On Realizability Of Gauss Diagrams And Constructions Of Meanders
aa r X i v : . [ m a t h . A T ] O c t ON REALIZABILITY OF GAUSS DIAGRAMS ANDCONSTRUCTIONS OF MEANDERS
Andrey Grinblat ∗ and Viktor Lopatkin † A BSTRACT . The problem of which Gauss diagram can be realized by knotsis an old one and has been solved in several ways. In this paper, we presenta direct approach to this problem. We show that the needed conditions forrealizability of a Gauss diagram can be interpreted as follows “the number ofexits = the number of entrances” and the sufficient condition is based on Jordancurve Theorem. Further, using matrixes we redefine conditions for realizabilityof Gauss diagrams and then we give an algorithm to construct meanders.
Mathematics Subject Classifications : 57M25, 14H50.
Key words : Gauss diagrams; Gauss code; realizability; plane curves. I NTRODUCTION
In the earliest time of the Knot Theory C.F. Gauss defined the chord diagram(= Gauss diagram). C.F. Gauss [3] observed that if a chord diagram can berealized by a plane curve, then every chord is crossed only by an even number ofchords, but that this condition is not sufficient.The aim of this paper is to present a direct approach to the problem of whichGauss diagram can be realized by knots. This problem is an old one, and hasbeen solved in several ways.In 1936, M. Dehn [2] found a sufficient algorithmic solution based on theexistence of a touch Jordan curve which is the image of a transformation of theknot diagram by successive splits replacing all the crossings. A long time afterin 1976, L. Lovasz and M.L. Marx [5] found a second necessary condition andfinally during the same year, R.C. Read and P. Rosenstiehl [7] found the thirdcondition which allowed the set of these three conditions to be sufficient. Thelast characterization is based on the tripartition of graphs into cycles, cocyclesand bicycles.In [8] the notation of oriented chord diagram was introduced and it wasshowed that these diagrams classify cellular generic curves on oriented surfaces.As a corollary a simple combinatorial classification of plane generic curves wasderived, and the problem of realizability of these diagrams was also solved. ∗ [email protected] † [email protected] (please use this e-mail for contacting.) owever all these ways are indirect; they rest upon deep and nontrivial aux-iliary construction. There is a natural question: whether one can arrive at theseconditions in a more direct and natural fashion?We believe that the conditions for realizability of a Gauss diagram (by someplane curve) should be obtained in a natural manner; they should be deducedfrom an intrinsic structure of the curve.In this paper, we suggest an approach, which satisfies the above principle.We use the fact that every Gauss diagram G defines a (virtual) plane curve C ( G ) (see [4, Theorem 1.A]), and the following simple ideas:(1) For every chord of a Gauss diagram G , we can associate a closed pathalong the curve C ( G ) .(2) For every two non-intersecting chords of a Gauss diagram G , we canassociate two closed paths along the curve C ( G ) such that every chordcrosses both of those chords correspondences to the point of intersectionof the paths.(3) If a Gauss diagram G is realizable (say by a plane curve C ( G ) ), thenfor every closed path (say) P along C ( G ) we can associate a coloringanother part of C ( G ) into two colors (roughly speaking we get “inner”and “outer” sides of P cf. Jordan curve Theorem). If a Gauss diagramis not realizable then ([4, Theorem 1.A]) it defines a virtual plane curve C ( G ) . We shall show that there exists a closed path along C ( G ) forwhich we cannot associate a well-defined coloring of C ( G ) , i.e., C ( G ) contains a path is colored into two colors.Using these ideas we solve the problem of which Gauss diagram can be real-ized by knots. We then give a matrix approach of realization of Gauss diagramsand then we present an algorithm to construct meanders.1. P RELIMINARIES
Recall that classically, a knot is defined as an embedding of the circle S into R , or equivalently into the 3-sphere S , i.e ., a knot is a closed curve embeddedon R (or S ) without intersecting itself, up to ambient isotopy.The projection of a knot onto a 2-manifold is considered with all multiplepoints are transversal double with will be call crossing points (or shortly cross-ings ). Such a projection is called the shadow by the knots theorists [1, 9], fol-lowing [8] we shall also call these projections as plane curves. A knot diagram is a generic immersion of a circle S to a plane R enhanced by information onoverpasses and underpasses at double points. IGURE
1. The plane curve and its Gauss diagram are shown.1.1.
Gauss Diagrams.
A generic immersion of a circle to a plane is character-ized by its Gauss diagram [6].
Definition 1.1.
The Gauss diagram is the immersing circle with the preimagesof each double point connected with a chord.On the other words, this natation can be defined as follows. Let us walk on apath along the plane curve until returning back to the origin and then generate aword W which is the sequence of the crossings in the order we meet them on thepath. W is a double occurrence word. If we put the labels of the crossing on acircle in the order of the word W and if we join by a chord all pairs of identicallabels then we obtain a chord diagram (=Gauss diagram) of the plane curve (seeF IGURE
A virtual knot diagram [4] is a generic immersion of the circle into the plane,with double points divided into real crossing points and virtual crossing points,with the real crossing points enhanced by information on overpasses and under-passes (as for classical knot diagrams). At a virtual crossing the branches are notdivided into an overpass and an underpass. The Gauss diagram of a virtual knotis constructed in the same way as for a classical knot, but all virtual crossings aredisregarded.
