadiagramof K 2 Proof. ThisfollowsfromDenition2.1.1andFigure2.4.Also,the k =1casewasproved in[KVKT]byKauman. Theorem2.1.10 Fortwovirtualknots K 1 and K 2 ,if K 2 isobtainedfrom K 1 bya k -swap move,thentheirJonespolynomialcoincide: V K 1 ( t )= V K 2 ( t ) 17

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x 1 x 2 x 3 x k +1 x k x 1 x 2 x 2 x k +2 x k +1 x k +1 x k +2 x 01 x 02 x 0k x 0k +1 x 0k +2 x 03 x 02 x 0k +1 Figure2.4:A k -swaponthefundamentalinvolutoryquandle Proof. Givenatwist T inaclassicalknot K ,usethevirtualReidemeisterII'movetomake aknot K 0 .Thisknot K 0 isstillanunreducedclassicalknot,soperforma k -swaptoobtain avirtualknot K 00 .ByLemma2.1.11,thisknot K 00 hasthesameJonespolynomialas K Thisfollowsfromthenextlemma. Lemma2.1.11 Let h D 1 i bethebracketpolynomialofavirtualknotdiagram D 1 .If D 2 is obtainedfrom D 1 bya k -swapmove,then h D i = h D 0 i x 1 x 2 x 3 x k +1 x k x 1 x 2 x 2 x k +2 x k +1 x k +1 x k +2 x 01 x 02 x 0k x 0k +1 x 0k +2 x 03 x 02 x 0k +1 Figure2.5:A k -swap'seectonthebracketpolynomial Proof. UseDenition2.1.1andlookatFigure2.5.Morespecically ,applytheaxiomin Fig.1.15atacrossinginbetweenvirtualcrossingsintheto pofFig.2.5andacrossing ofthebottomofthegure,thenafterReidemeistermoves,we obtainthesameinvariant 18

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values.Also,notethatKaumanprovedthe k =1casein[KVKT],andthisproofisa directanalogue. Thefollowingsectionswillgivemanyexamplesoftheaboves tatements,sowewillnot delveintoanyexampleshere. 2.2SomeFamiliesofAlternatingVirtualKnotDiagrams 2.2.1Closed2-StringVirtualBraidDiagramsInordertoshowwhatthecoloringsofclosed2-stringvirtua lbraidsare,weneedsome background.Denition2.2.1 Forapostiveinteger n ,let k i 2 Z for i =1 ;:::;n .A virtualtwist [ x 1 ;k 1 ;k 2 ;:::;k n ;x 2 ]isasequenceofclassicaltwistsandvirtualcrossingssuc hthatthefollowingholdstrue.First,each k i correspondstoaclassicaltwistwith j k i j classicalcrossings, positivecrossingsif k i > 0andnegativeif k i < 0.Betweenthe i -thand( i +1)-thtwistsis avirtualcrossingfor i =1 ; 2 ;:::;n ,andeach x ` ,where ` =1 ; 2,representseitheravirtual crossing(denotedby x ` = v )ornovirtualcrossing(denotedby x ` = ; ).An alternating virtualtwist [ x 1 ;k 1 ;k 2 ;:::;k n ;x 2 ],where x ` iseither v or ; ,isavirtualtwistsuchthat k i > 0 forodd i and k j < 0foreven j ,or k i < 0forodd i and k j > 0foreven j .Wealsowrite [ k 1 ;k 2 ;:::;k n ;x ` ]for[ ; ;k 1 ;k 2 ;:::;k n ;x ` ].Forexamplesofthis,seeFigure2.6. Theabovedenitionagreeswithourintuitionanduseofthew ordalternatingbecause eitherstrandinanalternatingvirtualtwistisseentoalte rnateinoverarcsandunderarcs. Denition2.2.2 If[ k 1 ;k 2 ;:::;k n ;x 1 ]isavirtualtwist,thentheclosed2-stringvirtualbraid diagramisdenotedby T (2 ; [ k 1 ;k 2 ;:::;k n ;x 1 ]).Also, T (2 ; [ k 1 ])istheclassical(2 ;k 1 )-torus knot([CP]),alsodenotedby T (2 ;k 1 ). Figure2.6:Examplesofvirtualtwists:thetopis[ v; 3 ; 2 ;v ]andthebottomis[2 ; 3 ; 1] 19

