Kauffman-Jones polynomial of a curve on a surface

0.17 MB·

by  Shinji Fukuhara

KAUFFMAN-JONESPOLYNOMIAL OF A CURVE ON A SURFACE
7 SHINJIFUKUHARAANDYUSUKEKUNO
1
0
2 ABSTRACT. WeintroduceaKauffman-JonestypepolynomialLγ(A)foracurveγ on
anorientedsurface,whoseendpointsareontheboundaryofthesurface. Thepolynomial
n Lγ(A)isaLaurent polynomial inonevariable Aandisaninvariant ofthehomotopy
a classofγ. Asanapplication, weobtainanestimateintermsofthespanofLγ(A)for
J
theminimumself-intersectionnumberofacurvewithinitshomotopyclass.Wethengive

9 achorddiagrammaticdescriptionofLγ(A)andshowsomecomputationalresultsonthe
2 spanofLγ(A).

]
T
G
. 1. INTRODUCTION
h
t LetS beanorientedC∞-surfacewithnon-emptyboundary∂S. ByacurveonS, we
a
m meanaC∞-immersionγ fromtheunitintervalI =[0,1]toS,whichhasonlytransverse
doublepointsasitssingularitiesandsatisfiesγ−1(∂S)={0,1}withγ(0)6=γ(1).
[
In this article, we consider curves on S from the view point of virtual knots [6] or

1 equivalently, abstract link diagrams [4], with emphasis on their invariants coming from
v theKauffmanbracket[5]. Moreconcretely,weintroduceLaurentpolynomialshD iand
γ
8
L (A) in one variable A. We show that the span of these polynomials can be used for
1 γ
estimating the number of double points of γ. In fact, the polynomialshD i and L (A)
4 γ γ
8 dependonlyoncombinatoricsoftheimageofthecurveγinitsregularneighborhoodinS.
0 Based on this fact, we then givea chorddiagrammaticdescriptionof these polynomials.
1. AnadvantageofbeingfreefromtheambientsurfaceS isthatitbecomeseasytoprovide
0 andcomputeexamples. In§4, weshowsome computationalresultsonthespanofhD i
γ
7 fromthispointofview.
1
Intherestofthissection,wedescribemainconstructionsandresults. Someproofswill
:
v bepostponedto§2.
i We beginwithterminology. LetX beacompact1-manifold. Namely,X isadisjoint
X
unionoffinitelymanyI’sandcircles:
r
a X =I ⊔···⊔I ⊔S1⊔···⊔S1.
AC∞-immersionf : X → S iscalledgenericifithasonlytranseversedoublepointsas
itssingularities,f−1(∂S) = ∂X,andf| isinjective. AgeneralizedlinkdiagramonS
∂X
isasubsetofS oftheformD =f(X)forsomegenericimmersionf :X →S,endowed
withachoiceofcrossingtoeachdoublepointofD. SeeFigures1and2.
TwogeneralizedlinkdiagramsDandD′ arecalledequivalent(resp. regularlyequiva-
lent)ifDistransformedintoD′ byafinitesequenceofambientisotopiesofS relativeto
∂S,andthethreeReidemeistermovesR ,R ,andR (resp. R andR )showninFigure
1 2 3 2 3
3. WewriteD ∼D′(resp.D ∼ D′)whenDisequivalent(resp.regularlyequivalent)to
r
D′.
2010MathematicsSubjectClassification. Primary57M25;Secondary57N05.
Keywordsandphrases. Kauffman-Jonespolynomial,curvesonsurfaces,linearchorddiagrams.
1
2 SHINJIFUKUHARAANDYUSUKEKUNO
(cid:81)(cid:84)
FIGURE 1. a choice FIGURE 2. an exam-
of under- and over- ple of a generalized
crossing linkdiagram
(cid:52) (cid:52) (cid:52)
(cid:19) (cid:19) (cid:21)
(cid:52)
(cid:52) (cid:21)
(cid:20)
FIGURE 3. Reidemeistermoves
(cid:44) (cid:44)
(cid:19) (cid:20)
(cid:44) (cid:44)
(cid:20) (cid:19)
FIGURE 4. replacingdoublepointswithcrossings
Let γ be a curveon S. For each doublepointp of γ, there is a neighborhoodU of p
such that U ∩γ(I) consists of two arcs J and J intersecting at p, and J is traversed
1 2 1
first when we go along γ from γ(0). Then we replace p with a crossing with J being
1
overcrossing(seeFigure4). LetD denotethegeneralizedlinkdiagramonS obtainedin
γ
thisway.Inotherwords,D istheprojectiondiagramintheusualsenseoftheembedding
γ
I →S×I,t7→(γ(t),1−t)bytheprojectionS×I →S×{0}∼=S,(x,t)7→x.
