On a move reducing the genus of a knot diagram

0.16 MB·English

by  Kenji Daikoku

ON A MOVE REDUCING THE GENUS OF A KNOT DIAGRAM
KENJIDAIKOKU,KEIICHISAKAI,ANDMASAMICHITAKASE
Abstract. For a knot diagram we introduce an operation which does not increase the
genusofthediagramanddoesnotchangeitsrepresentingknottype. Wealsodescribea
conditionforthisoperationtocertainlydecreasethegenus.Theproofinvolvesthestudyof
2 arelationbetweenthegenusofavirtualknotdiagramandthegenusofaknotoiddiagram,
1 theformerofwhichhasbeenintroducedbyStoimenow,TchernovandVdovina,andthe
0 latterbyTuraevrecently.OuroperationhasasimpleinterpretationintermsofGausscodes
2 andhencecaneasilybecomputer-implemented.

n
a
J

4 1. Introduction
2
Seifert[3]gaveanalgorithmtoconstructfromadiagram DofaknotK anorientable

]
T surface bounded by the knot (see Definition 2.1). We call the surface constructed by
G Seifert’s algorithm the canonical Seifert surface for D and its genus the genus g(D) of
D. Thecanonicalgenusg (K)ofaknotK isdefinedtobetheminimalgenusofallpos-
. c
h siblediagrams. Itisanimportantknotinvariantandextensivelystudied(seeforexample
t
a [4]). By definition the canonical genus of a knot K gives an upper bound for the genus
m g(K)ofK,thatistheminimumofgeneraofallpossibleSeifertsurfacesforK.
[ In thispaper, we introducean operation,called the bridge-replacingmove, fora knot
diagramwhichdoesnotchangeitsrepresentingknottypeanddoesnotincreasethegenus
2 ofthediagram(seeDefinition3.9andTheorem3.11).Anecessaryandsufficientcondition
v
fortheoperationtoactuallydecreasethegenusofthediagramisgiveninProposition3.14.
5
1 OurmoveisderivedfromTuraev’sideausingthenotionknotoid[6].Aknotoiddiagram
2 D◦ is a diagram which differs from usual knot diagrams in that the underlying curve is
3 an immersed interval rather than an immersed circle (see Definition 2.15). Turaev has
.
1 constructed in [6, §2.5] the canonical surface for a given knotoid diagram by using an
1 analogousprocedureto the usualSeifertalgorithm(see Definition 2.16). The genusofa
1 knotoiddiagramisdefinedtobethegenusofitscanonicalsurface.In[6,§2.5]ithasbeen
1
suggestedthatthestudyofknotoiddiagramscanbeusedtoobtainagoodestimateforthe
:
v genusofaknot.For,ifwehaveadiagramDofanon-alternatingknotK whichcontainsa
i
X consecutivesequence Bofk over-orunder-crossingsegments(k ≥ 2),thenbyremoving
B from D we can obtain the knotoid D◦ with genus g(D◦) ≥ g(K). In this context, we
r B B
a willproveinProposition3.5thattheinequalityg(D)≥g(D◦)alwaysholds,whichhasnot
B
been explicitly provenin [6, §2.5]. Moreover,the bridge-replacingmovefor (D,B)(see
Definition3.9)provestoproducetheknotdiagramD˜ withthepropertyg(D˜ ) = g(D◦).
B B B
Thisimpliesg(D)≥ g(D˜ ),thatis,thebridge-replacingmoveisindeedusefultoobtaina
B
betterestimateforg (K)thantheonegivenbytheinitialdiagramD.
c
Fortheproof,wewillstudyin§3arelationbetweenthecanonicalsurfaceassociatedto
aknotoiddiagramandthesurfaceassociatedtoacertainvirtualknotdiagram,whichhas
beenintroducedbyStoimenow,TchernovandVdovina[5]throughGaussdiagrams.
In§4,wediscussanexamplewhichsuggeststhatthebridge-replacingmoveisactually
usefulforthedeterminationofthegenusofaknot.
2000MathematicsSubjectClassification. 57M25;68R15.