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Notethatif D = T (2 ; [ k 1 ;k 2 ;:::;k n ]),thenthereare n 1virtualcrossingsin D .Also,if D = T (2 ; [ k 1 ;k 2 ;:::;k n ;v ]),thenthereare n virtualcrossingsin D Theproofofthefollowinglemmaarestraightforwardfromth edenition,sotheyare omitted.Lemma2.2.3 Thevirtualknotdiagram T (2 ; [ k 1 ;k 2 ;:::;k n ]) isalternatingi n isoddand k i ispositiveforodd i andnegativeforeven i ,orpositiveforeven i andnegativeforodd i Similarly,thevirtualknotdiagram T (2 ; [ k 1 ;k 2 ;:::;k n ;v ]) isalternatingi n isevenandthe sameconditionissatised.InFigure2.6,thetorusknotclo sureofthetopvirtualtwistis notalternatingbutthebottomoneis. NotethatLemma2.2.3impliesthatif D = T (2 ; [ k 1 ;k 2 ;:::;k n ;x ])isalternatingthenit hasanevennumberofvirtualcrossings,where x iseither v or ; .Here x = ; meansthat D = T (2 ; [ k 1 ;:::;k n ]). Lemma2.2.4(ReducingLemma) If T =[ x 1 ;k 1 ;k 2 ;:::;k n ;x 2 ] isanalternatingvirtual twist,where x ` iseither v or ; ,thenthevirtualtwistbecomes T 0 = x ( T ) ; ( T ) ( x 2 ) n X i =1 ( 1) i k i ; where x ( T )= 8<: v if ( T ) ( x 1 ) ( x 2 )= 1 ; if ( T ) ( x 1 ) ( x 2 )=1 ; ( T )= 8<: 1 if n isodd 1 if n iseven ; ( x i )= 8<: 1 if x i = v 1 if x i = ; ; afterasequenceof k -swapmovesandReidemeister II 0 moves. Proof. Suppose T =[ x 1 ;k 1 ;k 2 ;:::;k n ;x 2 ]isanalternatingvirtualtwist.Weprovethe lemmabyinductionon n For n =1,wehave T =[ x 1 ;k 1 ;x 2 ].First,weprovethecasewhere T =[ v;k 1 ;v ]. Weperforma k 1 -swapandweobtain[ v;v;k 1 ].Sincetwovirtualcrossingsareadjacent,we canperformaReidemeistertypeII 0 movetoget T 0 =[ k 1 ].Usingtheformulagivenin 20

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thestatementofthelemma,wenoticethat ( T )=1,since n =1.Also, ( x i )=1forboth i =1 ; 2.SeeFigure2.7.Thus,bytheformula, T 0 =[ ; ; (1)(1)(1)(( 1) 1 k 1 )]=[ k 1 ].The othercasesaresimilarandweobtain [ ; ;k 1 ; ; ]=[ k 1 ] ; [ ; ;k 1 ;v ]=[ v; k 1 ] ; [ v;k 1 ; ; ]=[ v;k 1 ] : k 1 k 1 k 1 Figure2.7:Thecase n =1oftheReducingLemma Forthenextpartoftheinduction,thereare8casestotal,bu ttwosetsoffourcanbe checkedinasimilarmanner.Fortherstpart,weprovetheca seswhere x 2 = ; .Thus,the caseswecheckrstare: n iseven, x 1 = v x 2 = ; :[ v;k 1 ;:::;k n 2 ;k n 1 ;k n ; ; ] ; n iseven, x 1 = ; x 2 = ; :[ ; ;k 1 ;:::;k n 2 ;k n 1 ;k n ; ; ] ; n isodd, x 1 = v x 2 = ; :[ v;k 1 ;:::;k n 2 ;k n 1 ;k n ; ; ] ; n isodd, x 1 = ; x 2 = ; :[ ; ;k 1 ;:::;k n 2 ;k n 1 ;k n ; ; ] : Foranillustrationofthesecases,seeFigure2.8. 21

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Figure2.8:Thecasewhen n isodd, x 1 = v and x 2 = ; So,assumetheresultholdsfor n 2classicaltwists.Weusethenotation T =[ v;k 1 ;:::;k n 2 ; ; ] ks x ( T ) ; ( T )( 1) n 2 X i =1 ( 1) i k i ; toexpresstheconditionthat T ischangedto x ( T ) ; ( T )( 1) P n 2 i =1 ( 1) i k i bya k -swap move.Startingwithanalternatingvirtualtwist S =[ v;k 1 ;:::;k n 2 ;k n 1 ;k n ; ; ] ; weperforma( k n 1 )-swapandaReidemeisterII 0 movetoyield S 0 =[ v;k 1 ;:::;k n 2 k n 1 + k n ] : If n iseven,then S 0 stillhasanevennumberofclassicaltwists(similarilyif n isodd).This meansthat ( S )= ( S 0 ).Because S isalternating, S 0 isaswell,and k n 1 isthesame parityas k n 2 and k n .Nowlet k m = k n 2 k n 1 + k n .Noticethatthe k m twistisactually the( n 2)-thtwist.Thismeansthat S 0 = x ( T ) ; ( S )( 1) m X i =1 ( 1) i k i ; bytheinductionhypothesis. Nowfortheotherfourcases: n iseven, x 1 = v x 2 = v :[ v;k 1 ;:::;k n 2 ;k n 1 ;k n ;v ] ; n iseven, x 1 = ; x 2 = v :[ ; ;k 1 ;:::;k n 2 ;k n 1 ;k n ;v ] ; n isodd, x 1 = v x 2 = v :[ v;k 1 ;:::;k n 2 ;k n 1 ;k n ;v ] ; n isodd, x 1 = ; x 2 = v :[ ; ;k 1 ;:::;k n 2 ;k n 1 ;k n ;v ] : Foranillustrationofthesecases,seeFigure2.9. 22