Thefollowingfactiscrucialinourargument:
Theorem 1.1. Suppose that two curves γ and γ′ on S are homotopic (resp. regularly
homotopic)relativeto∂S. ThenDγ ∼Dγ′ (resp. Dγ ∼r Dγ′).
TheKauffmanbracket[5]isextendedtolinkdiagramsonsurfaces[2]. Thisextension
isstraightforwardandappliestoourgeneralizedlinkdiagramsalso. Forthesakeofdefi-
niteness,letusrecalltheconstruction. LetDbeageneralizedlinkdiagramonS. Wecan
splitDateachcrossingintwoways. WewilldistinguishthesesplittingsasatypeAsplit-
tingandatypeBsplitting,respectively(seeFigures5and6,accordingtotheorientation
ofS). A state of D is a choiceof splitting typefor eachcrossing of D. Fora state s of
KAUFFMAN-JONESPOLYNOMIALOFACURVE 3
(cid:36)
(cid:35)
(cid:35)
(cid:36)
FIGURE 6. splitting
FIGURE 5. splitting
withtheotherorienta-
withanorientation
tion
(cid:38) (cid:38) (cid:38) (cid:38) (cid:38) (cid:38)
(cid:35) (cid:35)(cid:15)(cid:19) (cid:35)(cid:15)(cid:19) (cid:35)
FIGURE 7. threediagrams
D,letD(s)bethecompact1-submanifoldofS obtainedbysplittingD bys. IfD hasn
crossings,thereare2nstatesofD.
ToeachstatesofD,weassignthefollowingthreenumbers:
α(s):=thenumberoftypeAsplittings,
β(s):=thenumberoftypeBsplittings,
µ(s):=thenumberofconnectedcomponentsofD(s).
ThenwedefinethebracketpolynomialofDby
hDi:= Aα(s)−β(s)(−A2−A−2)µ(s)−1,
s
X
wheresrunsoverallstatesofD.
Abasicpropertyofthebracketpolynomialisthefollowingskeinrelation,whoseproof
isthesameasthatoftheclassicalcase[5].
Lemma1.2. LetDbeageneralizedlinkdiagramonS.
(1) Pick a crossing of D and consider the two splittings of it as shown in Figure 7.
Then
hDi=AhDAi+A−1hDA−1i.
(2) LetT beageneralizedlinkdiagramwhichisconnectedandhasnocrossing.
(a) WehavehTi=1.
(b) IfDandT aredisjoint,thenhD⊔Ti=(−A2−A−2)hDi.
AssumethatageneralizedlinkdiagramD = f(X)isoriented. Thatis,X isoriented
andDinheritsthisorientation. Forinstance,ifγ isacurveonS,thenD canbeoriented
γ
fromthenaturalorientationofI. Letw(D)denotethewrithenumberofD. Thatis,
w(D):= ε ,
p
p
X
4 SHINJIFUKUHARAANDYUSUKEKUNO
(cid:493)(cid:31)(cid:19) (cid:493)(cid:31)(cid:15)(cid:19) (cid:493)(cid:31)(cid:15)(cid:19) (cid:493)(cid:31)(cid:19)
(cid:50) (cid:50) (cid:50) (cid:50)
FIGURE 8. signsofcrossings
wherep runsoverallcrossingsofD andε ∈ {±1}isthesignofthecrossingatp (see
p
Figure8). ThenwedefinetheKauffman-JonespolynomialofDby
L (A):=(−A)−3w(D)hDi.
D
The following result is an analogy of the result for ordinary link diagrams given by
Kauffman[5],whereLemma1.2playedacentralrole. Hisargumentcanalsobeapplied
tothecaseofgeneralizedlinkdiagrams,soweomittheproof.
