Kirby diagrams and 5-colored graphs representing compact 4-manifolds
KKirby diagrams and 5-colored graphs representing compact4-manifolds
Maria Rita Casali and Paola Cristofori Department of Physics, Mathematics and Computer Science, University of Modena and Reggio EmiliaVia Campi 213 B, I-41125 Modena (Italy), [email protected] Department of Physics, Mathematics and Computer Science, University of Modena and Reggio EmiliaVia Campi 213 B, I-41125 Modena (Italy), [email protected]
February 18, 2021
Abstract
It is well-known that any framed link (
L, c ) uniquely represents the 3-manifold M ( L, c ) obtainedfrom S by Dehn surgery along ( L, c ), as well as the PL 4-manifold M ( L, c ) obtained from D byadding 2-handles along ( L, c ), whose boundary coincides with M ( L, c ). In this paper we study therelationships between the above representation tool in dimension 3 and 4, and the representationtheory of compact PL manifolds of arbitrary dimension by edge-coloured graphs: in particular, wedescribe how to construct a (regular) 5-colored graph representing M ( L, c ), directly “drawn over” aplanar diagram of (
L, c ). As a consequence, the combinatorial properties of the framed link (
L, c ) yieldupper bounds for both the invariants gem-complexity and (generalized) regular genus of M ( L, c ). Keywords : framed link, Kirby diagram, PL 4-manifold, handle decomposition, edge-colored graph,regular genus, gem-complexity. : 57K40 - 57M15 - 57K10 - 57Q15.
Among combinatorial tools representing PL manifolds, framed links (and/or
Kirby diagrams ) turn outto be a very synthetic one, both in the 3-dimensional setting and in the 4-dimensional one, while edge-colored graphs have the advantage to represent all compact PL manifolds and to allow the definitionand computation of interesting PL invariants in arbitrary dimension (such as the regular genus , whichextends the Heegard genus, and the gem-complexity , similar to Matveev’s complexity of a 3-manifold).Previous works exist establishing a connection between the two theories, both in the 3-dimensionaland 4-dimensional setting ([23], [4], [6]): they make use of the so called edge colored graphs withboundary , which are dual to colored triangulations of PL manifolds with non-empty boundary, andfail to be regular. More recently, a unifying method has been introduced and studied, so to representby means of regular colored graphs all compact PL manifolds, via the notion of singular manifold associated to a PL manifold with non-empty boundary.Purpose of the present work is to update the relationship between framed links (and Kirby di-agrams) and colored graph in dimension 4, by making use of regular 5-colored graphs representingcompact PL 4-manifolds. The new tool turns out to be significantly more efficient than the classic one,both with regard to the simplicity of the procedure and with regard to the possibility of estimatinggraph-defined PL invariants directly from the Kirby diagram. a r X i v : . [ m a t h . G T ] F e b s it is well-known, a framed link is a pair ( L, c ), where L is a link in S with l ≥ c = ( c , c , . . . , c l ), is an l -tuple of integers. ( L, c ) represents - in dimension 3 - the 3-manifold M ( L, c ) obtained from S by Dehn surgery along ( L, c ), as well as - in dimension 4 - the (simply-connected) PL 4-manifold M ( L, c ), whose boundary coincides with M ( L, c ), obtained from D byadding 2-handles along ( L, c ).Moreover, in virtue of a celebrated result by [26] and [22], in case M ( L, c ) = r ( S × S ) (with r ≥ L, c ) represents also the closed PL 4-manifold M ( L, c ) obtained from M ( L, c ) by adding - in a unique way - r M admits a framed link ( L, c ) so that M = M ( L, c ), it is an open question whether or not each closed simply-connected PL 4-manifold M may be represented by a suitable framed link (or, equivalently, if M admits a so called specialhandle decomposition , i.e. a handle decomposition lacking in 1-handles: see [21, Problem 4.18] , [25],[9]).As far as general compact PL 4-manifolds (with empty or connected boundary) are concerned, it isnecessary to extend the notion of framed link, so to comprehend also the case of trivial (i.e. unknottedand unlinked) dotted components , which represent 1-handles of the associated handle decompositionof the manifold: in this way, any framed link ( L, d ) possibly admitting trivial dotted components -which is properly called a