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Figure2.9:Thecasewhen n isevenand x i = v Westartwith T =[ x ( T ) ;k 1 ;:::;k n 2 ;k n 1 ;k n ;v ] ; andweperforma k n -swapandaReidemeistertypeII 0 movetoobtain T 0 =[ x ( T ) ;k 1 ;:::;k n 2 ;k n 1 k n ] : Usingthepreviousresultfromtheinductionstep, T 0 ks !T 00 = x ( T ) ; ( T 0 )( 1) n X i =1 ( 1) i k i = x ( T ) ; ( T ) n X i =1 ( 1) i k i ; because ( T 0 )=( 1) ( T ).Thisistruebecauseif T hasanoddnumberofclassicaltwists, then T 0 hasanevennumberofclassicaltwists.Thisiswhatwewouldh aveobtainedifwe used x ( T ) ; ( T ) ( x 2 ) n X i =1 ( 1) i k i ontheoriginal T ,where ( x 2 )=1.Hence,theresultfollows. Lemma2.2.5 If D = T (2 ; [ k 1 ;k 2 ;:::;k n ;x 2 ]) isalternating,thenoneofthefollowingholds: a ) n isevenand x 2 = v b ) n isoddand x 2 = ; Proof. Since D isalternating,thetwist[ k 1 ;k 2 ;:::;k n ;x 2 ]mustbealternatingtoo,andits strandsmustconnectinsuchawaythat D isstillalternating.SeeFigure2.10.Nowsince D isatorusknot, x 1 = ; becauseif x 1 = v ,itcanbemovedtothebottomofthetorus knot.ByLemma2.2.5, D isalternatingifithasanevennumberofvirtualcrossings. Soif n iseven,thenthereare n 1virtualcrossingsbetweeneachclassicaltwistso x 2 = v .A similarargumentholdsforodd n 23

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K 1 K 2 Figure2.10:ThetwocasesfromLemma2.2.5: D 1 haseven n and x 2 = v ,while D 2 hasodd n and x 2 = ; Theorem2.2.6 Let T (2 ; [ k 1 ;k 2 ;:::;k n ;x 2 ]) bealternating, p beprime, T bethevirtualtwist in K ,and x 2 beeither v or ; ( accordingtowhether n isevenorodd,respectively ) .Then Col p ( T (2 ; [ k 1 ;k 2 ;:::;k n ;x 2 ]))=Col p T 2 ; ( T ) ( x 2 ) n X i =1 ( 1) i k i ; where ( T ) and ( x 2 ) areasintheReducingLemma. Proof. Let T bethevirtualtwistin D .ApplyingtheformulafromtheReducingLemma D 0 = x ( T ) ; ( T ) ( x 2 ) n X i =1 ( 1) i k i to D ,weseethat x 1 = ;) ( x 1 )= 1 : Itsucestocheckwhat x ( T )is.UsingLemma2.2.5, wehavetwocasestocheck.If n iseven,then ( T )= 1and x 2 = v ) ( x 2 )=1 : Thus, x ( T )= ; because ( T ) ( x 1 ) ( x 2 )=1 : Nowif n isodd,then ( T )=1and x 2 = ;) ( x 2 )= 1 : Thus, x ( T )= ; because ( T ) ( x 1 ) ( x 2 )=1 : SobyCorollary2.1.7,theresultis obtained. 24

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Theorem2.2.7 Thedeterminantofanalternatingknot D = T (2 ; [ k 1 ;k 2 ;:::;k n ;x 2 ]) ,where x 2 iseither v or ; ,isgivenby Det( D )= n X i =1 ( 1) i k i : Proof. UsingtheReducingLemmaandTheorem2.2.6,weobtainthekno t D ks D 1 = T 2 ; ( T ) ( x 2 ) n X i =1 ( 1) i k i : ByCorollary2.1.6,weknowthatDet( D )=Det( D 1 ).Hence,usingthefactthatDet( T (2 ;n ))= n (see[CP],forexample),weseethat Det( T (2 ; [ k 1 ;:::;k n ;x 2 ]))= n X i =1 ( 1) i k i ; where x 2 iseither v or ; Figure2.11:The2-stringvirtualbraiddiagram D = T (2 ; [ v; 2 ; 1 ; 2 ;v ]) Example2.2.8 Considerthe2-stringvirtualbraiddiagram D = T (2 ; [ v; 2 ; 1 ; 2 ;v ]).By inspectionofFigure2.11andLemma2.2.5, D isalternating.So,performingtwo2-swap moveson D andthenusingaReidemeisterII 0 movegivesus D 0 = T (2 ; [ 5]).Usingthe ReducingLemmagivesusthesame D 0 .Westartwith n beingodd, x 1 = v;x 2 = v ,sothen ( x 1 )=1 ; ( x 2 )=1 ; ( T )=1 ;x ( T )= ; ; where T =[ v; 2 ; 1 ; 2 ;v ].BytheReducingLemma,weget D 0 = T 2 ; ; ; (1)(1) 3 X i =1 ( 1) i k i = T (2 ; [( 2 1 2)])= T (2 ; [ 5]) : ThisshowsthattheReducingLemmaagreeswiththegure.ByT heorem2.2.6,weknow 25