Theorem1.3. LetDandD′begeneralizedlinkdiagramsonS.
(1) IfDandD′areregularlyequivalent,hDi=hD′i.
(2) AssumefurtherthatDandD′areoriented.IfDandD′areequivalent,L (A)=
D
LD′(A).
Tosimplifynotation,wedenoteL (A):=L (A)foracurveγ.CombiningTheorems
γ Dγ
1.1and1.3,weobtain
Theorem1.4. Letγ andγ′becurvesonS.
(1) Ifγ andγ′areregularlyhomotopicrelativeto∂S,thenhDγi=hDγ′i.
(2) Ifγ andγ′arehomotopicrelativeto∂S,thenLγ(A)=Lγ′(A).
For a Laurent polynomial f(A) ∈ Z[A,A−1], the span of f, denoted by spanf, is
defined to be the difference of the maximal and the minimal degrees of f. Note that
spanhD i = spanL (A) for any curve γ. We denote by d(γ) the number of double
γ γ
pointsofacurveγ. Thenwehavethefollowingestimateford(γ),whichisanalogousto
[8]and[9].
Theorem1.5. Foracurveγ onS,itholdsthat
(1.1) spanhD i≤4d(γ).
γ
Wedefinetheminimumself-intersectionnumberc(γ)ofacurveγ by
c(γ):=min{d(γ′)|γ′isacurveonS homotopictoγ relativeto∂S}.
Corollary1.6. Foranycurveγ onS,itholdsthat
spanhD i
γ
≤c(γ).
4
WegiveexamplesofusingCorollary1.6forestimatingc(γ).
Example1.7. Letγ bethecurveshowninFigure9. Thebracketpolynomialofγ is
1 1
hD i=A−A−3−A−5.
γ1
WeseethatspanhD i=6and6/4≤c(γ ). Henceweobtain2≤c(γ )≤3.
γ1 1 1
Example1.8. Letγ bethecurveshowninFigure10. Thebracketpolynomialofγ is
2 2
hD i=−A5+A+A−1−A−3−A−5.
γ2
SincespanhD i=10,wehave10/4≤c(γ ). Therefore,c(γ )=3.
γ2 2 2
KAUFFMAN-JONESPOLYNOMIALOFACURVE 5
FIGURE 10. a curve
FIGURE 9. acurveγ1 γ on a punctured
onapuncturedtorus 2
torus
(cid:513) (cid:513) (cid:513)
(cid:19) (cid:20) (cid:21)
FIGURE 11. Reidemeistermovesofacurveγ
(cid:21)
(cid:22) (cid:22)
(cid:560) (cid:560) (cid:560)
(cid:20) (cid:20) (cid:21) (cid:22)
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
(cid:53) (cid:53) (cid:53)
(cid:20) (cid:21) (cid:22)
FIGURE 12. Reidemeistermovesofγ andDγ
2. PROOFS OFTHEOREMS 1.1AND 1.5
Inthissection,weproveTheorems1.1and1.5.
ProofofTheorem1.1. Iftwocurvesγandγ′arehomotopicrelativeto∂S,thenγistrans-
formedintoγ′ byusingafinitesequenceofambientisotopiesofS relativeto∂S andthe
threelocalmovesω ,ω ,ω ,showninFigure11. Seee.g.,[1]Lemma5.6.
1 2 3
It is easily seen that if γ is transformed into γ′ by ω (i = 1,2,3), then D can be
i γ
transformedintoDγ′ byRi (i = 1,2,3)respectively(seeFigure12). Thiscompletesthe
proof. (cid:3)
Next, we proveTheorem 1.5. Recall that a generalized link diagram D has the form
D = f(X) for a generic immersionf : X → S, endowedwith a choice of crossing to
eachdoublepoint.WesaythatDisconnectedifitisconnectedasasubsetofS. Letd(D)
bethenumberofcrossingsofD.
LetusconsiderthefollowingconditionforageneralizedlinkdiagramD =f(X):
(2.1) thenumberofconnectedcomponentsofX homeomorphictoI isatmostone.
SinceD isconnectedforanycurveγ,Theorem1.5isaspecialcaseofthefollowing:
γ
6 SHINJIFUKUHARAANDYUSUKEKUNO
Proposition2.1. LetDbeaconnectedgeneralizedlinkdiagramsatisfyingcondition(2.1).