Theorem 1.2. [4, Theorem 1.A]
A Gauss diagram defines a virtual knot diagramup to virtual moves.
Arguing similarly as in the real knot case, one can define a shadow of thevirtual knot (see F
IGURE x y IGURE
2. The chord diagram and the shadow of the virtual knotare shown. Here x and y are the virtual crossing points.1.2. Conway’s Smoothing.
We frequently use the following notations. Let K be a knot, C its shadow and G the Gauss diagram of C . For every crossing c of C we denote by c the corresponding chord of G .If a Gauss diagram G contains a chord c then we write c ∈ G . We denote by c , c the endpoints of every chord c ∈ G . We shall also consider every chord c ∈ G together with one of two arcs are between its endpoints, and a chosen arcis denoted by c c .Further, c × denotes the set of all chords cross the chord c and c k denotes theset of all chords do not cross the chord c . We put c c × , and c ∈ c k . Throughout this paper we consider Gauss diagrams such that c × = ∅ for every c ∈ G . As well known, John Conway introduced a “surgical” operation on knots,called smoothing , consists in eliminating the crossing by interchanging the strands(F
IGURE
IGURE
3. The Conway smoothing the crossings are shown.We aim to specialize a Conway smoothing a crossing of a plane curve to anoperation on chords of the corresponding Gauss diagram. et K be a knot, C its shadow, and G the Gauss diagram of C . Take acrossing point c of C and let D c be a small disk centered at c such that D c ∩ C does not contain another crossings of C . Denote by ∂ D c the boundary of D c .Starting from c , let us walk on a path along the curve C until returning back to c .Denote this path by L c and let cc la c lz c be the sequence of the points in the orderwe meet them on L c , where { c la , c lz } = L c ∩ ∂ D c . After returning back to c letus keep walking along the curve C in the same direction as before until returningback to c . Denote the corresponding path by R c and let cc ra c rz c be the sequenceof the points in the order we meet them on R c , where { c ra , c rz } = ∂ D c ∩ R c . c c la c lz c ra c rz c la c lz c ra c rz c c IGURE
4. The Conway smoothing the crossing c and the chord c are shown.Next, let us delete the inner side of D c ∩ C and attach c al to c ar , and c rz to c lz . We thus get the new plane curve b C c (see F IGURE K by Conway’ssmoothing the crossing c . Let b G c be the Gauss diagram of b C c . We shall say hat the Gauss diagram b G c is obtained from the Gauss diagram G by Conway’ssmoothing the chord c . As an immediate consequence of the preceding discussion, we get the fol-lowing proposition.
Proposition 1.1.
Let G be a Gauss diagram and c be its arbitrary chord. Then b G c is obtained from G as follows: (1) delete the chord c , (2) if two chords a , b ∈ c × intersect (resp. do not intersected) in G then they do not intersect in b G c (resp.intersected), (3) another chords keep their positions.Proof. Indeed, let W be the word which is the sequence of the crossings in theorder we meet them on the curve C . Since c × = ∅ , W can be written as follows W = W cW cW , where W , W , W are subwords of W and at least one of W , W is not empty. Define W R as the reversal of the word W . Then, from the precedingdiscussion, the word b W c : = W W R W gives b G c (see F IGURE
4) and the statementfollows. (cid:3)
2. P
ARTITIONS OF G AUSS D IAGRAMS
In this section we introduce notations, whose importance will become clearas we proceed.
Definition 2.1.
Let G be a Gauss diagram and a a chord of G . A C-contour ,denoted C ( a ) , consists of the chord a , a chosen arc a a , and all chords of G such that all their endpoints lie on the arc a a . We call a chord from the set a × the door chord of the C-contour C ( a ) .Let us consider a plane curve C : S → R and let G be its Gauss diagram.Every chord c ∈ G correspondences to the crossing c of C . Thus for every C -contour C ( c ) , we can associate a closed path C ( c ) along the curve C . We call C ( c ) the loop of the curve C . It is obviously that there is the one-to-one corre-spondence between self-intersection points of C ( c ) and all chords from C ( c ) . Example 2.2.
In F
IGURE C and its Gauss diagram are shown.Consider the (cyan) C -contour C ( ) . We see that C ( ) is the closed path alongthe curve. It is the self-intersecting path and we see that the crossing 6 cor-respondences to the chord from the set 5 k . Further, the red closed path C ( ) correspondences to the red C -contour C ( ) . Definition 2.3.
Let G be a Gauss diagram, a , b its intersecting chords. An X -contour , denoted X ( a , b ) , consists of two non-intersecting arcs a b , a b andall chords of G such that all their endpoints lie on a b or on a b . A chordis called the door chord of the X -contour X ( a , b ) if only one of its endpointsbelongs to X ( a , b ) . We say that the X -contour X ( a , b ) is non-degenerate if it hasat least one door chord, and it does not contain all chords of G . IGURE
5. Every C -contour of the Gauss diagram correspon-dences to the closed path along the plane curve and vise versa.We see that the chord 6 correspondences to the self-intersectionpoint 6 of the dotted loop. Example 2.4.
