ADVANCES IN SOVIET MATHEMATICS Volume 21, 1994

On the Enumerationof Curvesfrom Infinity to Infinity S. M. GUSEIN.ZADE AgsrRAcr. A combinatorial invariant is describedwhich enumeratesconnected componentsof the spaceof smooth plane curves with only simple (i.e. double points) self-intersectionscoinciding with the x-axis outside a boundeddomain. This invariantpermits to enumeratesuchcurves(components) with different numbersof self-intersectionsin the consecutiveorder.

The aim of the paper is the description of a combinatorial invariant enumerating connectedcomponents of the spaceof smooth plane curves coinciding with the x-axis outside a bounded domain and having only simple self-intersections(i.e., double points). This invariant permits to enumerate such curves (components) with different numbers of self-intersections in consecutiveorder. The enumeration of closed plane curves is a more complicated problem (see []). Known invariants (the index, Gauss diagrarhs, ) are not in (i.e., one-to-onecorrespondencewith classesof closedcurves. The enumeration of curves from infinity to infinity can be regarded as the enumeration of closed curyes with a distinguished (non-singular) point on their outlines. The describedcombinatorial invariant resemblesthe Vassiliev diagram (see[1]), but is finer. DnrrulrroN 1. We say that an immersion of an one-dimensionalmanifold (i.e. a union of lines and circles) into the plane is genericif it has only simple self-intersections(i.e., double points) as singularities. DrrwrrtoN 2. A curve from infinity to infinity (or simply a long cume) is ageneric C*-immersion ,L: lR.l--'lR2of theline lRl intotheplane R2 that coincideswith the map x -' (x, 0) for lxl large enough. DenNtrroN 3. A reduciblelong curve is a genericimmersion

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x --+(x, 0) outside of compact set and whoseimage is connected. DrrwlrtoN 4. We say that two long curves (reducible or not) are equivalent (or, if this does not lead to misunderstanding,simply equal) if they belong to the same connectedcomponent of the spaceof all long curves, i.e. if they can be deformed one into another inside the classof generic immersions. Rela.q.nr. There exists a natural integer invariant of long curves: the index, i.e., the number of rotations of the tangent vector. Equivalent curves have equal indices. Any two long curves with the same index can be deformed one into the other inside the class of immersions (generallyspeaking with the appearanceof singularities of the image more complicated than simple self-intersections). SurrunN"r l. There existsa surjectton of the set of classesof long curves onto the set of classesof closed curvesand a bijection of the set of classesof long curvesonto the set of closedcurveswith a distinguished nonsingalar point on the outline. Pnoor. The map from the set of long curves onto the set of closed curves with a distinguished point on the outline can be defined in the way shown in Figure 1. The map in the opposite direction (from the set of long curves onto the set of closed curves) can be obtained from this one by forgetting the distinguishedpoint on the outline.

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Flcuns 1. RnueRr. Under the surjectionindicatedabovethe numberof preimages of a closedcurveis equalto the numberof topologicallydifferent choicesof one of the sidesof its outline (which is equalto the numberof sidesof the outline divided by the order of the cyclicsymmetryof the curve). Exnuplr 1. The curye in Figure 2 has two preimages.The curve in Figure 3 hasone preimage. DnnrNnloN 5. An attachingdiagram with n self-intersectionpoints is a diagramof the type shownin Figure4. It consistsof 2n pointson a straight o'upper"and "lower" line and of n nonintersecting arcsconnectingn pairs points. of these DrrnrrrroN 6. An (admissible)path in a diagramis a (continuous)path,

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Frcune 5. DnrINrrtoN 7. A diagramis irreducibleif it doesnot contain closedpaths. This means that the admissiblepath that begins with the extreme left

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the line' haifline, passesall (2n + 1) segrnentsof is irreducible (the only admissible Ex.q.t"{ple3. The diagram in Figure 5 is reducible' path is shown there)' The diagram in Figure 6

ilu*"u' (or that it is an [attaching] DnrwrrroN 8. we saythat a diagram is normal diagram of a long curve) tf: (1) it is irreducible; ( 2 )th e d i a g ra mo b ta i n edbyetim inatingthear cthatgoesfr om theex of this arc) is normal' treme lefr point (and eliminating both ends without intersectionpoints) Rprr{.4nr.The empty diagram (i.e., the diagram is consideredother normal by defrnition' normal' The diagram in Figure 7 Exevrpr.E4. The diagramin Figure 5 is path in it), but is not ngrmal is irreducible (one .un ,.. the only admissible the (reducible) diagram in (the elimination of the extreme left arc leads to Figure 6.

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the setof diagrams onto the SrnreueN.r 7. There existsa surjection of that maps a diagram into the curve set of alt (possiblyreducible)tong curves geometric curve in the way shown in obtained by replacing each arc by the Figare 8. curve with a self-intersection Pnoon. Let L be any (possiblyreducible)long point p in one of the ways point p . Letus substituii a neighborhoodof the of the curve outside it)' The shown in Figure 9 (without any modification resorutionof the self-intersection resurtof such a reconstructionis caled the point there point p. It is not difficult to see that for any self-intersection

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always exists a resolution which is a (possibly reducible) long curve, i.e., at least one of two possibleresolutionsdoes not destroy the connectivity of the image of the curye. The resolved curve has one self-intersectionpoint less than the initial one. The initial curve can be obtained from the resolved one by pasting in the fragment shown in Figure 8. Hence the proof can be obtained by induction over the number of self-intersectionpoints.