Thenitholdsthat
spanhDi≤4d(D).
Proof. ThebracketpolynomialofDiswrittenas
hDi= hD|siδµ(s)−1,
s
X
wheresrunsoverallstatesofDandwesethD|si:=Aα(s)−β(s),δ :=−A2−A−2.
LetsbeastateofDhavingatypeAsplitting,andlets′denotethestateofDobtained
fromsbyreplacingthetypeAsplittingwithatypeBsplitting. Thenwehave
hD|s′i=hD|siA−2, µ(s′)≤µ(s)+1, µ(s)≤µ(s′)+1.
Hencewehave
max deghD|s′iδµ(s′)−1 ≤max deghD|siδµ(s)−1,
min deghD|s′iδµ(s′)−1 ≤min deghD|siδµ(s)−1.
Lets (resp.s )denotethestateofDwhosesplittingateachcrossingisoftypeA(resp.
A B
oftypeB).Thenwehave
max deghDi≤max deghD|s iδµ(sA)−1 =d(D)+2(µ(s )−1),
A A
min deghDi≥min deghD|s iδµ(sB)−1 =−d(D)−2(µ(s )−1).
B B
Fromtheseinequalities,wehave
spanhDi≤2d(D)+2(µ(s )+µ(s )−2).
A B
Lemma2.2. Wehaveµ(s )+µ(s )≤d(D)+2.
A B
Proof. Ifd(D) = 0,theinequalityisobvious. Letd(D) > 0andchooseacrossingofD
and consider the two splittings of it as shown in Figure 7. Then, at least one of them is
connectedandsatisfiescondition(2.1)byvirtueoftheassumption(2.1)onD. LetD′ be
suchageneralizedlinkdiagramandassumethatD′ isobtainedfromthetypeAsplitting
(theothercaseistreatedsimilarly).Lets′ ands′ bethestatesofD′definedbythesame
A B
way as we introduces and s to D. Then µ(s ) = µ(s′ ) and µ(s ) ≤ µ(s′ )+1,
A B A A B B
henceµ(s )+µ(s )≤µ(s′ )+µ(s′ )+1.Thentheassertionisprovedbyinductionon
A B A B
d(D). (cid:3)
ByLemma2.2weconclude
spanhDi≤2d(D)+2(µ(s )+µ(s )−2)≤4d(D).
A B
ThiscompletestheproofofProposition2.1. (cid:3)
3. CHORD DIAGRAMMATIC DESCRIPTION
For a curve γ on S, the bracketpolynomialhD i is actually determinedby a regular
γ
neighborhoodofγ(I)inS. Inthissection,westudyhD ifromthispointofview.
γ
Let d be a positive integer. An oriented linear chord diagram of d chords is a set
C = {(i ,j ),...,(i ,j )} of d ordered pairs of integers such that {i } ∪ {j } =
1 1 d d k k k k
{1,...,2d}. EachelementofC iscalledachordofC. Achord(i,j)iscalledpositiveif
i < j, andnegativeotherwise. Finally, a state ofC is a maps: C → {A,B}, whereA
andB arefixedsymbols.
KAUFFMAN-JONESPOLYNOMIALOFACURVE 7
Letγ beacurvewithd(γ) = d. Thentheinverseimageofthedoublepointsofγ are
2dpointsonI. Wenamethem{p } sothat0< p <p < ··· <p <1. Theoriented
i i 1 2 2d
linearchorddiagramC is definedbytheconditionthatan orderedpair (i,j)isa chord
γ
ofC ifandonlyifγ(p )=γ(p )andthepair(dγ/dt(p ),dγ/dt(p ))oftangentvectors
γ i j i j
matchestheorientationofS.
Remark3.1. Conversely,foranyorientedlinearchorddiagramC, thereisa curveγ on
someorientedsurfaceS suchthatC =C .
γ
Let C be an oriented linear chord diagram of d chords and s a state of C. For each
chordc = (i,j) ∈ C, wedefinea subsetR ⊂ S ofpermutationsof2d+1letters
c 2d+1
{0,1,...,2d}inthefollowingway.
• If s(c) = A andc is positive, or s(c) = B and c is negative, thenwe set R =
c
{(i,j−1),(i−1,j)}.