Let us consider the Gauss diagram in F
IGURE
6. The black chordsare the door chords of the black X -contour X ( , ) . We see that the door chordscorrespondence to “entrances” and “exits” of the black closed path along thecurve. We also see that this X -contour is non-degenerate.The previous Example implies a partition of a Gauss diagram ( resp. a planecurve) into two parts. Definition 2.5 ( An X -contour coloring ) . Given a Gauss diagram G and its an X -contour. Let us walk along the circle of G in a chosen direction and color allarcs of G until returning back to the origin as follows: (1) we don’t colors thearcs of the X -contour, (2) we use only two different colors, (3) we change a colorwhenever we meet an endpoint of a door chord.Similarly, one can define a C -contour coloring of a Gauss diagram G . Remark 2.6.
Let G be a Gauss diagram and C the corresponding (may be vir-tual) plane curve, i.e., G determines the curve C . For every X -contour X ( a , b ) in G , we can associate the closed path along the curve C . We call this path the X -contour and denote by X ( a , b ) . Similarly one can define door crossing for X ( a , b ) .Further, for the X ( a , b ) -contour coloring of G , we can associate X ( a , b ) -contour coloring of the curve C .Next, let G be a realizable Gauss diagram determines the plane curve C andlet X ( a , b ) be an X -contour of G such that X ( a , b ) is the non-self-intersecting ath (= the Jordan curve). Then the X ( a , b ) -contour coloring of C divides thecurve C into two colored parts, cf. Jordan curve Theorem.7 8 92101 30 456
IGURE
6. For the X ( , ) -contour coloring of the Gauss dia-gram, we associate the plane curve coloring. We see that the X -contour X ( , ) (= black loop) divides the plane curve into twoparts. 3. T HE E VEN AND T HE S UFFICIENT C ONDITIONS
If a Gauss diagram can be realized by a plane curve we then say that thisGauss diagram is realizable, and non-realizable otherwise. So, in this section,we give a criterion allowing verification and comprehension of whether a givenGauss diagram is realizable or not. Moreover, we give an explanation allowingcomprehension of why the needed condition is not sufficient for realizability ofGauss diagrams.3.1.
The Even Condition.Proposition 3.1.
Let C : S → R be a plane curve and G its Gauss diagram.Then (1) | a × ∩ b × | ≡ for every two non-interesting chords a , b ∈ G , (2) | c × | ≡ for every chord c ∈ G .Proof. Let a , b ∈ G be two non-intersecting chords of G . Take two C -contours C ( a ) , C ( b ) such that their arcs a a , b b do not intersect. It is obvious that forthe loops C ( a ) , C ( b ) , we can associate the one-to-one correspondence betweenthe set C ( a ) ∩ C ( b ) and the set a × ∩ b × . ecause, by Proposition 1.1, all chord from the set a × ∩ b × keep their po-sitions in Gauss diagram b G c (= Conway’s smoothing the chord c ) for every c ∈ a k ∩ b k \ { a , b } , it is sufficient to prove the statement in the case a k ∩ b k = { a , b } , i.e., the loops C ( a ) , C ( b ) are non-self-intersecting loops (= the Jordan curves).From Jordan curve Theorem, it follows that the loop C ( a ) divides the curve C into two regions, say, I and O . Assume that b ∈ O and let us walk alongthe loop C ( b ) . We say that an intersection point c ∈ C ( a ) ∩ C ( b ) is the entrance( resp. the exit) if we shall be in the region I ( resp. O ) after meeting c withrespect to our walk. Since a number of entrances has to be equal to the numberof exits, then | a × ∩ b × | ≡ | c × | ≡ c . (cid:3) As an immediate consequence of Proposition 3.1 we get the following.
Corollary 3.1 ( The Even Condition).
If a Gauss diagram is realizable then thenumber of all chords that cross a both of non-intersecting chords and every chordis even (including zero).
We conclude this subsection with an explanation why the even condition isnot sufficient for realizability of Gauss diagrams.Roughly speaking, from the proof of Proposition 3.1 it follows that everyplane curve can be obtained by attaching its loops to each other by given points.Conversely, if a Gauss diagram satisfies the even condition then it may be non-releasible. Indeed, when we attach a loop, say, C ( b ) to a loop C ( a ) , where b ∈ a k , by given points (=elements of the set a × ∩ b × ) then the loop C ( b ) can be self-intersected curve, which means that we get new crossings (= virtual crossings),see F IGURE
Proposition 3.2.
Let G be a non-realizable Gauss diagram which satisfies theeven condition. Let G defines a virtual plane curve C (up to virtual moves).There exist two non-intersecting chords a , b ∈ G such that there are paths c → x → d, e → x → f on a loop C ( b ) , where c , d , e , f ∈ a × ∩ b × are different chordsand x is a virtual crossing of C .Proof. Let a , b ∈ G be two non-intersecting chords. Take non-intersecting C -contours C ( a ) , C ( b ) . Hence we may say that the loop C ( b ) attaches to the loop C ( a ) by the given points p , . . . , p n , where { p , . . . , p n } = a × ∩ b × . Since G isnot realizable and satisfies the even condition then a virtual crossing may ariseonly as a self-intersecting point of, say, the loop C ( b ) . Indeed, when we attach C ( b ) to C ( a ) by p , . . . , p n we may get self-interesting points, say, q , . . . , q m of the loop C ( b ) . If G contains all chords q , . . . , q m for every such chords a , b ,then G is realizable. Thus, a virtual crossing x does not belong to { p , . . . , p n } = C ( a ) ∩ C ( b ) for some non-intersecting chords a , b ∈ G . Then we get two paths → x → d , e → x → f , where c , d , e , f ∈ C ( a ) ∩ C ( b ) are different chords, asclaimed. (cid:3) x x x IGURE
7. It shows that the even condition is not sufficient forrealizability of the Gauss diagrams. We see that the plane curvecan be obtained by attaching the white-black loop to the dottedone by the points 3 , , ,
6, and thus the dotted loop has to have“new” crossings (= self-intersections) x , x , x .3.2. The Sufficient Condition.Definition 3.1.