Kirby diagram - uniquely represents the compact PL 4-manifold M ( L, d )obtained from D by adding 1-handles according to the trivial dotted components and 2-handles alongthe framed components. Note that the boundary of M ( L, d ) coincides with M ( L, c ), (
L, c ) beingthe framed link obtained from the Kirby diagram (
L, d ) by substituting each dotted component with a0-framed one); hence, in case M ( L, c ) = r ( S × S ) (with r ≥ L, d ) uniquelyrepresents also the closed PL 4-manifold M ( L, d ) obtained from M ( L, d ) by adding - in a uniqueway - r M ( L, c ) directly“drawn over” a self-framed diagram of (
L, c ); as a consequence, the procedure yields an upper bound- significantly improving the one obtained in [4] - for the regular genus of M ( L, c ), together withupper bounds for the gem-complexity of both M ( L, c ) and M ( L, c ). Various examples are presented,where the above upper bounds turn out to be sharp. Moreover, the process is applied in order toobtain a pair of 5-colored graphs representing an exotic pair of compact PL 4-manifolds (i.e. a pair of4-manifolds which are TOP-homeomorphic but not PL-homeomorphic), thus opening the search forpossibile graph-defined PL invariants distinguishing them.We hope soon to be able to extend the procedure to the case of Kirby diagrams including dottedcomponents, too, as well as to the associated closed PL 4-manifolds.
In this section we will briefly recall some basic notions about the representation of compact PLmanifolds by regular colored graphs ( crystallization theory ). For more details we refer to the surveypapers [17] and [10].From now on, unless otherwise stated, all spaces and maps will be considered in the PL category,and all manifolds will be assumed to be compact, connected and orientable .Crystallization theory was first developed for closed manifolds; the extension to the case of non-empty boundary, that is more recent, is performed by making use of the wider class of singularmanifolds. Definition 1 A singular (PL) n -manifold is a closed connected n -dimensional polyhedron admittinga simplicial triangulation where the links of vertices are closed connected ( n − h -simplices of the triangulation with h > n − h − Actually all concepts and results exist also, with suitable adaptations, for non-orientable manifolds; however, since thepresent paper focuses on the relationship between Kirby diagrams and colored graphs, we will restrict to the orientable case. emark 1 If N is a singular n -manifold, then a compact n -manifold ˇ N is easily obtained by deletingsmall open neighbourhoods of its singular vertices. Obviously N = ˇ N iff N is a closed manifold,otherwise ˇ N has non-empty boundary (without spherical components). Conversely, given a compact n -manifold M , a singular n -manifold (cid:99) M can be constructed by capping off each component of ∂M bya cone over it.Note that, by restricting ourselves to the class of compact n -manifolds with no spherical bound-ary components, the above correspondence is bijective and so singular n -manifolds and compact n -manifolds of this class can be associated to each other in a well-defined way.For this reason, throughout the present work, we will make a further restriction considering onlycompact manifolds without spherical boundary components. Obviously, in this wider context, closed n -manifolds are characterized by M = (cid:99) M .