Keywordsandphrases. knot,canonicalgenus,bridge-replacingmove,knotoid,virtualknot,Gaussdiagram,
Gausscode.
1
2 KENJIDAIKOKU,KEIICHISAKAI,ANDMASAMICHITAKASE
Figure1. Smoothingofthecrossings
Figure2. Half-twistedbandsum
2. Preliminaries
2.1. Knots. Herewerecallsomewell-knownnotionsconcerningclassicalknotsinR3.
ASeifertsurfaceforaknotKisaconnected,orientedsurfaceΣembeddedinR3whose
boundary∂ΣcoincideswiththeknotK.RecallSeifert’salgorithmwhichproducesaSeifert
surfaceΣ foraknotKfromagivendiagramD= D ofK:
D K
Definition2.1(Seifert’salgorithm[3]). GivenadiagramD= D ofaknotK,thecanon-
K
icalSeifertsurfaceΣ forK isthesurfaceobtainedinthefollowingway.
D
(i) DrawDinR2×{0}andorientDinanarbitraryway.
(ii) Smoothallthecrossingsof DasinFigure1toobtainadisjointunionofembedded
circlesintheplane(calledSeifertcircles).
(iii) FilltheSeifertcircleswiththedisjointdisksinR3.
(iv) Takethehalf-twistedbandsumsalongtheoriginalcrossings(Figure2)ofD.
Thefollowingisobvious.
Lemma2.2. GivenaknotdiagramD = D withncrossings,thegenusofthecanonical
K
SeifertsurfaceΣ ofDisgivenbytheformula
D
n−s +1
(1) g(Σ )= D ,
D
2
where s isthenumberoftheSeifertcirclesobtainedinthestep(ii)inDefinition2.1.
D
AccordingtoStoimenow,TchernovandVdovina[5],fortheGaussdiagramofaknot
diagramDwecanconstructasurface,whichturnsouttobehomeomorphictothecanoni-
calSeifertsurfaceΣ ,asexplainedbelow.
D
Definition 2.3. A Gauss diagram is an orientedcircle equippedwith some numbern of
signed,orientedchordseachofwhichconnectsdistincttwopointsonthecircle(allthe2n
pointsaredistinctwitheachother).
ToeachknotdiagramD = D withncrossings,wecanassigntheGaussdiagramG
K D
withnchordsasfollows:
(i) ConnectthepreimagesofeachcrossingofDbyachord.
(ii) Choosetheorientationofeachchordfromtheoverpassbranchtotheunderpassone.
(iii) Givetoeachchordthesign+or−dependingonwhetherthecorrespondingcrossing
ispositive(theleftcrossinginFigure1)ornegative,respectively.
Example2.4. Figure3showsaknotdiagram D ofthetrefoilknot3 anditsGaussdia-
1
gram.Theendpoint¯iofGcorrespondstotheover-arcofthecrossingiofD.
Definition2.5. LetG beaGaussdiagram. ThecanonicalsurfaceΣ ofG isthesurface
G
(withexactlyoneboundarycomponent)obtainedinthefollowingway.
(i) ConsideranannulusA=S1×[0,1].
(ii) TakeabandB =[0,1]×[0,1]foreachchordcofG.
c
ONAMOVEREDUCINGTHEGENUSOFAKNOTDIAGRAM 3
1 (cid:22)3 2
(cid:0)
2
1 (cid:0) (cid:22)1
3
(cid:0)
(cid:22)2 3
Figure3. AknotdiagramDof3 anditsGaussdiagramG
1
Figure4. Σ′ fortheGaussdiagramGofFigure3
G
(iii) Foreachchordcandǫ ∈ {0,1},glue{ǫ}×[0,1] ⊂ B to[x −δ,x +δ]×{0} ⊂ ∂A
c ǫ ǫ
(where x ,x ∈ S1 = R/Z are the endpointsof c, and δ > 0 is a sufficiently small
0 1
number)sothatΣ′ := A∪(cid:0)S B (cid:1)isoriented.
G c c
(iv) GluealltheboundarycomponentsofΣ′ butS1×{1} ⊂ ∂Awithdiskstoobtainan
G
orientedsurfaceΣ withoneboundarycomponent.