Let G be a Gauss diagram (not necessarily realizable) and X ( a , b ) its X -contour. Take the X -contour coloring of G . A chord of G is called colorfulfor X ( a , b ) if its endpoints are in arcs which have different colors.Similarly, one can define a colorful chord for a C-contour C ( a ) of G . xample 3.2. Let us consider the Gauss diagram, which is shown in F
IGURE X -contour X ( , ) and the X ( , ) -coloring of G . The chord with theendpoints 5 is colorful for the X -contour X ( , ) . It is interesting to consider thecorresponding coloring of the virtual plane curve: one can think that we forget tochange color when we cross the gray loop, i.e., the gray loop “does not divide”the curve into two parts. We shall show that this observation is typical for everynon-realizable Gauss diagram. 7 x y z IGURE
8. This Gauss diagram satisfies even condition but isnon-realizable. There are colorful chords ( e.g. the chord withendpoints 5).We have seen that if a Gauss diagram is realizable then there is no color-ful chord, with respect to every X -contour. We shall show that it is sufficientcondition for realizability of a Gauss diagram. Proposition 3.3.
Let G be a non-realizable Gauss diagram but satisfy the evencondition. Then there exists an X -contour and a colorful chord for this X -contour.Proof. By Theorem 1.2, G defines a virtual curve C (= the shadow of a virtualknot diagram) up to virtual moves. Starting from a crossing, say, o , let us walkalong C till we meet the first virtual crossing, say, x . Next, let us keep walkingin the same direction till we meet the first real crossing, say d . Denote this pathby P . Just for convinces, let us put the labels, say, x , x of the virtual crossing x on the circle of G in the order we meet them on P .We can take two real crossings, say, a , b such that the path a → x → b doesnot contain another real crossings and a , b ∈ o × . Indeed, from Proposition 3.2 it ollows that for a chord o ∈ G we can find n ≥ o , . . . , o n such that, forevery 1 ≤ i ≤ n , we have: (1) the chords o , o i do not intersect, (2) the loop C ( o ) contains the following paths a i → x → b i , c i → x → d i where a i , b i , c i , d i ∈ o × ∩ o i × are different chords. Thus, for some 1 ≤ i , j ≤ n we have an arc, say, a i b j contains only one of x or x , no endpoints of another chords, and a i , b j ∈ o × .Denote this arc by a b and assume that x lies on a b .We next construct an X -contour contains the arc a b . To do so, we have toconsider the following two cases: Case 1.
The chords a , b intersect. a bc xdo o o a a c c b b d d x x F IGURE
9. Since the curve C is determined up to virtual movesthen the cyan line has to cross the olive line.Take the X -contour X ( a , b ) which contains the arc a b . It is easy to see thatthis X -contour has at least one real door chord, because its another arc a b hasno virtual crossing thus it has to have at least one real crossing (see F IGURE
Case 2.
The chords a , b do not intersect. e o bxo a d x c o o e o e o b b a x F IGURE
10. The “general” position of chords a , b , c , d is shown.Our walk along the path P correspondences to o → e o → b → · · · → d . et us walk along the circle of G in the direction b → x → a (F IGURE e o such that a , b , o ∈ e o × . If we cannot find suchchord we set e o : = o . Take the X -contour X ( e o , b ) contains the arc a b . This X -contour is non-degenerate. Indeed, let e o : = o and endpoints of all chords, whichstart in the arc b x d , lie on the arc o b . We then get the situation is shown inF IGURE xcdo b F IGURE
11. The dotted blue line gives another path from c to d .It follows that we can take another path P without the virtual crossing x .This contradiction implies that the X -contour X ( o , b ) is non-degenerate.So, we have a non-degenerate X -contour of the curve C such that x is itsvirtual door. If the X -contour of G contains another chords such that all theirendpoints lie on an arc of this X -contour, then Proposition 1.1 implies that Con-way’s smoothing all such chords does not change the X -contour coloring of P . Thus we may assume that this X -contour is the closed curve without self-intersections. Hence, by Jordan curve Theorem, it divides P into two parts. Letus color these parts into two different colors. If the crossing x would be a realdoor of this X -contour, then by Definition 2.5, we can take such X -contourcoloring of P as before. But since x is not real door then after meeting it at thesecond time we do not change the color and thus we get an intersection point oftwo lines which have different colors, i.e., the corresponding chord is colorful.This completes the proof. (cid:3) Lemma 3.1.
Let G be a Gauss diagram. Consider a C ( a ) -contour coloring of G for some chord a ∈ G . If there exists a colorful chord for the C-contour C ( a ) then the diagram G does not satisfy the even condition.Proof. Indeed, let b be a colorful chord for the C -contour C ( a ) . First note that,the chord b cannot cross a because otherwise b should be a door chord of the C -contour C ( a ) . Next, if the chord b is colorful then it crosses an odd number ofdoor chords of C ( a ) . Hence | a × ∩ b × | ≡ (cid:3) Example 3.3.