Definition 2
An ( n +1) -colored graph ( n ≥
2) is a pair (Γ , γ ), where Γ = ( V (Γ) , E (Γ)) is a multigraph(i.e. multiple edges are allowed, but no loops) which is regular of degree n + 1, and γ : E (Γ) → ∆ n = { , . . . , n } is a map which is injective on adjacent edges ( edge-coloration ).In the following, for sake of concision, when the coloration is clearly understood, we will drop itin the notation for a colored graph.For every { c , . . . , c h } ⊆ ∆ n let Γ { c ,...,c h } be the subgraph obtained from Γ by deleting all the edgesthat are not colored by { c , . . . , c h } . Furthermore, the complementary set of { c } (resp. { c , . . . , c h } )in ∆ n will be denoted by ˆ c (resp. ˆ c · · · ˆ c h ). The connected components of Γ { c ,...,c h } are called { c , . . . , c h } -residues of Γ; their number will be denoted by g { c ,...,c h } (or, for short, by g c ,c , g c ,c ,c and g ˆ c if h = 2 , h = 3 and h = n respectively).For any ( n + 1)-colored graph Γ, an n -dimensional simplicial cell-complex K (Γ) can be constructedin the following way: • the n -simplexes of K (Γ) are in bijective correspondence with the vertices of Γ and each n -simplexhas its vertices injectively labeled by the elements of ∆ n ; • if two vertices of Γ are c -adjacent ( c ∈ ∆ n ), the ( n − n -simplices that opposite to the c -labeled vertices are identified, so that equally labeled verticescoincide. | K (Γ) | turns out to be an n -pseudomanifold and Γ is said to represent it.Furthermore, in virtue of the bijection described in Remark 1, an ( n + 1)-colored graph Γ is saidto represent a compact n -manifold M with no spherical boundary components (or, equivalently, tobe a gem of M , where gem means Graph Encoding Manifold ) if Γ represents its associated singularmanifold, i.e. if | K (Γ) | = (cid:99) M .Note that, by construction, Γ can be seen as the 1-skeleton of the dual complex of K (Γ). As aconsequence there is a bijection between the { c , . . . , c h } -residues of Γ and the ( n − h )-simplices of K (Γ) whose vertices are labeled by ˆ c · · · ˆ c h .In particular, given an ( n + 1)-colored graph Γ, each connected component of Γ ˆ c ( c ∈ ∆ n ) isan n -colored graph representing the disjoint link of a c -labeled vertex of K (Γ), that is also (PL)homeomorphic to the link of this vertex in the first barycentric subdivision of K. Therefore, we can characterize ( n + 1)-colored graphs representing singular (resp. closed) n -manifolds as satisfying the condition that for each color c ∈ ∆ n any ˆ c -residue represents a connectedclosed ( n − (resp. the ( n − Given a simplicial cell-complex K and an h -simplex σ h of K , the disjoint star of σ h in K is the simplicial cell-complexobtained by taking all n -simplices of K having σ h as a face and identifying only their faces that do not contain σ h . The disjoint link , lkd ( σ h , K ), of σ h in K is the subcomplex of the disjoint star formed by those simplices that do not intersect σ h . In case of polyhedra arising from colored graphs, the condition about links of vertices obviously implies the one aboutlinks of h -simplices, with h ≥ . he following theorem extends to the boundary case a well-known result - originally stated in [28]- founding the combinatorial representation theory for closed manifolds of arbitrary dimension viaregular colored graphs. Theorem 1 ([12])
Any compact orientable (resp. non-orientable) n -manifold with no spherical bound-ary components admits a bipartite (resp. non-bipartite) ( n + 1) -colored graph representing it. If Γ is a gem of a compact n -manifold, an n -residue of Γ will be called ordinary if it represents S n − , singular otherwise. Similarly, a color c will be called singular if at least one of the ˆ c -residues ofΓ is singular.An advantage of colored graphs as representing tools for compact n -manifolds is the possibility ofcombinatorially defining PL invariants.One of the most important and studied among them is the (generalized) regular genus extendingto higher dimension the classical genus of a surface and the Heegaard genus of a 3-manifold. Spheresare characterized by having null regular genus, while classification results according to regular genusand concerning 4- and 5-manifolds can be found in [13], [5], [10], [7] both for the closed and for thenon-empty boundary case.The definition of the invariant relies on the following result about the existence of a particulartype of embedding of colored graphs into closed surfaces. Proposition 2 ([18])