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ON THE ENUMERATION OF CURVESFROM INFINITY TO INFINITY

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is a s s ig n e dth e (3) a segmentwith the index i f o r wh ic h l< i< 2 k - l in d ex(2k +l -t); (4) the segmentwith the tmdex i > 2k - I , ls assignedthe index (i + 2) : Reuenr. lf k : I , then the segmentsof the extremeleft half-line are assignedthe indices I , 2 , 3 (from left to right), the indicesof all the other are increasedbY 2. segments Rnua.Rr. The number of normal diagrams (and hence the number of classesof long curves)that can be obtained from a given normal diagram with (with (n - l) points of intersection)by addingan arc with one of the ends in the extremeleft half-line is equal to the number of .segmentswith odd indecesthat are "open" (i.e., not closedby arcs) from aboveor from below (if a segmentis open both from aboveor from below, then it has to be countedtwice). In particular,this number is at least 4 and at most 2n . Exeupu 6. The enumerationof long curveswith small numbersof points of self-intersection.Let #, be the number of classesof long curveswith n points' we have: #o : 1 , #r:2 ' #z: 8 (seeFigure12)' self-intersection

FIcunn 12. In the set of normal diagrams(and hencein the set of classesof long curves)there is a free action of the involution given by the reflection with respectto the straiglrttine (in Figure 12 the right diagramsare reflectionsof the left ones). We can decreasethe number of picturesdrawing only "left" diagrams.

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13' Neareachof the diagrams We have #u * 42 (:2'21); seeFigure that are openfrom aboveor from we indicatethe numberof odd segments with 4 self-intersectionpoints is equal below. The numbet #o of long curves +5.2+6.8+7.4+8.4):260. to thes umofthes enumbe r s#: q : 2 . ( 4 . 3 points :0, 1 , and 2 self.intersection A considerationof diagramswith n odd segmentsalternate' The diagram can createthe illusion that even and (*) showi that in generalthis is not the with n = 3 markedby the asterisk ca s e.Thenumber#,ofl o n g c u r v e s wit h 5 s e lf . in t e r s e c t io n p o in t si se q u a l to 1 796. points)is a par(with n self-intersection DnnNrrtor\i9. A quasi-diagram 2rr into pairs' tition of the setof integers| ,2, "' ' the straight b. regardedasa partitionof 2n pointsin A quasi-diagramrun of ru arcsconnectingthesewithline into pairs or (equivalently)as a system lower arcsand without the noninterout any distinction betweenupper and map from the set of diagramsto the sectionrequirement.There is a natural set of quasi-diagrams' Dnrwtrtoxl0.Anembeddingofaquasi-diagramisitspreimageunder is embeddabletf it has at least one the describ.o *up. A quasi-diagram embedding. of embeddingsof an embeddable It is not difficult to seethat the number diagramis equalto 2k with k > 1 ' alongcurve L:lR'l -*R'2 isthe DnRurrroN11. TheGaussdiagramof partition of the preimageof the set quasi-diagru* J.,.r.ined by the riatural of doublePointsinto Pairs' Rnuenr.Generallyspeaking,theGaussdiagramofacurvedoesnotcoto its diagramof the gluing' incide with the quasi-diagramcolresponding Letus s uppos ethatwe h a v e a q u a s i. d ia g r a m a ndefine d a f ixae new d a r cquasiin i t( sa y us i:-)'Let i i, ir: u"d i, indices with betweenpoints tn! points t'-t 1'-.': ' i'- | to the diagramby changingthe "o*t"i"g Jf t tespectitt"ty;'This meansthat if inverseone (i.e., from i, - l' to i, + an arc joining the points 7, and i' in the initial quasi-diagramthere was then in the new diagram i'with indic.s ,lr andj, betweenl, + 1 !' * iz If there was tr.- 'r1.and ,i' i, + points joini#the iZ' urc an is there an! iz an arc joining the points 7r and'i, ytll it ""]:11"-]1:.segment joining the the nt* q"u'i-diagrim ther6is an arc in then 1 + ,iz-l], [i, was an arc joining the points points with the samenumbers.Finally if there and i' outsidethesegment i,+1 and i'-] ,11and /, with J'1between joining the points newquasi-diagr;nthereis an arc the in then iz-ll, 1 + , [i, i'+ 1 ' then the new quasi-diagram ir' tn particular'if irl i, + i, - i, ""icoincideswith the initial one' procedurewill be calledthe described the of result The 12. DnrmrrroN arc' with respectto the distinguished(fixed) quasi-diagram initial ofthe flip

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DgpenrrurNrorGeocnerHY,MoscowSrerrUNrvgnsl.ry,Moscow,l19899Russn su E-mail address:sabir@nath ' imu 'msk ' Translatedby the author