Let us consider the Gauss diagram which is shown in F
IGURE C -contour C ( ) and the C ( ) -contour coloring of the Gauss x y z
10 456 01271 0 6 5 4 2 5674F
IGURE
12. The Gauss diagram does not satisfy the even condi-tion; both the chords 1, 6 are crossed by only one chord 2.diagram and the corresponding (virtual) curve. We see that there are two chords(namely 5 and 6) which are colorful and 5 × ∩ × = × ∩ × = { } . Lemma 3.2.
Let G be a Gauss diagram and a , b ∈ G be its intersecting chords.Suppose that there exists a colorful chord c for an X -contour X ( a , b ) . Then thereexists a C-contour of the Gauss diagram b G b (= Conway’s smoothing the chord b ) such that the chord c is colorful for this C-contour in b G b .Proof. Indeed, consider the Gauss diagram b G b . From Proposition 1.1 it followsthat after Conway’s smoothing the chord b , the chord c does not intersect a andintersects the same door chords of the X -contour X ( a , b ) as in G . Further, let usconsider the C -contour C ( a ) in b G b such that it does not contain the chord c . ByProposition 1.1, the chord a crosses in b G b only the chord that are door chords ofthe X -contour X ( a , b ) . Hence, by Definition 2.5, we may take the C ( a ) -contourcoloring of b G b such that c is the colorful chord for this C -contour. (cid:3) Proposition 3.4.
Let a Gauss diagram G satisfy the even condition. Then G isrealizable if and only if b G c satisfies the even condition for every chord c ∈ G , b G c .Proof. Indeed, let G be a non-realizable Gauss diagram and let G satisfy theeven condition. By Proposition 3.3, there exists a colorful chord (say c ) for a X -counter X ( a , b ) of G . By Lemma 3.2, the chord c is the colorful chord in b G b .Hence from Lemma 3.1 it follows that b G b does not satisfy the even condition,and the statement follows. (cid:3) We can summarize our results in the following theorem. heorem 3.4. A Gauss diagram G is realizable if and only if the following con-ditions hold: (1) the number of all chords that cross a both of non-intersecting chords andevery chord is even (including zero), (2) for every chord c ∈ G the Gauss diagram b G c (= Conway’s smoothing thechord c ) also satisfies the above condition.
4. M
ATRIXES OF G AUSS D IAGRAMS
In this section we are going to translate into matrix language the conditionsof realizability of Gauss diagrams. We consider all matrixes over the field Z . Definition 4.1.
Given a Gauss diagram G contains n chords, say, c , . . . , c n . Con-struct a n × n matrix M ( G ) = ( m i , j ) ≤ i , j ≤ n as follows: (1) put m ii =
0, 1 ≤ i ≤ n ,(2) m i j = c i , c j intersect and m i j = M ( G ) is a symmetric matrix with respect to the maindiagonal.Next, let M ( G ) be a n × n matrix of a Gauss diagram G . Let ∪ ≤ i ≤ n { m i =( m i , . . . , m in ) } be all its strings. Define a scalar product of strings as follows: h m i , m j i : = m i m j + · · · + m in m jn . It is clear that we can define “scalar product” of any number of strings asfollows h m i , . . . , m i k i : = m i · · · m i k + · · · + m i n · · · m i k n . Lemma 4.1.
Let G be a Gauss diagram contains n chords c , . . . , c n . Considerthe corresponding matrix M ( G ) . Let the chords c , . . . , c n correspondence to thestrings m , . . . , m n , respectively. | c i × ∩ · · · ∩ c i k × | = h m i , . . . , m i k i , for every { i , . . . , i k } ⊆ { , , . . . , n } . Proof.
Indeed, let m i t · · · m i k t =
0, for some 1 ≤ t ≤ n , then the chord c t intersectchords c i , . . . , c i k . This completes the proof. (cid:3) Theorem 4.2.