Let Γ be a bipartite ( n + 1) -colored graph of order p . Then for each cyclicpermutation ε = ( ε , . . . , ε n ) of ∆ n , up to inverse, there exists a cellular embedding, called regular , of Γ into an orientable closed surface F ε (Γ) whose regions are bounded by the images of the { ε j , ε j +1 } -colored cycles, for each j ∈ Z n +1 . Moreover, the genus ρ ε (Γ) of F ε (Γ) satisfies − ρ ε (Γ) = (cid:88) j ∈ Z n +1 g ε j ,ε j +1 + (1 − n ) p. (1) Definition 3
The regular genus of an ( n + 1)-colored graph Γ is defined as ρ (Γ) = min { ρ ε (Γ) | ε cyclic permutation of ∆ n } ;the (generalized) regular genus of a compact n -manifold M is defined as G ( M ) = min { ρ (Γ) | Γ gem of M } . Within crystallization theory a notion of “complexity” of a compact n -manifold arises naturallyand, similarly to other concepts of complexity (for example Matveev’s complexity for 3-manifolds) isrelated to the minimum number of n -simplexes in a colored triangulation of the manifold itself: Definition 4
The (generalized) gem-complexity of a compact n -manifold M is defined as k ( M ) = min { p − | ∃ a gem of M with 2 p vertices } Important tools in crystallization theory are combinatorial moves transforming colored graphswithout affecting the represented manifolds (see for example [17], [16], [3], [23], [24]); we will recallonly those that will be necessary in the following sections.
Definition 5 An r -dipole ( ≤ r ≤ n ) of colors c , . . . , c r in an ( n + 1)-colored graph Γ is a subgraphof Γ consisting in two vertices joined by r edges, colored by c , . . . , c r , such that the vertices belongto different ˆ c . . . ˆ c r -residues of Γ. An r -dipole can be eliminated from Γ by deleting the subgraph andwelding the remaining hanging edges according to their colors; in this way another ( n + 1)-coloredgraph Γ (cid:48) is obtained. The addition of the dipole to Γ (cid:48) is the inverse operation.The dipole is called proper if | K (Γ) | and | K (Γ (cid:48) ) | are PL-homeomorphic. Since this paper concerns only orientable manifolds, we have restricted the statement only to the bipartite case, althougha similar result holds also for non-bipartite graphs. roposition 3 ([19, Proposition 5.3]) An r -dipole ( ≤ r ≤ n ) of colors c , . . . , c r in an ( n +1) -coloredgraph Γ is proper if and only if at least one of the two connected components of Γ ˆ c ... ˆ c r intersectingthe dipole represents the ( n − r ) -sphere. Remark 2
Note that, if Γ represents a compact n -manifold M (or equivalently its associated singular n -manifold (cid:99) M ), then all r -dipoles with 1 < r ≤ n are proper; further, if M has either empty orconnected boundary, then 1-dipoles are proper, too.Given an arbitrary ( n + 1)-colored graph representing a compact n -manifold M with empty orconnected boundary, then by eliminating all possible (proper) 1-dipoles, we can always obtain an( n + 1)-colored graph Γ still representing M and such that for each color c ∈ ∆ n , Γ ˆ c is connected.Such a colored graph is called a crystallization of M. Moreover, it is always possible to assume - up topermutation of the color set - that any gem (and, in particular, any crystallization) of such a manifold,has color n as its (unique) possible singular color.Let us now recall a further type of move, which is limited to the closed 3-dimensional case. Definition 6
Let Γ be a 4-colored graph representing a closed 3-manifold. If there exists in Γ an { i, j } -colored cycle of length m + 1 and a { k, l } -colored cycle of length n + 1 (with { i, j, k, l } = ∆ )having exactly one common vertex ¯ v , then Γ is said to contain an (m,n)-generalized dipole of type { i, j } (or, equivalently, of type { k, l } ) at vertex ¯ v .An ( m, n )-generalized dipole can always be eliminated from the graph, though increasing its num-ber of vertices, by performing its cancellation (see [16] for details); an example of this move is depictedin Figure 1 for the case m = 3 and n = 5. a abb cc dd ee ff gg hh ii ll mm nn oo pp qq rr Figure 1:
The cancellation of a (3 , We will refer to the cancellation of a ( m, n )-generalized dipole and to its inverse procedure as generalized dipoles moves .It is well-known ([16]) that generalized dipoles moves do not change the represented closed 3-manifolds.