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that Col p ( D )=Col p ( D 0 )=Col p ( T (2 ; [ 5])) : Furthermore,usingTheorem2.2.7,wealsoknowthat Det( D )=Det( D 0 )=5 : 2.2.2VirtualPretzelKnotDiagramsDenition2.2.9 Avirtualpretzelknotdiagramisdenotedby P ( T 1 ; T 2 ;:::; T n ) ; whereagiventwist T i =[ x i; 1 ;k i; 1 ;k i; 2 ;:::;k i;j ;x i; 2 ]isinsertedintothe i -thboxinFigure2.12, where n 3.SeeFigure2.13foranexample. 12 in Figure2.12:Thegeneralsetupofapretzelknot. Remark2.2.10 If n =1or n =2,wehavea T (2 ; T )torusknotdiagram,where T issome twist.Denition2.2.11 Avirtualtwist T is even ifithasanevennumberofvirtualcrossings ornovirtualcrossings,anditis odd ifithasanoddnumberofvirtualcrossings. Lemma2.2.12 If D = P ( T 1 ; T 2 ;:::; T n ) isanalternatingpretzellinkdiagramsuchthat T r =[ x r; 1 ;k r; 1 ;k r; 2 ;:::;k r;m ;x r; 2 ] ,where r =1 ;:::;n ,thenoneofthefollowingpropertieswill hold:a) T r =[ x r; 1 ;k r; 1 ;k r; 2 ;:::;k r;m ;x r; 2 ] forall r =1 ; 2 ;:::;n isanalternatingevenvirtualtwist 26

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Figure2.13:Thealternatingvirtualpretzelknot P ([ v; 1 ; 1 ;v ] ; [ 1 ; 3] ; [ v; 1 ; 1 ;v ]) : in D (seeFigure2.12),or b) T r isanalternatingoddvirtualtwistin D forall r =1 ; 2 ;:::;n (seeFigure2.12). Conversely,foranysequence ( m 1 ;m 2 ;:::;m n ) ofallevenoralloddnon-negativeintegers, thereisanalternatingdiagram P ( T 1 ; T 2 ;:::; T n ) suchthatthenumberofvirtualcrossingsof ( T 1 ; T 2 ;:::; T n ) are ( m 1 ;m 2 ;:::;m n ) Proof. Assume T r isanevenalternatingtwistin D T r +1 isthetwisttotherightof T r and connectedto T r .BytheReducingLemma,let T 0 r and T 0 r +1 beobtainedfrom T r and T r +1 respectively.Since T r iseven, T 0 r isaclassicaltwist,sothatif T r +1 iseven,then T 0 r +1 is classicalaswell.Nowweshowthat T r +1 cannotbeodd.Ifitwas,then T 0 r +1 wouldhave onevirtualcrossingatthetop.Withoutlossofgenerality, assumethat T 0 r hasallpositive classicalcrossings.Thentheupperrightstrandof T 0 r isanoverarcanditforces T 0 r +1 to haveallnegativeclassicalcrossingsinordertobealterna ting.Butlookingatthebottom crossingsof T 0 r and T 0 r +1 ,wendacontradiction.Indeed,thebottomstrandbetween T 0 r and T 0 r +1 willforce T 0 r +1 tohaveallpositiveclassicalcrossings.Thesitutationis depictedin Figure2.14.Conversely,ifallvirtualtwists T r areeven,thenthereisanalternatingdiagram 27