G
Example 2.6. Figure 4 shows Σ′ for the Gauss diagramG of Figure 3. Attaching two
G
disks along the boundary components of Σ′ other than S1 × {1} (the outermost circle),
G
weobtainthecanonicalsurfaceΣ ofG,whichishomeomorphictothecanonicalSeifert
G
surfaceofDshowninFigure3(asproveninProposition2.8).
Remark2.7. ThecanonicalsurfaceforaGaussdiagramconstructedbyStoimenow,Tch-
ernovandVdovinain[5]isaclosedsurface. Removingadisk(attachedalongS1×{1})
fromtheirsurface,weobtainthesurfacedefinedinDefinition2.5.
Proposition2.8([5, Theorem2.5]). Foraknotdiagram D,thesurfacesΣ andΣ are
D GD
homeomorphictoeachother.
Fortheproof,weintroducethenotionofacycleinaGaussdiagramwhichcorresponds
toaSeifertcircleinSeifert’salgorithm.
Definition2.9. LetGbeaGaussdiagram.AcycleinGisthecycleobtainedbyrepeating
thefollowingsteps: startingfromsomeendpointxofachordc,
(i) gototheotherendpointofcalongc,
(ii) gotothenextendpointyalongthecirclewithrespecttoitsorientation,
(iii) fromy,repeat(i)and(ii)aboveuntilcomingbackto xinthestep(ii).
We will denote a cycle as a cyclic sequence {i ,i ,...,i } of endpointswhich appear in
1 2 2k
theabovesteps(i)and(ii).
Example 2.10. The Gauss diagram G of Figure 3 has two cycles {1¯,1,2¯,2,3¯,3} and
{1,1¯,2,2¯,3,3¯}.
Lemma2.11. GivenaGaussdiagramGwithnchords,thegenusofthecanonicalsurface
Σ ofGisgivenbytheformula
G
n−s +1
(2) g(Σ )= G ,
G
2
where s isthenumberofthecyclesinG.
G
4 KENJIDAIKOKU,KEIICHISAKAI,ANDMASAMICHITAKASE
Proof. This formula is a consequence of the following facts: a cycle corresponds to a
boundarycomponentofΣ′ towhichwegluetheboundaryofadiskinthefinalstep(iv)in
G
Definition2.5. Thusthenumberofdisksweglueinthestep(iv)inDefinition2.5isequal
tothenumberofcyclesinG. (cid:3)
ProofofProposition2.8. Itiseasytoobservethat
• thenumberofthecrossingsofDandthatofthechordsinG arethesame,
D
• s = s (underthenotationsinLemmas2.2,2.11).
D GD
Thesefactsandformulas(1),(2)implythatthegenusofΣ isequaltothatofΣ . This
D GD
completestheproof,sincebothΣ andΣ haveexactlyoneboundarycomponent. (cid:3)
D GD
Remark2.12. Theformula(1)suggeststhat,toconstructaknotdiagramofsmallergenus
fromagivenknotdiagram,weneedanoperationwhichdecreasesthenumberofthecross-
ingsorincreasesthenumberofSeifertcircles. Theformula(2)givesasimilarsuggestion
forGaussdiagrams.
2.2. VirtualknotsandtheGaussdiagrams.
Definition2.13. AvirtualknotdiagramisagenericimmersionS1 #R2withonlytrans-
verse doublepointsas its singularities, some of which are endowedwith over-or under-
crossingdatabutothersarenot(see Figure6). A crossingendowedwithover-orunder-
crossinginformationiscalledarealcrossing,andonewithoutsuchinformationiscalled
avirtualcrossing.
Seifert’salgorithmcannotbe appliedto a virtualknotdiagramasit is. Butthe Gauss
diagramG ofavirtualknotdiagramD•canstillbedefinedinthesamewayasexplained
D•
afterDefinition2.3,exceptthatnochordsareassignedtovirtualcrossings(see[5,§2.1]).
ThereforeinviewofProposition2.8,thefollowingdefinitionwouldbenatural.
