Main Theorem.

Mathematica Pannonica 7

/2 (1996), 177 { 183

COLOURED KNOTS AND PERMU-

TATIONS REPRESENTING 3-MANI-

FOLDS

LuigiGras s elli

Dipartimento di Matematica, Universita, Piazza Porta San Do- nato 5, 40126 Bologna, Italia

Received: October 1994 MSC 1991: 57 Q 15, 57 M 12

Keywords: 3-manifolds, branched coverings, coloured knots, coloured graphs.

Abstract: It is known that every closed orientable 3-manifold can be rep- resented by coloured knots (4], 5]), edge-coloured graphs (2]) or transitive permutation pairs (6]). The present paper describes some relations between these representation theories: in particular, it is shown how to obtain a transi- tive permutation pair representing a 3-manifold^M3, starting from a coloured knot representing^M3.

1. Introduction

Throughout this paper, all spaces and maps are piecewise-linear manifolds are always supposed to be closed and connected.

In 6], Montesinos proves that every orientable 3-manifoldM³ can be represented by a transitive pair of permutations () of ^P_h, the symmetric group on Nh = ^f12:::h^g. In fact, M³ is a covering of the 3-sphere S³, branched over the graph G of Fig. 1 thus, M³ is

Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (Na- tional Research Council of Italy) and nancially supported by MURST.

Fig. 1.

determined by a monodromy map : ^Q1(S^{3 ;}G) ^7;!P_h dened by sending the meridians m¹ and m² to and respectively. If M³ is represented by (), we writeM³ =M().

A well-known theorem of Hilden 4] and Montesinos 5] states that every orientable 3-manifold M³ is a simple 3-fold covering of S³, branched over a knot^KS³ (where \simple"means that the associated monodromy! :^Q1(S^3;K)^7;!P3 sends meridiansto transpositions).

In other words,M³ is representable by a pair (^K!), where ^Kis a knot and ! :^Q1(S^{3 ;K})^7;!P3 is a simple monodromy map. Such a pair is called a coloured knot, since it can be visualized by a planarcoloured diagram ^D of ^K, in which each arc is coloured by k ² Z³ = ^f012^g if and only if the transposition associated to the corresponding meridian xes k. Moreover, we can always suppose (see, for example, 1]) that such a diagram is 3-coloured, i.e. at each crossing, the three incident arcs have distinct colours. If the 3-manifold M³ is represented by the 3-coloured diagram^D, we write M³ =M(^D).

In the present paper, by making use of results contained in 1]

and 3], we show how to obtain a transitive permutation pair () representing an orientable 3-manifold M³, starting from a 3-coloured diagram^D representing M³.

Coloured knots and permutations 179

2. Main results

Let ^D be a 3-coloured diagram of a knot, draw in the oriented euclidean plane ^E2. Let ^C = ^fc¹:::cn^g denote the set of the n 3 crossings of ^D and, for each k ² Z³ = ^f012^g, let ^k1:::_ks^k be the k-coloured arcs of ^D, with sk 1 and s⁰+s¹+s² =n. For eachk ²Z³ and each r ² Ns^k, we are going to dene a cyclic permutation on the set ^CZ³, associated to thek-coloured arc_kr. Denote the endpoints of _kr by the corresponding crossingschcj of ^D and suppose that _kr has t = t_kr undercrossings (where t can eventually be zero). If S¹ denotes the standard circle and d is a diameter of S¹, consider an embedding

kr :S¹ d^7;!E2 such that

a) _kr(d) =_kr (and hence _kr(d) =^fchcj^g)

b) the intersection of _kr(S^1;d) with the diagram^D is given by 2t points, one for each arc incident with _kr.

If cp is an undercrossing of _kr, denote by the same symbol cp the intersection point of _kr(S¹) with the (k + 1)-coloured arc incident to _kr in cp. Denote by ^C_kr the set given by the union of these intersection points together with the endpointschcj of _kr hence,^C_{kr kr}(S¹) and Card^C_kr = t+ 2. Finally, let (cq¹:::cq^t+2) be the ordered sequence obtained by reading the elements of ^C_kr while walking along _kr(S¹), coherently with the orientation induced on it by the xed orientation of

E

2 the cyclic permutation_kr on^CZ³ is dened in the following way:

_kr=(cq¹k):::(cq^t+2k): With these notations, we have the following result.