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byCorollary2.1.8. Figure2.14:Onenon-alternatingpretzelknotdiagramcons truction,andthetwoalternatingpretzel knotdiagrams Nowassume T r isanoddalternatingtwistin D .Weclaimthat T r +1 isoddaswell.We knowthat T r +1 cannotbeevenbythepreviousargument.Itneedstobeshownt hatthere arealternatingdiagramsifallof T r areodd.BytheReducingLemma, T 0 r canhaveone virtualcrossingatthetop.Withoutlossofgenerality,ass umethat T 0 r 'sclassicalcrossings areallpositive.Thenthetoprightstrandconnectstotheto pleftstrandof T 0 r +1 ,where T 0 r +1 isobtainedfrom T r +1 bytheReducingLemma.Thenthatstrandmustbeanoverarc, sotheclassicalcrossingsin T 0 r +1 areallpositiveaswell.Now,thebottomrightstrandof T 0 r isanunderarcandin T 0 r +1 ,itisanoverarc,asexpected.SeeFigure2.14. Theorem2.2.13 If D = P ( T 1 ; T 2 ;:::; T n ) isanalternatingvirtualpretzelknotdiagram, where T ` =[ x `; 1 ;k `; 1 ;k `; 2 ;:::;k `;m ;x `; 2 ] isanalternatingevenvirtualtwistfor ` =1 ; 2 ;:::;n p isprime,and x i;j iseither v or ; ,then Col p ( D )=Col p ( P ([ a 1 ] ; [ a 2 ] ;:::; [ a n ])) ; where [ a ` ]= ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i ; for ` =1 ; 2 ;:::;n ,andwhere and aredened asintheReducingLemma.Proof. ByLemma2.2.12,weknowthateachtwist T ` from D haseitheranevennumberof virtualcrossings,ornovirtualcrossings.Ineithercase, weusetheReducingLemmatosee 28

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that T ` ks ( T ` ) ( x `; 2 ) n X i =1 ( 1) i k `;i : Letting[ a ` ]= ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i givesusthedesiredresultbyCorollary2.1.5. Corollary2.2.14 Thedeterminantofanalternatingpretzelknotdiagram D = P ( T 1 ; T 2 ;:::; T n ) ; where T ` =[ x `; 1 ;k `; 1 ;k `; 2 ;:::;k `;m ;x `; 2 ] isanalternatingevenvirtualtwist for ` =1 ; 2 ;:::;n ,and x i;j iseither v or ; ,isgivenby Det( D )= n X j =1 a 1 a 2 a j 1 a j +1 a n ; where a ` = ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i ,for ` =1 ; 2 ;:::;n Proof. Fromtheprevioustheorem, D ks P ([ a 1 ] ; [ a 2 ] ;:::; [ a n ]) ; where a ` = ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i ; for ` =1 ; 2 ;:::;n .ByConway'sformula[CJ],any pretzelknothasthedeterminantgiveninthestatementofth etheorem,therefore D doesas well. Example2.2.15 Forthisexample,weshallinvestigatethevirtualevenpret zelknotdiagram D = P ([ v; 2 ; 2] ; [ v; 1 ; 3] ; [ 6]).ByinspectionofFigure2.15andLemma2.2.12,the diagram D isalternating.Usinga2-swapanda1-swapon D ,thenusingtwoReidemeister II 0 moves,weget D 0 = P ([ 4] ; [ 4] ; [ 6]) : OurgoalistoshowthattheReducingLemmagivesthesameresu ltfor D = P ( T 1 ; T 2 ; T 3 ), where T 1 =[ v; 2 ; 2] ; T 2 =[ v; 1 ; 3] ; and T 3 =[ 6].Wecalculateonly T 1 andleave T 2 to thereader.For T 1 =[ v; 2 ; 2],wehavethefollowingdata: n iseven, x 1 = v; and x 2 = ; Thenweobtain ( x 1 )=1 ; ( x 2 )= 1 ; ( T 1 )= 1 ;x ( T 1 )= ; : 29

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Figure2.15:Thevirtualevenpretzelknotdiagram D = P ([ v; 2 ; 2] ; [ v; 1 ; 3] ; [ 6]) Thus, T 0 1 = ; ; ( 1)( 1)( 2 2) =[ 4] : Also, T 2 =[ 4]byasimilarcalculation.Thus, D 1 ks D 0 1 = P ([ 4] ; [ 4] ; [ 6]) ; whichwasexactlywhatwasobtainedbyanalyzingthediagram .So,byTheorem2.2.13, Col p ( D )=Col p ( D 0 )=Col p ( P ([ 4] ; [ 4] ; [ 6])) : AquickapplicationofCorllary2.2.14showsthat Det( D )=Det( D 0 )=( 4)( 6)+( 4)( 6)+( 4)( 4)=64 : Example2.2.16 Forthenextexample,weconsidertheoddvirtualpretzelkno tdiagram D = P ( T 1 ; T 2 ; T 3 ),where T 1 =[ v; 5] ; T 2 =[ v; 1 ; 1 ; 1] ; and T 3 =[ v; 5].Noticethatby inspectionofFigure2.16andLemma2.2.12,thediagram D isalternating.Performinga ( 1)-swapinthemiddleof T 2 andthenusingaReidemeisterII 0 move,weget D 0 = P ([ v; 4] ; [ v; 3] ; [ v; 4]) : 30