Remark 3
If Γ is a crystallization of a closed 3-manifold M , Heegaard diagrams of M can beassociated to Γ for each choice of a pair of colors (see [18] for details). The cancellation from Γ of ageneralized dipole of type { i, j } corresponds to a Singer move of type III (cid:48) involving a pair of curvesin one of the diagrams associated to the pair ( i, j ) . From framed links to -colored graphs In this section we will present a construction that enables to obtain 5-colored graphs representing everycompact (simply-connected) 4-manifold admitting a special handlebody decomposition ([25, Section3.3]), i.e. a handle decomposition containing no 1- and 3-handles. Equivalently such manifolds canbe represented by framed links, i.e. Kirby diagrams without “dotted” components.Given a framed link (
L, c ) ( c = ( c , . . . , c l ), with c i ∈ Z ∀ i ∈ { , . . . , l } , l being the number ofcomponents of L ), let M ( L, c ) denote the 4-manifold with boundary obtained from D by adding l L, c ). The boundary of M ( L, c ) is the closed orientable3-manifold M ( L, c ) obtained from S by Dehn surgery along ( L, c ). In case M ( L, c ) ∼ = S , we willconsider, and still denote by M ( L, c ) , the closed 4-manifold obtained by adding a further 4-handle.Let D be a diagram of ( L, c ); in the following we will always suppose D to be self-framed , i.e. foreach i ∈ { , . . . , l } , the writhe of the i -th component of L equals c i ; this hypothesis is not restrictivesince any diagram can be made so by possibly adding a suitable number of “additional curls“ (seeFigure 2).Let us now describe the steps of the construction of a 5-colored graph Γ( D , c ) associated to D :1. consider a 4-colored graph Λ( D , c ) constructed as follows:each crossing of D gives rise to the order eight graph in Figure 3, while each possible curl givesrise to one of the order four graphs of Figure 4-left or Figure 4-right according to the curl beingpositive or negative. Figure 2:
Positive (left) and negative (right) curls
Figure 3:
Figure 4:
The 0- and 1-colored edges of the above graphs should be “pasted” together so that every regionof D (having r crossings on its boundary) gives rise to a { , } -colored cycle of length 2 r , while P P P P Q Q Figure 5:
The 4-colored graph Λ( D , c ) representing M ( L, c ), for c = +1 and L = trefoil P P P P P P P P P P P P Figure 6: main step yielding Γ( D , c ) each component L i ( i ∈ { , . . . , l } ) of D , having s i crossings and t i additional curls, gives riseto two { , } -colored cycles of length 2( s i + t i ) . Actually Λ( D , c ) can be directly “drawn over” D as can be seen in Figure 5, which shows anexample where L is the trefoil knot and c = +1 .