Definition2.14([5]). LetD• beavirtualknotdiagram. ThecanonicalsurfaceΣ ofD•
D•
is defined to be the canonicalsurface Σ of the Gauss diagramG of D•. The genus
GD• D•
g(D•)ofD• isdefinedtobethegenusofthecanonicalsurfaceΣ ofD•.
D•
2.3. Turaev’sKnotoids.
Definition2.15([6]). A knotoiddiagramis a genericimmersion f : [0,1] # R2 which
hasonlytransversedoublepoints(endowedwithover-orunder-crossingdata)asitssin-
gularities. Theendpoints f(0)and f(1)arecalledrespectivelythelegandtheheadofthe
knotoiddiagram.
SeethecentraldiagraminFigure6foranexampleofaknotoiddiagram.Inthediagram,
PisthelegandQisthehead. In[6]asurfaceisproducedfromaknotoiddiagramD◦,in
ananalogouswaytoSeifertalgorithmforknotdiagrams.
Definition2.16([6]). Thecanonicalsurface Σ ofa knotoiddiagram D◦ is the surface
D◦
obtainedasfollows.First,drawD◦ inR2×{0}⊂R3. Then:
(i) OrientD◦ inanarbitraryway.
(ii) SmoothallthecrossingsofD◦ asinFigure1toobtainadisjointunionofembedded
circlesintheplane(alsocalledSeifertcircles)andanembeddedinterval(denotedby
J). WecallJ theSeifertinterval. Jhasthesameendpointsx,yasD◦.
(iii) Fill the Seifertcircleswith the disjointdiskslyingaboveR2 ×{0}, andtake a band
J×[0,1]lyingbelowR2×{0}andmeetingR2×{0}alongJ×{0}≈ J.
(iv) Take thehalf-twistedbandsumsalongtheoriginalcrossings(Figure2)of D◦. The
resultingsurfaceΣ isthecanonicalsurfaceofD◦.
D◦
TheboundaryofΣ istheunionofD◦with J×{1}∪{x,y}×[0,1].
D◦
Definition2.17. The genusg(D◦) of a knotoiddiagram D◦ isdefinedto be the genusof
thecanonicalsurfaceΣ ofD◦.
D◦
ONAMOVEREDUCINGTHEGENUSOFAKNOTDIAGRAM 5
3. Thebridge-replacingmove
Themainoperationofthispaper,thebridge-replacingmove,isintroducedinthissec-
tion. It is clear by definition that this operation does not change the representing knot
types. TheobservationgiveninRemark2.12isakeyinprovingthatthisoperationdoes
notincrease,andsometimescertainlydecreases,thegenusofaknotdiagram.
Definition3.1. Anover-bridge(resp.under-bridge)oflengthk ofaknotdiagramDisa
consecutivesequence B ofk over-crossing(resp.under-crossing)segmentsof D (see the
leftmostdiagraminFigure6).
Notation3.2. LetDbeadiagramofaknotKandBbeanover-orunder-bridgeofD. We
denoteby D◦ the knotoiddiagramobtainedbyremoving(theinteriorof) Bfrom D. We
B
denotebyD• thevirtualknotdiagramobtainedfromDbyturningeachcrossingalong B
B
intoavirtualcrossing.SeeFigure6.
Proposition3.3. UnderNotation3.2,wehave
g(D◦)=g(D•).
B B
Proof. Ifwedenotebyδthenumberofdisksthatweregluedinthefinalstep(thestep(iv)
inDefinition2.5)oftheconstructionofΣD• andbyγ thenumberofchordsintheGauss
diagramGD•B, thentheEulercharacteristicχB(ΣD•B)of D•B isequalto δ−γ. We easily see
thatδ equalsthe numberof Seifertcircles in Turaev’sconstructionof the Seifertsurface
ΣEDu◦BlearncdhathraacttγereiqstuicalassthΣeDn•uamndbehreonfcecriosshsoinmgesoomfDor◦Bp.hTichtuosΣitDi•s.clearthatΣD◦B hasthesam(cid:3)e
B B
Remark 3.4. Given a knotoid diagram D◦, we can construct a virtual knot diagram by
connectingthetwoendpointsofD◦ withanarbitrarypathalongwhichonlyvirtualcross-
ings occur; then it is a natural idea to define the Gauss diagram of D◦ to be the Gauss
diagramofsuchavirtualknotdiagram. AccordingtoProposition3.3,itturnsoutthatwe
can alternativelydefinethe genusof a knotoiddiagram(Definition2.17)to be the genus
ofitsGaussdiagram(justasStoimenow,TchernovandVdovina[5]definethegenusofa
virtualknotdiagram).