Main Theorem.

^{Let C} = ^fc¹:::cn^g be the set of the n crossings of a 3-coloured diagram ^D of a knot and denote by sk, for each k ² Z³, the number of the k-coloured arcs of ^D. If are the permutations on

C Z³ dened by

= ^Y

k²Z^31k:::_ks^k = ^Y

i²N

(c¹0)(c¹1)(c¹2)

then()is a transitive permutation pair such that M() =M(^D). The proof of this theorem, given in Section 3, makes use of the possibility of representing manifolds by means of edge-coloured graphs and is performed by joining two constructions, respectively described in 1] and 3].

Fig. 2

Fig. 2 contains a 3-coloured diagram representingS¹S² (see 7], Fig. 21) the application of the above algorithm produces the following transitive permutation pair () onN⁹Z³ such that M() =S¹

S²:

=(c¹0)(c²0)(c⁷0)(c⁸0)(c³0)(c⁹0)(c⁴0)(c⁶0)(c⁵0)

(c¹1)(c²1)(c³1)(c⁴1)(c⁸1)(c⁹1)(c⁵1)(c⁶1)(c⁷1)

(c¹2)(c⁶2)(c⁵2)(c²2)(c³2)(c⁴2)(c⁷2)(c⁸2)(c⁹2) = ^Y

i²N⁹

(ci0)(ci1)(ci2):

Corollary.

LetM³ be an orientable 3-manifold which is a simple 3-fold covering of S³ branched over a knot ^K and suppose that a 3-coloured diagram ^D of ^K representing M³ has n crossings. Then M³ is also a (3n)-fold covering of S³ branched over the graph G.

Proof.

The Main Theorem states thatM³ =M(^D) can be represented by a permutation pair () acting on^CZ³. Since Card(^CZ³) = 3n, M³ is determined by a monodromy map :^Q1(S^{3 ;}G)^7;!P3_n and henceM³ is a (3n)-fold covering of S³ branched over G.

Coloured knots and permutations 181

3. The proof

Let us recall some notations and results about the manifold repre- sentation theory by means of edge-coloured graphs for a general survey on it, see 2] or 9].

An (m+1)-coloured graph is a pair (;), where ; = (V(;)E(;)) is a nite regular multigraph of degree m+1 and :E(;)^7;!Zm⁺¹ =

= ^f01:::m^g is a map such that (e) ⁶= (f), for each pair ef of adjacent edges. For every ^F Zm⁺¹, set ;^F = (V(;)^;1(^F)) each connected component of ;^F is often called an ^F-residue. Note that, for every distinct ij ² Zm⁺¹, the ^fij^g-residues of (;) are cycles alternatively coloured by i and j.

Anm-dimensionalball complexK(;), triangulatinganm-pesudo- manifold, can be associated to a given (m+1)-coloured graph (;) by the following rules:

| take an m-simplex (v) for each v ² V(;) and label its vertices by Zm⁺¹

| if vw ² V(;) are joined by a k-coloured edge, identify the (m^;1)-faces of (v) and (w) opposite to the vertices labelled by k, so that equally labelled vertices are identied together.

Even if its ballsare simplexes, the resulting complexK(;) is not in gen- eral a simplicial one, since the intersection of two simplexes may be the union of morethan one maximalface nevertheless, it isapseudocomplex (2], p. 122). The graph (;) is said to represent K(;), ^jK(;)^j and every homomorphic space. Note that every h-simplex of K(;), whose vertices are labelled by the distinct coloursk⁰:::kh, corresponds to a unique (Zm^+1;fk⁰:::kh^g)-residue of (;) and viceversa.

Everym-manifold M^m is representable by(m+1)-coloured graphs (8]). If (;) represents M^m, then M^m is orientable if and only if ; is bipartite.