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Figure2.16:Theoddvirtualpretzelknotdiagram D = P ([ v; 5] ; [ v; 1 ; 1 ; 1] ; [ v; 5]) NowweshowthesameresultusingtheformulagivenintheRedu cingLemma.Since T 1 and T 3 arealreadyreduced,weonlyneedtolookat T 2 =[ v; 1 ; 1 ; 1].Thedatawehavethenis n isodd, x 1 = v; and x 2 = ; .Thus, ( x 1 )=1 ; ( x 2 )= 1 ; ( T 2 )=1 ;x ( T 2 )= v; sonallywehave T 2 ks !T 0 2 =[ v; (1)( 1)( 1 1 1)]=[ v; 3] ; whichiswhatweobtainedbefore.Now,wecomeuponasituatio nthatwecannothandle asofyetandneedsmoreresearch:whatisCol p ( P ([ v; 4] ; [ v; 3] ; [ v; 4]))?Ingeneral,whatis Col p ( E ),where E isanyoddvirtualalternatingpretzeldiagram?Similarly, whatisthe determinantoftheseknots?Obviously,theseareasneedmor eresearch. 2.2.3Virtual2-BridgeKnotDiagramsForournalfamilyofalternatingvirtualknotdiagrams,we turnourattentiontovirtualized 2-bridgeknotdiagrams.Denition2.2.17 Avirtual2-bridgelinkdiagramisdenotedby B ( T 1 ; T 2 ;:::; T n ) ; 31

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n i 2 1 Figure2.17:Thegeneral2-bridgevirtualknotdiagram whereagiventwist T i =[ x i; 1 ;k i; 1 ;k i; 2 ;:::;k i;j ;x i; 2 ]isinsertedintothe i -thboxinFigure2.17. SeeFigure2.18foranexample.Inthecasetherearenovirtua lcrossingsinany T i ,thenthe classical2-bridgeknotwillstillbedenotedby B ( T 1 ; T 2 ;:::; T n ),regardlessoftheconventional notations. k 1 ; 1 k 1 ; 2 k 2 ; 1 k 3 ; 1 Figure2.18:The2-bridgevirtualknotdiagram B ([ v; 2 ; 2 ;v ] ; [ v; 1 ;v ] ; [ 1 ;v ]) Lemma2.2.18 Ifavirtual2-bridgediagram D isalternating,thenallvirtualtwistsin D areevenandalternating.Conversely,foranysequence ( m 1 ;m 2 ;:::;m n ) ofallevennonnegativeintegers,thereisanalternatingdiagram B ( T 1 ; T 2 ;:::; T n ) suchthatthenumberof virtualcrossingsof ( T 1 ; T 2 ;:::; T n ) are ( m 1 ;m 2 ;:::;m n ) Proof. Assume D = B ( T 1 ; T 2 ;:::; T n )isalternating.UsingtheReducingLemma,wereduce alltwists T r to T 0 r sothat,foreachtwist T 0 r ,thereiseitheronevirtualcrossingornovirtual crossings.Notethatwecanplacethelonevirtualcrossinga nywhereinagiventwistthatwe choosebyusinga k -swapmove.Weclaimthatthereareonlyevenalternatingtwi stsin D Thiswouldmeanthatafterreducing,therewouldbenovirtua lcrossingsinanytwists.There arethreecasestocheckforthemiddletwists,asdepictedin Figure2.19.Furthermore,there aretwocaseseachforthecasewhen r = n andforthecasewhen r =1,asinFigure2.20. NotethatFigure2.20isarenementofFigure2.17. Fortherstcase,assumethat T 0 r +1 and T 0 r 1 arebothclassicaltwistsand T 0 r ispositive withonevirtualcrossingtotherightofitsclassicalcross ings.Thenthestrandbetween T 0 r 32

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c b a Figure2.19:Thethreecasestocheck and T 0 r +1 forces T 0 r +1 tobeapositivetwist.Thestrandbetween T 0 r and T 0 r 1 makes T 0 r 1 a negativetwist,sothestrandbetween T 0 r 1 and T 0 r +1 isnon-alternating.SeeFigure2.19,part a .TheothertwocasesaresimilarandaredepictedinFigure2. 19,parts b and c Thecasewhere r = n issimilartothemiddlecases.Weexaminethesubcaseof T n 1 beingclassical,and T n beingpositive.Since T n ispositive,thestrandbetween T n 1 and T n forces T n 1 tobeanegativetwist.However,theotherstrandbetweenthe mwouldforce T n 1 tobeapositivetwist.Thus,thissetupisnon-alternatinga ndtheothercasesaresimilar. SeeFigure2.20.Hence,therstpartoftheproofisdone. Theconversereliesonthefactthatwecanhaveanalternatin gclassical2-bridgeknot diagram,addpairsofvirtualcrossingstoanytwist,andapp lytheReducingLemmain reversetogetanalternatingvirtual2-bridgeknotdiagram .Thisispossiblebecauseof Corollary2.1.8. Theorem2.2.19 If D = B ( T 1 ; T 2 ;:::; T n ) isanalternatingvirtual2-bridgelinkdiagram, where T ` =[ x `; 1 ;k `; 1 ;k `; 2 ;:::;k `;m ;x `; 2 ] isanalternatingevenvirtualtwistfor ` =1 ; 2 ;:::;n 33

