boundary slopes of 2-bridge links determine the crossing number

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1、BoundarySlopesof2-BridgeLinksDeterminetheCrossingNumberJimHostePitzerCollegePatrickD.ShanahanLoyolaMarymountUniversityFebruary2,2008AbstractAdiagonalsurfaceinalinkexteriorMisaproperlyembedded,incompressible,boundaryincompressiblesurfacewhichfurthermorehasthesamenumberofboundarycompo

2、nentsandsameslopeoneachcomponentof∂M.Wederiveaformulafortheboundaryslopeofadiagonalsurfaceintheexteriorofa2-bridgelinkwhichisanalogoustotheformulafortheboundaryslopeofa2-bridgeknotfoundbyHatcherandThurston.Usingthisformulaweshowthatthediameterofa2-bridgelink,thatis,thedifferencebetwe

3、enthesmallestandlargestfiniteslopesofdiagonalsurfaces,isequaltothecrossingnumber.1IntroductionLetcr(K)denotetheminimalcrossingnumberofaknotKinthe3-sphere,andletD(K)bethediameterofthesetoffiniteboundaryslopesoftheknot.ItwasconjecturedbyIchiharathatarXiv:math/0603206v1[math.GT]9Mar20062

4、cr(K)≥D(K)(1)forallknotsK.Thisconjecturehasbeenprovenfor2-bridgeknotsbyMattman,Maybrun,andRobinson[10]andforMontesinosknotswiththreeormoretanglesbyIchiharaandMizushima[8].Moreover,foralternatingknots,thedifferencebetweentheboundaryslopesofthetwocheckerboardsurfaces(inthereducedaltern

5、atingdiagram)isalwaystwicethecrossingnumber.Hence,2cr(K)=D(K)(2)forallalternatingMontesinosknots.Neither[10]nor[8],however,discusspossibleextensionsofstatements(1)or(2)tolinkexteriors.Inthispaperwedothisbyconsideringarestrictedsetofessentialsurfacesinthelinkexteriorwhichwecall“diago

6、nal”surfaces.Ourmainresult,Theorem2,providesaformulafortheboundaryslopeofadiagonalsurfaceofa2-bridgelinkL1whichisanalogoustotheformulagivenbyHatcherandThurstonfortheboundaryslopeofa2-bridgeknot.Asanapplicationofthisformula,weproveTheorem11,thatcr(L)=D∆(L)whereLisa2-bridgelinkandD∆(L

7、)isthediameterofthefiniteslopesofdiagonalsurfaces.Inaddition,ifLisanon-split,ncomponent,alternatinglink,andifbothcheckerboardsurfacesarediagonal,thenweshowinProposition12that2cr(L)≤D(L).ThistogetherwithTheorem11n∆suggeststhat2cr(L)≥D∆(L),nwithequalityinthecaseofalternatinglinks,isapo

8、ssiblegeneralizatio