In 3], the following method is described for producing, starting from a 4-coloured graph (;) representing an orientable 3-manifold M³, a transitive permutation pair () such that M() =M³. Set Card(V(;)) = 2n. Let V⁰, V⁰⁰ be the two bipartition classes of V(;) and identify V⁰ with Nn. If v ² V⁰ and k ² Z³, denote by k(v) the vertex ofV⁰⁰ which isk-adjacent to vand denote by k(v) the vertex of V⁰ which is 3-adjacent to k(v). With these notations, dene the two permutations on NnZ³ in the following way:

=

v²Nⁿ (v0)(v1)(v2) ⁸v²Nn ⁸k ²Z³ (vk) = (k(v)k): Roughly speaking, the permutationisthe product of the disjointcycles obtained by \reading" the vertices of V⁰ in a suitable orientation of the

f

Proposition 1.

k3^g-residuesof (;3]Let). The pair ((;) be a4) is said to be-coloured graph representing theassociated to (;) orientable 3-manifold M³ and let () be the pair of permutations on NnZ³ (2n= CardV(;)) associated to (;). Then M() =M³.

On the other hand, the following algorithm, given in 1], produces a 4-colouredgraph representing the orientable3-manifoldM³ described by a given 3-coloured diagram ^D of a knot, draw in the oriented ^E2. Suppose, with the same notations of Section 2, that ^C = ^fc¹:::cn^g is the set of then3 crossings of ^Dand, for each k ²Z³, let^k1:::_ks^k be the k-coloured arcs of ^D. For everyr ²Ns⁰ (resp. r²Ns¹), thicken the arc ⁰_r (resp. ¹_r) to a \strip"^R0_r (resp. ^R1_r), so that all these strips have disjoint interiors and, if the arcs⁰_r⁰, ¹_r⁰⁰ are incident to the same crossing ci, then ^R0_r⁰ and ^R1_r⁰⁰ have a common edge, denoted by ei. For everyi ² Nn, denote by vi and v_i⁰ the endpoints of the edge ei, so that vi precedes v⁰_i while walking along^R0_r (r ²Ns⁰), coherently with the xed orientation of^E2. Set E³ =_i^S

2N^nfei^gand denote by E⁰ (resp.

E¹) the set of the edges of_r ^S

2N^s0(^R0_r) (resp. _r ^S

2N^s1(^R1_r)) not belonging to E³. For every i ² Nn, consider the 2-coloured arc ²_r⁽_i⁾ incident to the crossing ci then, draw an edge bi between v⁰_i and vj⁽i⁾ if and only if, while walking around ²_r⁽_i⁾ coherently with the xed orientation of

E

2, starting from the 0-coloured arc incident in ci, the rst incident 0-coloured arc is incident in cj⁽i⁾. Set E² =_i^S

2N^nfbi^g.

Let (;) = ;(^D) denote the 4-coloured graph dened by:

V(;) =

i²N^nfviv⁰_i^g E(;) =

k²Z⁴Ek where each edge of Ek is coloured by k.

Proposition 2.

1] For every 3-coloured diagram ^D of a knot, the 4- coloured graph ;(^D) represents M(^D).

Proof of the Main Theorem.

Let (;) = ;(^D) be the 4-coloured graph associated to ^D, so that ^jK(;)^j = M(^D). Orient each ^f03^g- residue of (;) coherently with the xed orientation of^E2 and denote

Coloured knots and permutations 183

byV⁰ the bipartition class ofV(;) containing the second vertex of each 3-coloured edge ei. Note that each crossingci of ^D corresponds to the 3-coloured edge ei of (;) hence, we can identify the sets ^C and V⁰ with Nn. By construction, for each k² Z³, the ^fk3^g-residues of (;) are in one-to-one correspondence with thek-coloured arcs^k1:::_ks^k of

D denote them by^Rk1:::^R_ks^k respectively. Moreover, for eachk ²Z³ and r ² Ns^k, there exists an orientation-preserving homeomorphism between _kr(S¹) and^R_kr sending each point cp of ^C_kr to the vertex of the 3-coloured edge ep belonging to V⁰. This proves that the permutation pair (), acting on the set ^CZ^{3 '}NnZ³, dened in Section 2, is precisely the transitive permutation pair, acting onV⁰Z^{3 '}NnZ³, associated to the 4-coloured graph (;). Hence, we have M() =

=^jK(;)^j=M(^D).

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