2. For each component of D , by possibly adding a trivial pair of opposite additional curls, aparticular subgraph - called quadricolor - can be selected in Λ( D , c ) . A quadricolor consists offour vertices { P , P , P , P } such that P i , P i +1 are connected by an i -colored edge (for each i ∈ Z ) and P i does not belong to the { i + 1 , i + 2 } -colored cycle shared by the other verticesof the quadricolor (see Figure 6-left). It is not difficult to see that such a situation arises with { P , P , P } belonging to the subgraph corresponding to a curl and P to an adjacent crossingor curl.The 5-colored graph Γ( D , c ) is obtained from Λ( D , c ) by adding, for each component of D ,4-colored edges between the vertices of the chosen quadricolor as in Figure 6 and “doubling”by 4-colored edges the 1-colored ones between the remaining vertices.Figure 7 shows the 5-colored graph Γ( D , c ) in the case of the trefoil knot with framing +1. The 5-colored graph Γ( D , c ), for c = +1 and L = trefoil Let us now consider a chess-board coloration of D , by colors α and β say, with the convention thatthe infinite region is α -colored. We will denote by m α the number of α -colored regions.Theorem 2 of [4] states that Λ( D , c ) represents M ( L, c ). Moreover, in the same paper a 4-coloredgraph is obtained from Λ( D , c ) still representing M ( L, c ) and having regular genus equal to m α . With regard to the 5-colored graph Γ( D , c ), the following result can be stated: Theorem 4
Let ( L, c ) be a framed link with l components and let D be a planar diagram of the link L. Then:(i) the bipartite -colored graph Γ( D , c ) represents the compact -manifold M ( L, c ); (ii) via a standard sequence of graph moves, a bipartite -colored graph, still representing M ( L, c ) ,can be obtained, whose regular genus is m α + l , while the regular genus of its ˆ4 -residue, repre-senting ∂M ( L, c ) = M ( L, c ) , is m α . Theorem 4 will be proved in the next section.As already hinted to in Section 1, the presented construction yields upper bounds for both theinvariants regular genus and gem-complexity of a compact 4-manifold represented by a framed link(
L, c ), in terms of the combinatorial properties of the link itself.With this aim, let us denote by s the number of crossings of L , and let us set, for each i ∈ { , . . . , l } :¯ t i = (cid:40) | w i − c i | if w i (cid:54) = c i otherwise where w i denotes the writhe of the i -th component of L. Theorem 5
Let ( L, c ) be a framed link with l components and let D be a self-framed diagram of ( L, c ) . Then: G ( M ( L, c )) ≤ m α + l oreover, if D is different from the standard diagram of the trivial knot, k ( M ( L, c )) ≤ s − l (cid:88) i =1 ¯ t i Proof.
The lower bound for the regular genus of M ( L, c ) trivially follows from Theorem 4 (ii).As far as the gem-complexity estimation is concerned, first note that, by construction, the 4-coloredgraph Λ( D , c ) has 8 s + 4 (cid:80) li =1 | w i − c i | vertices. As observed above, the presence of a curl near acrossing in a component of the diagram yields a quadricolor. Therefore, for each i ∈ { , . . . , l } , if | w i − c i | (cid:54) = 0, then the addition of curls, required to get a self-framed diagram, ensures the existenceof a quadricolor relative to L i , while if | w i − c i | = 0 a pair of opposite curls has to be added in orderto produce one. Since each curl contributes with 4 vertices to the final 5-colored graph, the statementis directly proved. (cid:50) The case of the trivial knot is discussed in the following example.
Example 1
Let (
L, c ) be the trivial knot with framing c ∈ N ∪ { } ; if c ≥