• If s(c) = A andc is negative,or s(c) = B and c is positive, thenwe set R =
c
{(i,j),(i−1,j−1)}.
ConsiderthesubgroupofS generatedby R ,andletΓ bethenumberoforbits
2d+1 c∈C c s
oftheactionofthisgroupon{0,...,2d}.
S
Weset
hC|si:=A|s−1(A)|−|s−1(B)|(−A2−A−2)Γs−1,
where|s−1(A)|denotesthecardinalityofthesets−1(A),andwedefine
hCi:= hC|si,
s
X
wherethesumrunsoverallstatesofC. Wealsodefine
L (A):=(−A)−3w(C)hCi,
C
wherew(C)isthenumberofpositivechordsminusthenumberofnegativechordsofC.
Proposition3.2. Letγ beacurveonS. ThenhD i=hC iandL (A)=L (A).
γ γ γ Cγ
Proof. Firstofall,thesecondformulafollowsfromthefirst,sincew(D )=w(C ).
γ γ
Now introduce 2d+ 1 points q , 0 ≤ i ≤ 2d. With respect to the parametrization
i
of γ, these points have the following interpretation: q = 0, q = (p + p )/2 for
0 i i i+1
1 ≤ i ≤ 2d−1, andq = 1. Fora state s of γ, let Γ(C ,s)be the graphwith the set
2d γ
ofverticesbeing{q } ,andthesetofedgesdeterminedbytheconditionthatq andq are
i i k l
connectedbyanedgeifandonlyif(k,l) ∈ R . ThenΓ(C ,s)ishomeomorphic
c∈Cγ c γ
tothespliceofD bys. SeeFigure13.(Hereandinwhatfollows,weassumethecounter-
γ
clockwiseorientationinanyfigure.)ThefirstSformulafollowsfromthisobservation. (cid:3)
Inthebelow,werecordelementarypropertiesofhD iintermsofchorddiagrams.
γ
LetC = {(i ,j ),...,(i ,j )}beanorientedlinearchorddiagram. Fix0 ≤ ℓ ≤ 2d.
1 1 d d
Fori∈{1,...,2d},weset
i ifi≤ℓ,
i′ :=
(i+2 ifi>ℓ.
Wedefine
Cℓ :={(i′,j′)} ∪{(ℓ,ℓ+1)},
+ k k k
Cℓ :={(i′,j′)} ∪{(ℓ+1,ℓ)}.
− k k k
8 SHINJIFUKUHARAANDYUSUKEKUNO
s(c)=A,cpositive s(c)=B,cpositive
qj qi qj qi qj qi qj qi
qi−1 qj−1 qi−1 qj−1 qi−1 qj−1 qi−1 qj−1
s(c)=A,cnegative s(c)=B,cnegative
qi qj qi qj qi qj qi qj
qj−1 qi−1 qj−1 qi−1 qj−1 qi−1 qj−1 qi−1
FIGURE 13. proofofProposition3.2
Also,wedefine
C∧ :={(i +1,j +1)} ∪{(1,2d+2)},
+ k k k
C∧ :={(i +1,j +1)} ∪{(2d+2,1)}.
− k k k
Proposition3.3(Birth/deathofmonogons). Wehave
hCℓi=hC∧i=(−A3)hCi,
+ +
hCℓi=hC∧i=(−A−3)hCi.
− −
Proof. If C = C for some curve γ, then Cℓ corresponds to a suitable insertion of a
γ +
negative monogon to γ. Therefore, from the behavior of the bracket polynomial under
the Reidemeister move R , we obtain hCℓi = (−A3)hCi. The other cases are treated
1 +
similarly. (cid:3)
Let
C ={(i ,j ),...,(i ,j )} and D ={(k ,ℓ ),...,(k ,ℓ )}
1 1 d d 1 1 e e
beorientedlinearchorddiagrams.WedefinethestackingofC andDby
C♯D :={(i ,j )} ∪{(k +2d,ℓ +2d)} .
a a a b b b
Proposition3.4(Stackingformula). WehavehC♯Di=hCihDi. Inparticular,spanhC♯Di=
spanhCi+spanhDi.