Proposition3.5. UnderNotation3.2,wehave
g(D)≥g(D◦).
B
Proof. ByProposition3.3,itsufficestocomparethegeneraoftheGaussdiagramsG and
D
ofGD•. NotethatGD• isobtainedbyremovingfromGD thechordscorrespondingtothe
B B
crossingsalongB.InLemma3.6,wewillprovethatthegenusofanyGaussdiagramdoes
notincreaseafteraremovalofachord.Thisimpliestheresult. (cid:3)
Lemma 3.6. Let G be any Gauss diagram and G′ be the Gauss diagram obtained by
removingfromGachordc. Thenwehaveg(G)≥g(G′).
Inmoredetail,ifcappearstwiceinasinglecycleofG (asCase(i)inFigure5),then
wehaveg(G)=g(G′)+1andhenceg(G)>g(G′). Otherwisewehaveg(G)=g(G′).
Proof. InordertocomparethegeneraofGandofG′,weneedonlytoknowthechangeof
thenumbersoftheirchordsandcyclesinconstructingG′fromGbyremovingthechordc
(seetheformula(2)inLemma2.11andRemark2.12).
Inthecasewherecappearstwiceinasinglecycle(sayα)ofG,thisαisoftheform
α={w ,a,b,w ,b,a}
1 2
fora,b ∈ ∂c and some wordsw , w in the endpoints. Let f(w) (resp.l(w)) be the first
1 2 i i
(resp. last) endpointcontained in w. Then the points a, b, f(w) and l(w) (i = 1,2) are
i i i
locatedonthecircleasCase(i)inFigure5. Afterthechordcisremoved(thenthepoints
aandbarealsoremoved),thecycleαsplitsintotwocycles{w }and{w },because f(w )
1 2 1
6 KENJIDAIKOKU,KEIICHISAKAI,ANDMASAMICHITAKASE
(cid:3)(cid:3) (cid:3) (cid:3) (cid:3)
l(w1)(cid:15) (cid:15)f(w2) l(w3)(cid:15) (cid:15)f(w3)
(cid:11) (cid:12)

a(cid:15) (cid:15)b a(cid:15) (cid:15)b
(cid:11) (cid:13)
f(w1)(cid:15) (cid:15)l(w2) f(w4)(cid:15) (cid:15)l(w4)
(cid:3)(cid:3) (cid:3) (cid:3)(cid:3) (cid:3)(cid:3)
Case (i) Case (ii)
Figure 5. Thethickenedcurvesare the partsof thecircle ofthe Gauss
diagramG.
(resp. f(w ))appearsjustafterl(w )(resp.l(w )). Thusthenumberofcyclesincreasesby
2 1 2
one. Since the number of chords decreases by one after c is removed, we have g(G) =
g(G′)+1bytheformula(2).
Nextconsiderthecasewheresuchacycleasabovedoesnotexist.Thenthereareexactly
twocycles(sayβ = {w ,a,b}andγ = {w ,b,a})ineachofwhichcappearsexactlyonce.
3 4
ThenβandγarelocatedasCase(ii)inFigure5. Thusaftertheremovalofc,thecycleβ
isunifiedwithγbyaband-sumalongc,andtheresultisasinglecycle{w ,w }. Thusthe
3 4
numberofcyclesdecreasesbyone.Thustheformula(2)impliesthatg(G)=g(G′). (cid:3)
Remark3.7. In[6,§1],itreadsthat“Thestudyofknotoiddiagramsleadstoanelementary
butpossiblyusefulimprovementofthestandardSeifertestimatefromaboveforthegenus
ofknot.” However,ithasnotbeenrigorouslyprovenin[6]thataknotoiddiagramalways
gives an estimate not worse than “the usual Seifert estimate” with respect to the genus.