2, the 5-colored graphΓ( D , c ), with 4 c vertices, which is obtained by applying the above construction to the standard diagramof ( L, c ) with c additional positive curls, turns out to coincide with the one that in [7] was proved torepresent exactly the D -bundle over S with Euler number c , as expected from Theorem 4.If c = 0 (resp. c = 1), the diagram D requires two positive and two negative (resp. two positiveand one negative) curls in order to get a quadricolor; however in this case the resulting graph Γ( D , c )admits a sequence of dipole moves consisting in three 3-dipoles and one 2-dipole (resp. consistingin two 3-dipoles) cancellations yielding a minimal order eight crystallization of S × D (resp. theminimal order eight crystallization of CP ) obtained in [7] (resp. in [20]).Note that all these graphs ( ∀ c ∈ N ∪ { } ) realize the regular genus of the represented 4-manifold,which is equal to 2 (= m α + l ), as proved in [7]. Hence, for this infinite family of compact 4-manifolds,the first upper bound of Theorem 5 turns out to be sharp.We will end this section with two further examples of the construction. Example 2
Let (
L, c ) be the Hopf link and c = (0 , S × S and realizes its regular genus (which is known to be equal to2, too: see [7] and references therein). This can also be proved directly since, by a sequence of dipolecancellations and a ρ -pair switching (see [8] for the definition), a 5-colored graph is obtained, whichbelongs to the existing catalogue of crystallizations of 4-manifolds up to gem-complexity 8 (see [8]). Example 3
The above construction applied to the framed link descriptions given in [27] of an ex-otic pair (see Figure 8), allows to obtain two regular 5-colored graphs representing two compactsimply-connected PL 4-manifolds W and W with the same topological structure that are not PL-homeomorphic: see Figures 9 and 10. Before starting the proof, we need to recall that a graph-based representation for compact PL mani-folds with non-empty boundary - different from the one considered in Section 2 - was already intro-duced by Gagliardi in the eighties (see [17]) by means of colored graphs failing to be regular. More details about such catalogue (together with other similar ones) can be found athttps://cdm.unimore.it/home/matematica/casali.mariarita/CATALOGUES.htm
Framed links representing the exotic pair W and W (pictures from [27]) Figure 9:
A 5-colored graph representing the compact simply-connected PL 4-manifold W More precisely, any compact n -manifold can be represented by a pair (Λ , λ ), where λ is still anedge-coloration on E (Λ) by means of ∆ n , but Λ may miss some (or even all) n -colored edges: such a(Λ , λ ) is said to be an ( n + 1) -colored graph with boundary, regular with respect to color n , and verticesmissing the n -colored edge are called boundary vertices .However, a connection between these different kinds of representation can be established throughan easy combinatorial procedure, called capping-off . Proposition 6 ([15])
Let (Λ , λ ) be an ( n + 1) -colored graph with boundary, regular with respect tocolor n , representing the compact n -manifold M . Chosen a color c ∈ ∆ n − , let (Γ , γ ) be the regular ( n + 1) -colored graph obtained from Λ by capping-off with respect to color c , i.e. by joining twoboundary vertices by an n -colored edge, whenever they belong to the same { c, n } -colored path in Λ . Then, (Γ , γ ) represents the singular n -manifold (cid:99) M , and hence M , too. Proposition 6 is actually the main tool to prove Theorem 4.
A 5-colored graph representing the compact simply-connected PL 4-manifold W Proof of Theorem 4. (i) Let us consider the 4-colored graph Λ( D , c ) constructed in in Section 3 and let us choose a quadri-color for each component of the link L. In [4, Theorem 3] it is proved that by substituting each quadricolor by the order ten 5-coloredsubgraph depicted in Figure 11-right, a 5-colored graph with boundary ˜Λ( D , c ), representing M ( L, c ),is obtained.If the “capping off” procedure described in Proposition 6 is applied to ˜Λ( D , c ) with respect to color1, the obtained regular 5-colored graph (which represents the singular 4-manifold (cid:92) M ( L, c )) turns outto admit, for each component of L , a sequence of three (proper) 2-dipoles involving only vertices of aquadricolor and never involving color 4: in fact, they consist of the pairs of vertices { P , R } , { R (cid:48) , R (cid:48) } , { P , R (cid:48) } . It is not difficult to see that, after these cancellations we obtain a (regular) 5-colored graphthat coincides with Γ( D , c ) up to the exchange of the colors 1 and 4.(ii) We point out that the 4-colored graph Λ( D , c ) admits a finite sequence of generalized dipoleeliminations not affecting the quadricolor structures, but reducing its regular genus.More precisely, each generalized dipole that is cancelled consists in a { , } -colored cycle of lengthfour, corresponding to a crossing of D , and one of its adjacent { , } -colored cycles, that does notcontain vertices of a quadricolor.Furthermore, by choosing a maximal tree of the graph corresponding to the β -colored regionsof the diagram, the sequence of generalized dipoles eliminations can be arranged so that a new 4-colored graph Ω( D , c ) is obtained having regular genus m α with respect to the cyclic permutation ε (cid:48) = (1 , , ,