Proof. SincethechordsofC♯D areinone-to-onecorrespondencewiththedisjointunion
of the chords of C and D, any state of C♯D is of the form s♯t, where s is a state of C
and t is a state of D. The assertion follows from the observation that |Γ(C♯D,s♯t)| =
|Γ(C,s)|+|Γ(D,t)|−1. (cid:3)
Proposition3.5. LetC beanorientedlinearchorddiagramofdchords.
• Ifdiseven,thenhCihasonlytermsofevendegree.
• Ifdisodd,thenhCihasonlytermsofodddegree.
Proof. Bydefinition,hC|sihasthisproperty,sodoeshCi. (cid:3)
Proposition3.6(Reversingallthechords). LetC ={(i ,j ),...,(i ,j )}beanoriented
1 1 d d
linearchorddiagramandsetC :={(j1,i1),...,(jd,id)}. ThenhCi=hCi|A7→A−1.
KAUFFMAN-JONESPOLYNOMIALOFACURVE 9
···
1 2 3 d
FIGURE 14. thecurveγd inExample4.1
Proof. There is a natural bijection ι from the set of chords of C to that of C given by
(i ,j )7→(j ,i ). Thismapspositive(resp. negative)chordstonegative(resp. positive)
k k k k
chords. Moreover, it induces a bijection from the set of states of C to that of C given
by s 7→ s, determined by the condition that {s(c),s(ι(c))} = {A,B} for any chord c
of C. Then, it holds that hC|si = hC|si|A7→A−1 for any state s of C. This proves the
formula. (cid:3)
4. THERANGE OF THESPAN
Inthissection,westudytherangeofspanhCi. ByTheorem1.5,spanhCi ≤ 4difC
hasdchords. Also, byProposition3.5, spanhCiisalwaysaneveninteger. Fixingd, let
usconsiderwhichevenintegersnotgreaterthan4darerealizedasspanhCi forsomeC
withdchords.
We say that an even integer l is d-realizable if there exists an oriented linear chord
diagramC ofdchordssuchthatspanhCi=l.
Ifd=1,C =C :={(1,2)}orC =C . ThushCi=−A±3andspanhCi=0.
1 1
Ifd = 2,byadirectcomputation,weseethat0and6are2-realizable,while2,4,and
8 are not. For example, C = {(1,3),(2,4)} satisfies hC i = A2 +1−A−4, so that
2 2
spanhC i=6.
2
If d = 3, we see that0, 6, 10, and 12 are 3-realizable, while 2, 4, and8 are not. For
example, the stacking C ♯C satisfies spanhC ♯C i = 6; C = {(1,5),(2,4),(6,3)}
1 2 1 2 3
satisfieshC i=−A5−A3+A+A−1−A−5,sothatspanhC i=10;thechorddiagram
3 3
C(3)inExample4.1belowsatisfiesspanhC(3)i=12.
Toseethecased≥4,weconsiderthefollowingtwoexamples.
Example4.1. Letd≥1beanoddinteger,andsetC(d):={(i,i+d)}d .Then
i=1
d−1
hC(d)i= (−1)i−1A−3d−2+4i−Ad+2.
i=1
X
Inparticular,ifd≥3,thenspanhCi=(d+2)−(−3d+2)=4d.
10 SHINJIFUKUHARAANDYUSUKEKUNO
···
1 2 3 4 d−1
FIGURE 15. thecurveγd inExample4.2
Proof. WehaveC(d) = C ,whereγ isthecurveasshowninFigure14. Letd ≥ 1be
γd d
anoddinteger.Then
···
1 2 3 d+2
hγ i=
d+2
···
2 3 d+2
=A
···
2 3 d+2
+A−1
···
3 d+2
=A2
···
3 d+2
+
+A−1·(−A−3)d+1
=A2hγ i+(−A−3)d+A−3d−4.
d
Now it is easy to see that hγ i = −A3, and the formula is proved by an inductive
1
argument. (cid:3)
Example4.2. Letd≥4beaneveninteger,andset
C(d):={(1,d),(d+1,2d)}∪{(2d−i,i+1)}d−2.
i=1
Then
d−4
hC(d)i=A−3d+4−A−3d+8+2 (−1)i−1A−3d+8+4i +Ad−4−Ad+Ad+4.
!
i=1
X
Inparticular,spanhC(d)i=(d+4)−(−3d+4)=4d.