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r = n Figure2.20:Thecaseswhen r = n p isprime,and x i;j iseither v or ; ,then Col p ( D )=Col p ( B ([ a 1 ] ; [ a 2 ] ;:::; [ a n ])) ; where [ a ` ]= ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i ; for ` =1 ; 2 ;:::;n ,andwhere and aredened asintheReducingLemma.Proof. ByLemma2.2.18,weknowthateachtwist T ` from D hasanevennumberofvirtual crossings.Thus,aquickapplicationoftheReducingLemmas howsthat T ` ks ( T ` ) ( x `; 2 ) n X i =1 ( 1) i k `;i : Letting[ a ` ]= ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i givesusthedesiredresult. Theorem2.2.20 Thedeterminantofanalternating2-bridgelinkdiagram D = B ( T 1 ; T 2 ;:::; T n ) where T ` =[ x `; 1 ;k `; 1 ;k `; 2 ;:::;k `;m ;x `; 2 ] isanalternatingeventwistfor ` =1 ; 2 ;:::;n ,and x i;j iseither v or ; ,isgivenby Det( D )= a n ( a 4 ( a 3 ( a 2 a 1 +1)+2)+3)+ )+ a n 1 ; 34

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where [ a ` ]= ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i : Proof. Fromtheprevioustheorem, D ks B ([ a 1 ] ; [ a 2 ] ;:::; [ a n ]) ; where a ` = ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i ; for ` =1 ; 2 ;:::;n .ByConway'sformula[CJ],any 2-bridgeknothasthedeterminantgiveninthestatementoft hetheorem,therefore D does aswell. Example2.2.21 Inthislastexample,weconsiderthevirtual2-bridgeknotd iagram D = B ( T 1 ; T 2 ; T 3 ),where T 1 =[ v; 2 ; 2] ; T 2 =[2 ; 2 ;v ] ; and T 3 =[ 3].ByLemma2.2.18, thediagram D isalternating.Againperformingtwok-swapson D andthenusingtwo ReidemeisterII 0 movesyields D 0 = B ([ 4] ; [4] ; [ 3]) : WewanttoshowthattheReducingLemmagivesthesame D 0 from D = B ( T 1 ; T 2 ; T 3 ).We calculateonly T 2 andleave T 1 tothereader.Startingwitheven n x 1 = ; ; and x 2 = v ,we seethat ( x 1 )= 1 ; ( x 2 )=1 ; ( T 2 )= 1 ;x ( T 2 )= ; : SobytheReducingLemma, T 0 2 = ; ; ( 1)(1) 2 X i =1 ( 1) i k i =[ (( 1)2+( 2))]=[4] : Also, T 1 =[ 4]byasimilarcalculation.ThustheReducingLemmaagreesw ithourmanipulationsofthediagram.ByTheorem2.2.19,weknowthat Col p ( D )=Col p ( D 0 )=Col p ( B ([ 4] ; [4] ; [ 3])) : Furthermore,usingTheorem2.2.20,wealsoknowthat Det( D )=Det( D 0 )=( 3)( 16+1)+2=47 : 35

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2.3Kauman-HararyConjectureforAlternatingVirtualKno tsandSwap Moves Forconvenience,werecalltheKauman-Hararyconjectures here. Conjecture2.3.1(Kauman-HararyConjecture1) Let D beareducedalternating knotdiagramwithaprimedeterminant p .TheneverynontrivialFox's p -coloringof D assignsdierentcolorstodierentarcsof D Conjecture2.3.2(Kauman-HararyConjecture2) Let D beareducedalternating virtualknotdiagramwithaprimedeterminant p .TheneverynontrivialFox's p -coloring of D assignsdierentcolorstodierentarcsof D Theorem2.3.3 Let D 2 beobtainedfrom D 1 byanitesequenceof k -swapmoves.Then theKauman-HararyConjecture(1.2.13)istrueforanalter natingvirtualknotdiagram D 1 itheKauman-HararyConjectureistrueforanalternating knotdiagram D 2 Proof. ByCorollary2.1.6andTheorem2.1.4,thearcsin D 1 areonlyrelabeledin D 2 after the k -swap.Theresultfollows. Corollary2.3.4 TheKauman-Hararyconjectureistrueforanalternatingkn otdiagram D = T (2 ; [ k 1 ;k 2 ;:::;k n ;x 2 ]) ,where x 2 iseither v or ; Proof. ByTheorem2.2.6, D ks D 0 = T 2 ; ( T ) ( x 2 ) n X i =1 ( 1) i k i ; where T =[ k 1 ;k 2 ;:::;k n ;x 2 ] : ByCorollary2.1.8andTheorem2.2.6, D 0 isanalternating classicaldiagram.In[KH],itwasproventhatalternating T (2 ; [ n ])classicalknotssatisfy theKauman-Hararyconjecture.Therefore,alternatingvi rtual2-stringbraidssatisfythe Kauman-Hararyconjecture,byTheorem2.3.3. Corollary2.3.5 TheKauman-Hararyconjectureistrueforalternatingvirt ualpretzelknot diagrams D = P ( T 1 ; T 2 ;:::; T n ) ; where T ` =[ x `; 1 ;k `; 1 ;k `; 2 ;:::;k `;m ;x `; 2 ] isanalternatingeven virtualtwistfor ` =1 ; 2 ;:::;n ,and x i;j iseither v or ; 36