3) of ∆ (see [4] for details).Obviously, since generalized dipole eliminations do not change the represented 3-manifold, Ω( D , c ) P P P RR RRRR P P P P
01 2 3
RR RRRR ' ''
Figure 11: main step from Λ( D , c ) to ˜Λ( D , c ) still represents M ( L, c ) . By operating the substitution of a quadricolor for each component of Ω( D , c ) as described in point(i) for Λ( D , c ), a new 5-colored graph with boundary is obtained, which, as proved in [4, Theorem1], represents M ( L, c ) . By capping it off with respect to color 1, a regular 5-colored graph ˜Ω( D , c ) isobtained representing the singular 4-manifold (cid:92) M ( L, c ) (hence M ( L, c ), too).Note that, by passing from Ω( D , c ) to ˜Ω( D , c ), the number of { ε (cid:48) i , ε (cid:48) i +1 } -colored cycles, with i ∈{ , , } , increases by l (i.e. by one for each substituted quadricolor), while the number of verticesincreases by 6 l. Furthermore, the following computations easily follow from the construction of ˜Ω( D , c ): g ε (cid:48) ,ε (cid:48) ( ˜Ω( D , c )) = g ε (cid:48) ,ε (cid:48) (Ω( D , c )) + l, g ε (cid:48) ,ε (cid:48) ( ˜Ω( D , c )) = p (Ω( D , c )) + l where V (Ω( D , c )) = 2 p (Ω( D , c )) . As a consequence, by considering the cyclic permutation ε = (1 , , , ,
4) of ∆ , formula 1 yields ρ ε ( ˜Ω( D , c )) = ρ ε (cid:48) (Ω( D , c )) + 2 l = m α + 2 l , while ρ ε (cid:48) ( ˜Ω ˆ4 ( D , c )) = m α + l. We point out that the previous eliminations of generalized dipoles do not involve vertices ofquadricolors while, on the other hand, for each component of L , the three 2-dipoles considered inpoint (i) involve only vertices of quadricolors and not color 4; therefore the 2-dipoles can be cancelledalso from ˜Ω( D , c ) producing exactly the same configuration as in Figure 6-right.Since one of the dipoles has colors non-consecutive in the permutation ε , it decreases by one theregular genus, while the other two dipoles have colors consecutive in the same permutation and so theirelimination does not affect the regular genus. Therefore, we finally obtain a regular 5-colored graph˜Γ( D , c ) (representing both (cid:92) M ( L, c ) and M ( L, c )) with regular genus ρ ε (˜Γ( D , c )) = ρ ε ( ˜Ω( D , c )) − l = m α + l. Moreover, it is not difficult to prove that its unique singular residue (if any) is ˜Γ ˆ4 ( D , c ) , whichrepresents M ( L, c ) and can be obtained from ˜Ω ˆ4 ( D , c ) by cancelling 2-dipoles that are induced bythe above ones. Therefore ρ ε (cid:48) (˜Γ ˆ4 ( D , c )) = ρ ε (cid:48) ( ˜Ω ˆ4 ( D , c )) − l = m α . This proves also point (ii).See Figures 13, 14 and 15 for the whole procedure from Λ( D , c ) to ˜Γ( D , c ) applied to the case ofthe trefoil knot with framing +1 . In this example, Ω( D , c ) is obtained from Λ( D , c ) by subsequentlyeliminating two generalized (3 , Q and Q respectively, andtwo 2-dipoles. (cid:50) Acknowledgements.
This work was supported by GNSAGA of INDAM and by the University ofModena and Reggio Emilia, project: “Discrete Methods in Combinatorial Geometry and GeometricTopology”.The authors thank Paolo Cavicchioli for drawing Figures 9 and 10. Q Q P P P P Figure 12:
The 5-colored graph with boundary ˜Λ( D , c ) representing M ( L, c ), for c = +1 and L = trefoil. P P P P Figure 13:
The 4-colored graph Ω( D , c ) representing M ( L, c ), for c = +1 and L = trefoil P P P P Figure 14:
The 5-colored graph with boundary from which the regular 5-colored graph ˜Ω( D , c ) representing M ( L, c ) (for c = +1 and L = trefoil) is obtained by “capping off” with respect to color 1 Figure 15:
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