Propositions3.5and3.3,whichdeducetheinequalityg(D)≥g(D◦),implementit. Notice
B
thatintroducingthenotionoftheGaussdiagramofaknotoidhasbeenthemainkeyinthe
proof(seeRemark3.4).
Remark3.8. Wedonotneedtoconsidertheremovalofany“sub-bridge”of Bsince,by
Lemma3.6,the removalofthe whole Balwaysgivesa betterestimate ofthe genusthan
anyremovalofasub-bridge.
Now we introduce the bridge-replacing move for a knot diagram which does not in-
crease(Theorem3.11),andinsomecasescertainlydecrease(Proposition3.13),thegenus
ofthediagram(cf.[6,§2.5]).
Definition3.9(thebridge-replacingmove). LetDbeadiagramofaknotK withanover-
bridge (resp. under-bridge) B. Then we define an operation, called the bridge-replacing
movefor(D,B),whichreplacestheover-bridge(resp.under-bridge)Bintoanotherover-
bridge(resp.under-bridge)B˜,asfollows.
TakeanorientationonthediagramDandconsiderthe(oriented)knotoiddiagram D◦
B
byremovingBfromD(seeFigure6).LetJbetheorientedSeifertintervalofD◦,obtained
B
by the smoothingprocess(the step (ii) in Definition 2.16), from the leg P to the head Q
(seeFigure7)inD◦.
B
NowthenewbridgeB˜ fromQtoPisconstructedjustalongJsothatitgoesontheright
(resp.left)sideofJifatallpossible,andisallowedtooverpass(resp.underpass)onlythe
arcsofwhich J consists. ThustheunionD◦ ∐B˜ providesanewknotdiagram,whichwe
B
denotebyD˜ (seeFigure8).ClearlyD˜ representsthesameknottypeK asDdoes.
B B
Remark 3.10. In Definition 3.9, in fact, it is not so important which side of J the new
bridgegoeson(althoughitisessentialthatthenewbridgedoesnotcausecrossingsoutside
J). We have adopted the convention here so that we can treat over-bridges and under-
bridgessymmetrically.ThissystematictreatmentwillbeconvenientinAlgorithmA.6.
ONAMOVEREDUCINGTHEGENUSOFAKNOTDIAGRAM 7
8 8
8
B B
QÆ Æ (cid:15) (cid:15)
4 5 4 5
P
1 1 1
3 2 3 2 3 2
7 6 7 6 7 6
Æ (cid:15)
D DB DB
Figure 6. A diagram D of 8 , a knotoid diagram D◦ obtained by re-
20 B
movinganover-bridgeB=(45)fromD,andadiagramD• ofthevirtual
B
knotobtainedbyturningthecrossings4,5alongBintovirtualcrossings.
ThedottedarrowinDisabypassforB(seeProposition3.14).
Q P
Figure 7. The diagramobtainedbysmoothingall the crossingsof D◦.
B
ThethickenedcurveistheSeifertintervalJ. Thedottedcurvewillbea
guideforthenewbridgeB˜ inthenewdiagramD˜.
B~
Figure8. ThediagramD˜ of8 obtainedbyreplacingBwithB˜
20
Regardingtheabovebridge-replacingmove,ourmainclaimisthefollowing.
Theorem3.11. Let Dbeaknotdiagramwithanover-orunder-bridge BandD˜ bethe
B
knotdiagramobtainedbythebridge-replacingmovefor(D,B).Thenwehave
g(D)≥g(D˜ ).
B
To prove Theorem 3.11, we just need to combine Lemma 3.12 below with Proposi-
tions3.3and3.5.
8 KENJIDAIKOKU,KEIICHISAKAI,ANDMASAMICHITAKASE
Lemma3.12. LetDbeaknotdiagramwithanover-orunder-bridgeB.LetD˜ betheknot
B
diagramobtainedbythebridge-replacingmovefor(D,B)andD◦ betheknotoiddiagram
B
obtainedbyremovingfromDtheinteriorofB. Thenwehave
g(D˜ )=g(D◦).