Let G be a realizable Gauss diagram contains n chords. Then itsmatrix M ( G ) satisfies the following conditions: (1) h m i , m i i ≡ ( ) , ≤ i ≤ n, (2) h m i , m j i ≡ ( ) , if the corresponding chords do not intersect, h m i , m j i + h m i , m k i + h m j , m k i ≡ ( ) , if the corresponding chordspairwise intersect.Proof. The first two conditions follow from the first condition of Theorem 3.4and Lemma 4.1.Assume that G does not contain three pairwise intersecting chords. Then G is realizable if and only if the first condition of Theorem 3.4 holds and we getthe first two conditions for M ( G ) .Next, let a , b , c be three pairwise intersecting chords. Consider the followingsets of chords (see F IGURE A : = a × \ { b × , c × } , B : = b × ∩ c × \ {{ a } , a × } ) . bbc ca a A ABB F IGURE
13. The sets A , B are roughly shown.By Proposition 1.1, chords b , c does not intersect in b G a , and the set b × ∩ c ofchords in b G a is equal to the set A ∪ B . Hence, by Theorem 3.4, | A | + | B | ≡ ( ) . Further, using Lemma 4.1, we obtain: | A | = h a , a i − h a , b i − h a , c i − h a , b , c i , | B | = h b , c i − − h a , b , c i . It follows that | A | + | B | = h a , a i − h a , b i − h a , c i − h a , b , c i + h b , c i − − h a , b , c i≡ h a , b i + h a , c i + h b , c i≡ ( ) , because, as we have already discussed, h a , a i ≡ ( ) . As claimed. (cid:3) emark 4.3. The number of matrix satisfy the above conditions is bigger than anumber of realizable Gauss diagrams. The reason is the following observation.A matrix which satisfies the above conditions “knows” only intersections butdon’t know positions of chords. However there is a very important sort of planecurves (=meanderes) such that there is a one-to-one correspondence betweenGauss diagrams of these curves and the corresponding matrixes.5. T
HURSTON G ENERATORS OF B RAID G ROUPS
We will use as generators for B n the set of positive crossings , that is, thecrossings between two (necessary adjacent) strands, with the front strand havinga positive slope. We denote these generators by σ , . . . , σ n − . These generatorsare subject to the following relations: ( σ i σ j = σ j σ i , if | i − j | > , σ i σ i + σ i = σ i + σ i σ i + . One obvious invariant of an isotopy of a braid is the permutation it induceson the order of the strands: given a braid B , the strands define a map p ( B ) fromthe top set of endpoints to the bottom set of endpoints, which we interpret as apermutation of { , . . . , n } . In this way we get a homomorphism p : B n → S n ,where S n is the symmetric group. The generator σ i is mapped to the transposi-tion s i = ( i , i + ) . We denote by S n = { s , . . . , s n − } the set of generators for thesymmetric group S n .Now we want to define an inverse map p − : S n → B n . To this end, we needthe following definition [10, p.183] Definition 5.1.
Let S = { s , . . . , s n − } be the set of generators for S n . Eachpermutation π gives rise to a total order relation ≤ π on { , . . . , n } with i ≤ π j if π ( i ) < π ( j ) . We set R π : = { ( i , j ) ∈ { , . . . , n } × { , . . . , n }| i < j , π ( i ) > π ( j ) } . Lemma 5.1. [10, Lemma 9.1.6]
A set R of pairs ( i , j ) , with i < j, comes fromsome permutation if and only if the following two conditions are satisfied: i) If ( i , j ) ∈ R and ( j , k ) ∈ R, then ( i , k ) ∈ R. ii) If ( i , k ) ∈ R, then ( i , j ) ∈ R or ( j , k ) ∈ R for every j with i < j < k. Now we will define ([10, p. 186]) a very important concept of non-repeatingbraid (=simple braid).
Definition 5.2.
Recall that our set of generators S n includes only positive cross-ings; the positive braid monoid is denoted by B + = B + n . We call a positivebraid non-repeating if any two of its strands cross at most once. We define D = D n ⊂ B + n as the set of classes of non-repeating braids. he following lemma summarizes all the above mentioned concepts and no-tations. Lemma 5.2. [10, Lemma 9.1.10 and Lemma 9.1.11]
The homomorphism p : B + n → S n is restricted to a bijection D → S n . A positive braid B is non-repeatingif and only if | B | = | p ( B ) | (here | ? | means the length of a word). If a non-repeating braid maps to a permutation π , two strands i and j cross if and only if ( i , j ) ∈ R π . Example 5.3.
Let us consider the following permutation π = (cid:18) (cid:19) ∈ S . We obtain ( < , π ( ) > π ( ) , ( < , π ( ) > π ( ) , ( < , π ( ) > π ( ) , ( < , π ( ) > π ( ) , ( < , π ( ) > π ( ) , ( < , π ( ) > π ( ) , ( < , π ( ) > π ( ) , ( < , π ( ) > π ( ) . hence R ( π ) = { ( , ) , ( , ) , ( , ) , ( , ) , ( , ) , ( , ) , ( , ) , ( , ) } , and we get a non-repeating braid is shown in F IGURE
IGURE
14. The simple braid R π is shown. IGURE
15. An Example of a closed meander.6. M
EANDERS AND ITS G AUSS D IAGRAMS
In this section we deal with closed meanders. We show that any closed mean-der defines a special sort of Gauss diagrams (= Gauss diagrams of meanders) andthen we will see that these diagrams are completely determined by its matrixes(= meander matrix) i.e., there is a one-to-one correspondence between mean-der matrixes satisfy the conditions of Theorem 4.2. It allows us to describe analgorithm constructs all closed meanders.
Definition 6.1.
A (closed) meander is a planar configuration consisting of a sim-ple closed curve and on orientied line, that cross finitely many times and intersectonly transversally. Two meanders are equivalent if there exists a homeomor-phism of the plane that maps one to the other.
Definition 6.2.
A matrix M = ( m i j ) ∈ Mat n + ( Z ) is called meander matrix ifthe following conditions hold:(1) its main diagonal contains only 0,(2) M is symmetric,(3) it has at leas one string m i = ( m i , . . . , m i , n + ) with m ik = ≤ k ≤ n + k = i ,(4) M satisfies the conditions of Theorem 4.2,(5) if m i j = m jk = m ik = ≤ i , j , k ≤ n + m ik =
1, then m i j = m jk = j with 1 ≤ i < j < k ≤ n + Proposition 6.1.
Every meander matrix determines a unique closed meander(up to the equivalence). IGURE
16. A Gauss diagram of the meander, which is shownin F
IGURE .15, is shown
Proof.