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Proof. ByTheorem2.2.13, D ks D 0 = P ([ a 1 ] ; [ a 2 ] ;:::; [ a n ]) ; where[ a ` ]= ( T ` ) ( x `; 2 ) P ni =1 ( 1) i k `;i .Since D isanevenalternatingvirtualdiagram, D 0 isaclassicalalternatingdiagram(Corollary2.1.8andThe orem2.2.13).In[AM],it wasshownthatalternatingpretzelknotssatisfytheKauma n-Hararyconjecture.Thus, Theorem2.3.3statesthat D satisesthevirtualKauman-Hararyconjecture. Remark2.3.6 Sincethereareanoddnumberofvirtualcrossingsineachvir tualtwistof theoddalternatingpretzeldiagramsfromLemma2.2.12,wec anusetheReducingLemma togetonecrossingineachtwist.Furtherresultsinthatare awouldneedmoreresearchin ordertoproveordisprovetheKauman-Hararyconjecture.Corollary2.3.7 TheKauman-Hararyconjectureistrueforalternatingvirt ual2-bridge linkdiagrams D = B ( T 1 ; T 2 ;:::; T n ) ; where T ` =[ x `; 1 ;k `; 1 ;k `; 2 ;:::;k `;m ;x `; 2 ] isanalternating evenvirtualtwistfor ` =1 ; 2 ;:::;n ,and x i;j iseither v or ; Proof. ThisissimilartoCorollary2.3.5.Notethat2-bridgeknots satisfytheKaumanHararyconjecture,asprovedin[KL]. Remark2.3.8 Itwasprovedin[KL]thattheKauman-Hararyconjecturehol dsforany 2-bridgeknotwithoutrestrictionsonthedeterminantofth eknot.However,thestatement oftheconjectureneedstobechangedfrom\everynontrivial n -coloring"to\thereexistsa n -coloring,"where n isthedeterminantofa2-bridgeknot K 37

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3Conclusion Afterdeningthe k -swapmove,weprovedthattheyinduceabijectionbetweenco lorings beforeandafterthemoveinordertoshowthattwoknotsarere latedbyasequenceof k -swap moves,thentheircoloringsarethesame.Thisalsoallowedu stoprovethatdeterminantsdo notchangeafter k -swapmovesareperformed.Furthermore,weprovedthatthe k -swapdoes notchangethefundamentalinvolutoryquandles,anditdoes notchangeJonespolynomials either. Followingtheproofsconcerningtheinvarianceofthecolor ingsanddeterminantsofthe k -swap,weshowedwhatthealternatingconditionswereforea chofthethreevirtualknot diagramsdiscussed.Onlyoneofthefamiliescanbevirtuala fterreducingviatheReducing Lemma:thevirtualpretzelknotdiagrams.Next,weshowedwh ateachofthealternating virtualknotdiagrams'(exceptfortheoddalternatingvirt ualpretzelknotdiagrams)colorings anddeterminantswereviatheReducingLemma.Finally,wepr ovedthattheKaumanHararyconjectureholdsforalloftheabovealternatingvir tualknotdiagrams,exceptforthe oddalternatingvirtualpretzelknotdiagrams.Inthatcase ,itisnotcleartouswhatthe coloringsarefortheseknotdiagramsorwhattheirdetermin antsaresowecannotconjecture whetherornottheysatisfytheKauman-Hararyconjecture. Therearemanyquestionsregardingthe k -swapandfutureworkwhichcouldbedone. Doesthe k -swapholdforAlexandercoloringsofvirtualknots?Ifnot, canthe k -swapbe generalizedtoincludeAlexandercolorings?Arethereanyo thercoloringsinwhichthis movehold?Thereisalsonoreasontorestrictthesemovestoc oloringsordeterminants,so arethereotherinvariantsthatthe k -swapcouldbeusedtoexamine?Forinstance,what happenstotheJonespolynomial(orotherpolynomials)afte rthe k -swapisperformed?In Kauman'sworkmentionedabove,itwasshownthatthe1-swap isaninvariantforJones polynomialssoisthistrueingeneral?Wewouldexpectthatt he k -swapdoessomething interestingtotheAlexanderpolynomialortheConwaypolyn omial,butagain,moreresearch isneeded. 38

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