B B
Proof. WeusethenotationsinNotation3.2andDefinition3.9.
SincethenewbridgeB˜ passesonlythearcswhichconfigurestheSeifertintervalJ,the
SeifertalgorithmforthenewknotdiagramD˜ = D◦ ∐B˜ providesthesameresultthatthe
B B
SeifertalgorithmforD◦ does,exceptinthepartwithwhichJ isinvolved.
B
Assume that the new bridge B˜ has length k, that is, B˜ passes k times the arcs which
configuresJ. Then,inapplyingtheSeifertalgorithmfortheknotdiagramD˜ ,considering
B
theorientationsofJ andofB˜,wehave(k+1)newSeifertcircles(insteadofJ)inthepart
withwhichJ isinvolved.Thisimpliesthatg(D˜ )isequaltog(D◦). (cid:3)
B B
Next we study the condition for the bridge-replacing move to certainly decrease the
genus.
Anover-bridge(resp.under-bridge)ofaGaussdiagramGoflengthkisaconsecutive
sequence of endpointsv0,...,vk+1 on the orientedcircle ofG such that v1,...,vk are the
initial(resp.terminal)pointsofthechordsofG.
Proposition3.13. LetG be aGaussdiagramwith anover-orunder-bridgev0,...,vk+1,
and letG′ be the Gauss diagramobtainedby removing all the chordsc with v ∈ ∂c for
i
some1 ≤ i ≤ k. Theng(G) > g(G′)ifandonlyifthereisatleastonecycleαofG ofthe
form
α={...vivi+1...vjvj+1...}
forsome0≤i, j≤k.
Proof. ThisisaconsequenceofLemma3.6.Ifsuchacycleαasaboveexists,thenCase(i)
inFigure5happensforatleastonechordchavingitsendpointbetweenvivi+1 andvjvj+1,
andhencethegenuscertainlydecreases.Conversely,supposeanytwooftheportionsvivi+1
(0 ≤ i ≤ k)donotbelongtothesamecycleinG. Thenanychordcwithv ∈ ∂cisplaced
i
asCase(ii)inFigure5. Thusanyremovalofthemdoesnotdecreasethegenus. (cid:3)
Inthecaseofknotdiagrams,theconditioninProposition3.13canbestatedasfollows.
Proposition3.14. LetDbeadiagramofaknotK withanover-orunder-bridgeB. Then
the bridge-replacingmove for B certainly decreases the genus if and only if there exists
a collectionoforientedarcs in D whichconstitute, afterthe smoothingprocess (step(ii)
in Seifert’s algorithm, Definition 2.1), a (well-oriented) path connecting two crossings
includedinB.Wecallsuchacollectionofarcsabypass.
ForexampleintheleftmostdiagramofFigure6,theexistenceofthebypass(dotted)for
the over-bridge B(thickened)ensuresthatthe bridge-replacingmovefor (D,B)decrease
the genus of the diagram. In fact, the diagram shown in Figure 8 obtained from that in
Figure6bythebridge-replacingmoveforBdetectsthegenus2of8 .
20
4. Anexample
Theknot16 ,whichisthe306917thnon-alternating16crossingknotintheHoste-
686716
Thistlethwaite table [1], is referred to in Stoimenow’s recent paper [4, §10.3] as a knot
whose (canonical) genus has not been determined (to be whether 2 or 3) yet. Here we
demonstrateanapproachtothisproblemusingbridge-replacingmoves. Indeeditsgenus
turnsouttobe2(seeRemark4.1).