Let M be a meander matrix. Take n + R and markthem by numbers 1 , , . . . , n +
1. Conditions (1) and (2) imply that the matrix M can be considered as a matrix of intersection of the lines: m i j = i , j intersect and m i j = r , such that it crossed by allother lines. Next, by Lemmas 5.1, 5.2, and by conditions (5), (6), all lines withexception of the line is marked by r can be considered as a non-repeating braid.This gives rise a Gauss diagram (see F IGURE
16. Condition (4) implies that thisGauss diagram is releasable by a plane curve, say C . Finally, the chord, whichis marked by r , correspondences to a loop C ( r ) of C . It is obviously that thisloop contains all crossing of the curve C . We then get the meander C up to theequivalence, as claimed. (cid:3)
7. C
ONSTRUCTION OF M EANDERS
We start with the following example.
Example 7.1.
Let us consider the meander is shown in F
IGURE
15, its Gaussdiagram is shown in F
IGURE
16. The correspondence meander matrix has thefollowing form
16 1 1 1 1 1 1 his matrix is constructed as follows. Take a table with seven rows and sevencolumns. Number its columns from left to right and rows from top to bottom bynumbers 0 , , . . . ,
6. Fill the main diagonal by zeros.0 1 2 3 4 5 60 Since we start from 0 and go then to 3, then the chord 3 intersects chords0 , , ,
6, and hence we get 0 1 2 3 4 5 60
02 1
03 1 0 0 Next, we go to 4 and then the chord 4 crosses chords 0 , , , and etc. Note that we changed the parity of numbers of strings.
Lemma 7.1.
To make by step-by-step a meander matrix, the following hold: (1) every string is filled as follows: every its empty cell is to the left of themain diagonal is filled by and every its empty cell is to the right of themain diagonal is filled by , j F IGURE
17. If i , j have the same parity there are odd number ofpoints between them then.(2) to fill the next string its number has to have another parity than a numberof the previous string.Proof. Indeed, assume that we go from i to j in a meander. If they have thesame parity then there are odd number of points between them F IGURE .17 . Thisimplies that the trajectory has to have self intersection points, i.e., we get a nomeander. Hence the numbers i , j have to have different parity. This gives condi-tion (2).Next, as we have already seen in the previous example, after choosing a stringwith number, say, i , the corresponding chord has to intersect chords with num-bers 0 , i + , . . . , n +
1, here n is the number of all chords. This gives condition(1). (cid:3) Definition 7.2.
A string which is filled as in condition (1) of Lemma 7.1 is called ∆ -filled. Lemma 7.2.
Let in a step-by-step filling of a matrix a string with number i is ∆ -filled. Then (cid:10) i , j (cid:11) ≡ ( ) , for any j > i.Proof. Indeed, by Lemma 7.1, m i j = j > i in the matrix M , and thenby condition (3) of Theorem 4.2, (cid:10) , i (cid:11) + (cid:10) , j (cid:11) + (cid:10) i , j (cid:11) ≡ ( ) . We thus get (cid:10) , i (cid:11) + (cid:10) , j (cid:11) + (cid:10) i , j (cid:11) = (cid:10) i , j (cid:11) , because of (cid:10) , i (cid:11) = (cid:10) i , i (cid:11) −
1, 1 ≤ i ≤ n , as claimed. (cid:3) Theorem 7.3.
Let M ∈ Mat n ( Z ) be a symmetric matrix with zero main diago-nal. M is a meander matrix if and only if the following hold: (1) there is a string m i such that m i j = if j = i and m ii = , (2) if m i j = , m jk = , then m ik = , for all i , j , k ∈ { , , . . . , n } , (3) if m ik = then m i j = or m jk = for all i ≤ j ≤ k, for every ≤ i , j ≤ n we have (cid:10) m i , m j (cid:11) ≡ ( ) , if i = j or m i j = , (7.1) (cid:10) m i , m j (cid:11) ≡ ( ) , if m i j = . (7.2) Proof.
By Definition 6.2 and Proposition 6.1, it is sufficient to prove that con-dition (3) of Theorem 4.2 is equivalent to the last condition. It is clear that anequality (cid:10) m i , m i (cid:11) ≡ ( ) implies that only an even number (including zero)of chords intersect chord correspondences to string m i . Next, an equality m i j = i do not intersect chord j , and an equality m i j = i , j . We thus get the firsttwo conditions of Theorem 4.2.Further, by Lemma 7.2, from the third condition of Theorem 4.2 it fol-lows (7.1). Assume now that (7.1) holds and consider three pairwise intersect-ing chords, say i , j , k . Then (cid:10) m i , m j (cid:11) ≡ ( ) , (cid:10) m i , m k (cid:11) ≡ ( ) , and (cid:10) m j , m k (cid:11) ≡ ( ) and we complete the proof. (cid:3) From this Theorem it follows the following meanders construction algorithm.
REQUIRE an even number N , S = { , , . . . , N − } , S = { , , . . . , N } ; ENSURE n , . . . , n N ∈ S ∪ S ; . take a ( N + ) × ( N + ) tableau with empty cells and fill the main diag-onal by zeros; . number strings from left to right, and number columns from top to bot-tom by numbers 0 , , . . . , N ; . ∆ -fill string with number 0; . choose a string with an odd number n ∈ S and ∆ -fill it and column withthe same number; . i = . PRINT n ; . S i : = S i \ { n } ; . IF S i = ∅ THEN GOTO ELSE GOTO ; i : = i + ( ) ; . choose a string (a column) with number m ∈ S i ; . IF the string (the column) can be ∆ -filled THEN GOTO ELSE chooseanother m ′ ∈ S i \ { m } GOTO ; . using (7.1), (7.2) get a system of equations for empty cells; . IF the system can by solved THEN GOTO ELSE choose a string (col-umn) with another number m ′ ∈ S i \ { m } GOTO ; . END
Example 7.4.
Let us construct a 9 × × (1) Choose a string and column 5 and ∆ -fill them0 1 2 3 4 5 6 7 80
02 1
03 1
04 1
05 1 0 0 0 0 By(7.2), (cid:10) , (cid:11) = + ( , ) + ( , ) ≡ ( ) , (cid:10) , (cid:11) = + ( , ) + ( , ) ≡ ( ) , (cid:10) , (cid:11) = + ( , ) + ( , ) ≡ ( ) , t implies that ( , ) = ( , ) = ( , ) = a ∈ { , } , and we thus get 0 1 2 3 4 5 6 7 80
02 1
03 1
04 1
05 1 0 0 0 0 a a a a a a (2) We have to choose a string (column) with an even number.(i) Take a string (column) 2, we then get ( , ) = , ( , ) = ( , ) = , ( , ) = ( , ) = ( , ) = , it follows that (cid:10) , (cid:11) ≡ ( ) . We have (cid:10) , (cid:11) = + ( , ) + ( , )+( , ) + ( , ) + ( , ) ≡ ( ) , and we do not get any contradiction, hence this string may be cho-sen.(ii) Similarly, by the straightforward verification, one can easy verifythat strings 4, 6 can be chosen.(iii) Take string 8. We get ( , ) = ( , ) = ( , ) = ( , ) = = a = , and hence (cid:10) , (cid:11) ≡ ( ) , but the tableau implies that (cid:10) , (cid:11) =
1. Therefore this string cannot be chosen in this step.(3) Chose string 2. We then get
04 1 1
05 1 0 0 0 0 a a a a a a by (7.2), (cid:10) , (cid:11) = + ( , ) + ( , ) + ( , ) + ( , ) ≡ ( ) , (cid:10) , (cid:11) = + ( , ) + ( , ) + ( , ) + ( , ) ≡ ( ) , (cid:10) , (cid:11) = + ( , ) + ( , ) + a + a ≡ ( ) , (cid:10) , (cid:11) = + ( , ) + ( , ) + a + a ≡ ( ) , (cid:10) , (cid:11) = + ( , ) + ( , ) + a + a ≡ ( ) , and we then obtain ( , ) = ( , ) = b ∈ { , } ( , ) = ( , ) = c ∈ { , } ( , ) = ( , ) = d ∈ { , } ( , ) = b + c + d mod ( ) . Next, using the condition of Theorem 4.2, (cid:10) , (cid:11) ≡ ( ) , we get ( , ) =
0. Similarly, one can get ( , ) =
0. Using (cid:10) , (cid:11) ≡ (cid:10) , (cid:11) ≡ (cid:10) , (cid:11) ≡ ( ) we get ( , ) = ( , ) = ( , ) = b c d b c d b b a a c c a a d d a a (4) We have to chose a string with an odd number 1 , ( , ) = ∆ -filled.Next, if we chose string 7 we then get ( , ) = ( , ) =
1, but ( , ) =( , ) = a , we then get a contradiction. Further, one can easy verify thatstring 3 can be chosen.(5) Take string 3. It follows that b = c = d = a a a a a a We see that strings 1 , ∆ -filled.So, we have two possibilities: 1) a =
0, and 2) a =
1. These cases cor-respondence to the following possibilities for our meander (see F
IGURE IGURE
18. The meanders which are correspondence to case a = a = EFERENCES [1] C. Adams,
The Knot Book,
New York: Freeman & Co, 1994.[2] M. Dehn, ¨Uber Kombinatorische Topologie,
Acta Math, (1936), 123–168.[3] C.F. Gauss, Werke,
Teubner, Leipzig,
VII (1900), 272, 282–286.[4] M. Goussarov, M. Polyak and O. Viro, Finite type invariants of classical and virtual knots,
Topology, (2000), 1045–1068.[5] L. Lovasz and M.L. Marx, A forbidden subgraph characterization of Gauss code, Bull.Amer. Math. Soc, (1976), 121–122.[6] M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, InternationalMathematics Research Notes, (1994), 445–453.[7] R.C. Read and P. Rosenstiehl, On the Gauss Crossing Problem, Colloquia MathematicaSocietatis Janos Bolyai , 1976.[8] A. Sossinsky, A Classification of Generic Curves on Oriented Surfaces,
Russian Journal ofMathematical Physics, (2) (1994), 251–260.[9] A. Sossinsky, Knots: Mathematics with a Twist,
Harvard University Press, 2002.
10] D.B.A. Epstein, I.W. Cannon, D.E. Holt, S.V.F. Levy, M.S. Paterson and W.P. Thurston,Word Processing in Groups, Jones and Bartlett Publishers, INC., 19910] D.B.A. Epstein, I.W. Cannon, D.E. Holt, S.V.F. Levy, M.S. Paterson and W.P. Thurston,Word Processing in Groups, Jones and Bartlett Publishers, INC., 199