Intheknotdiagramof16 showninFigure9,thatisdrawnbyKnotscape[1]and
686716
has genus 5, we can find three over-bridges (thickened) each of which has a favorable
bypass (see Proposition 3.14). In fact we can check that the bridge-replacing move for
ONAMOVEREDUCINGTHEGENUSOFAKNOTDIAGRAM 9
Figure 9. The knot 16 . This diagram has Dowker-Thistlethwaite
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code(DTcode)410-26-22-18220-26-32-281430-6-12-824
Figure10. Adiagramwithgenus5
Figure11. Aminimalcrossingdiagramwithgenus3of16 . ItsDT
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codeis-122622-1428-2-2030-248-32-1641018-6
one of the three bridges produces a new diagram with genus 4. In this case, however,
suchamoveforonebridgedestroysthebypassesoftheothertwobridgesandapparently
wecannotperformfurtherbridge-replacingmoves. Toavoidthisweprecookthediagram
intothediagramshowninFigure10. ThenwecanperformforthediagraminFigure10
bridge-replacingmovestwice successively, first with respect to the bridge A and second
withrespectto B,sothatweobtainadiagramwithgenus3,butwith20crossings,ofthe
knot16 (thenumberofthecrossingsofthisdiagramcanbepareddownto18bythe
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secondReidemeistermove).
Nowwecaninterpretthebridge-replacingmoveintermsofGausscodes,ofwhichwe
give a detailed account in Appendix A. By using the interpretation, we develop a small
Pythonprogramwhich,foraninputtedGausscode,performsallpossiblebridge-replacing
moves(andreducesexcessivecrossingsinpairsbythesecondReidemeistermoves).With
10 KENJIDAIKOKU,KEIICHISAKAI,ANDMASAMICHITAKASE
Figure12. ASeifertsurfacefor16 withgenus2
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theaidoftheprogram(andwithsomeheuristicprocedures),wehavefoundthe diagram
showninFigure11oftheknot16 ,thathasgenus3andminimal(thatis,16)crossings.
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Remark4.1. InspiredbyFigures10and11,MikamiHirasawafoundSeifertsurfacesof
genus 2 for the knot 16 (by probing for “compression disks”). Figure 12 depicts
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one of such (possibly non-canonical) Seifert surfaces with genus 2. By an estimate via
polynomialinvariants,thesesurfacesturnouttoattainthegenus2oftheknot1. Wedonot
know,however,if thecanonicalgenusoftheknotcangodownto 2. Namely,we donot
knowwhetheranyoftheseSeifertsurfacesfor16 withgenus2canberealizedasa
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canonicalone.
AppendixA. Thebridge-replacingmoveintermsofGausscodes
We give an interpretationof the bridge-replacingmove, introducedin §3, in terms of
Gausscodes. Wewillfollowtheconventionof[2]forGausscodes.
DefinitionA.1. Aunitisawordoflengththreeoftheform“αβγ,”where
• “α”iseitherthecharactersOorU(whichindicate“over”or“under”information),
• “β”isanaturalnumber(thelabelofacrossing),
• “γ”isthesign+or−(thesignofacrossing).
AGausscodeoflengthnisacyclicsequenceof2nunitsX ...X suchthat:
1 2n
(i) Itincludesndistinctnaturalnumbersandeachnumberappearsexactlytwice.
(ii) Ifthenumberk(whichappearstwice)appearsinsomeunitincludingthecharacterU
(resp.O),thenkmustappearintheotherunitincludingO(resp.U).
(iii) Foreachnumberk,thetwounitsincludingkhavethesamesign.
ToanorientedknotdiagramD= D withncrossings,wecanassociatetheGausscode
K
X = X ...X oflengthnasfollows:labelallthecrossingsas1,2,...,n,andgoalongD
D 1 2n
accordingtotheorientation(startingfromanarbitrarypointb).Wewillmeeteachcrossing
exactlytwicebeforewecomebacktob. Thei-thunitX consistsof
i
• thecharacterOorUdependingonwhetherweoverpassorunderpassthei-thcross-
ingrespectively,
• thenaturalnumberktakenfromthelabelofthei-thcrossing,and
• the sign + or − depending on whether the i-th crossing is positive or negative
respectively.
ThefollowingdefinitionismotivatedbythenotionofcyclesinGaussdiagrams(Defi-
nition2.9).
DefinitionA.2. LetC = X ...X beaGausscode. AcycleinC isacyclicsequenceof
1 2n
unitsinCobtainedinthefollowingsteps: startingfromsomeunitX,
i
1ThisshouldbededucedalsofromacalculationoftheknotFloerhomology.