A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice polytope, then we characterize the polytopes that arise as the lamination space of some graph on a pair of pants. This characterizes the image of a purely topological version of the spectral map for the vector bundle Laplacian for a flat connection on a pair of pants. The proof uses a graph exploration technique akin to the peeling of planar maps.

The combinatorial study of the determinant of the vector bundle Laplacian on graphs was initiated by Forman [5] followed by Kenyon [7] as a generalization of the classical matrix-tree theorem [12]. While the (reduced) determinant of the usual Laplacian operator on a graph enumerates spanning trees of this graph, the determinant of the vector bundle Laplacian enumerates cycle-rooted spanning forests (CRSFs), which are spanning forests where each connected component is a unicycle (a connected graph with as many vertices as edges). The weight of a CRSF is the product over its cycles of a quantity related to the monodromy of the connection along each cycle.

Of particular interest is the case of a flat SU(2,ℂ)ݑ†ݑˆ2ℂSU(2,\mathbbC)italic_S italic_U ( 2 , blackboard_C ) connection on a graph embedded on a surface [7], page_seo_titlely the case when the parallel transports are in SU(2,ℂ)ݑ†ݑˆ2ℂSU(2,\mathbbC)italic_S italic_U ( 2 , blackboard_C ) and the monodromy of the connection along each cycle of the graph which is contractible on the surface has to be trivial. In that case, the only CRSFs which contribute to the determinant of the vector bundle Laplacian are those which have no contractible cycles. Such CRSFs are called incompressible CRSFs and the cycles of an incompressible CRSF form a lamination of the surface, i.e. a collection of pairwise disjoint non-contractible simple loops. The determinant of the vector bundle Laplacian in the flat connection case can be written as a polynomial in variables of the form 2-Tr(w)2ݑ‡ݑŸݑ¤2-Tr(w)2 - italic_T italic_r ( italic_w ), where wݑ¤witalic_w is the monodromy along a non-contractible cycle on the surface [7]. Moreover these variables are free [3].

The most basic non-simply connected surfaces to consider are the annulus and the torus and was done in [7, 8, 6, 11, 9]. The next simplest case is probably the one of the pair of pants (aka three-holed sphere), briefly mentioned in [7]. It is one of the simplest surfaces for which the fundamental group is non abelian. A non-contractible cycle on a pair of pants can be of three possible topological types, thus the determinant of the vector-bundle Laplacian associated with a flat SU(2,ℂ)ݑ†ݑˆ2ℂSU(2,\mathbbC)italic_S italic_U ( 2 , blackboard_C ) connection on a graph embedded on that surface is a polynomial P(X,Y,Z)ݑƒݑ‹ݑŒݑP(X,Y,Z)italic_P ( italic_X , italic_Y , italic_Z ) of three independent variables.

The map which to a graph on a pair of pants associates the polynomial P(X,Y,Z)ݑƒݑ‹ݑŒݑP(X,Y,Z)italic_P ( italic_X , italic_Y , italic_Z ) is interesting to understand. We shall call it the spectral map, extending the terminology of the torus case [8] (this is a slight abuse of terminology, since the image of a graph under the spectral map should be the zero-locus of the polynomial together with a certain divisor on that algebraic variety [6]). Important questions include determining the image of the spectral map as well as the fiber of the spectral map above a given polynomial. This provides information on the probabilistic model associated with the uniform measure on incompressible CRSFs on the graph [7]. The polynomial P(X,Y,Z)ݑƒݑ‹ݑŒݑP(X,Y,Z)italic_P ( italic_X , italic_Y , italic_Z ) also plays an important role in relation with integrable systems, where it serves as the generating function of the integrals of motion [6]. The case of the annulus and the torus have been thoroughly investigated by Kenyon [7, 8]. For a different probabilistic model, the dimer model on bipartite graphs, the spectral map in the torus case is completely understood [10, 6, 4].

To any polynomial in nݑ›nitalic_n variables one can associate its Newton polytope, which is the convex hull in ℤnsuperscriptℤݑ›\mathbbZ^nblackboard_Z start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT of the nݑ›nitalic_n-tuples of integers (i1,…,in)subscriptݑ–1…subscriptݑ–ݑ›(i_1,\ldots,i_n)( italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) such that the monomial X1i1…Xninsuperscriptsubscriptݑ‹1subscriptݑ–1…superscriptsubscriptݑ‹ݑ›subscriptݑ–ݑ›X_1^i_1\ldots X_n^i_nitalic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT … italic_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT has a nonzero coefficient in the polynomial. We define the topological spectral map, which to a graph on a surface associates the Newton polytope of the polynomial produced by applying the spectral map. While the image under the spectral map depends on some weights that the edges of the graph may carry, the image under the topological spectral map only depend on the topological graph. The same questions can be asked about the topological spectral map : what is its image and what is the fiber above a given polytope ? These questions were answered in the case of the annulus and the torus [7, 8, 6]. In this article, we characterize the image of the topological spectral map for the pair of pants. The answer is much more involved than the annulus and torus cases. The next step would be to understand the fiber of this topological spectral map above a given polytope. Answering these questions for the spectral map itself in the pair of pants case seems to be much harder.

A monomial XiYjZksuperscriptݑ‹ݑ–superscriptݑŒݑ—superscriptݑݑ˜X^iY^jZ^kitalic_X start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_Y start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_Z start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT appears in P(X,Y,Z)ݑƒݑ‹ݑŒݑP(X,Y,Z)italic_P ( italic_X , italic_Y , italic_Z ) the determinant of the vector bundle Laplacian of a graph GݐºGitalic_G on a pair of pants if and only if GݐºGitalic_G has a lamination of type (i,j,k)ݑ–ݑ—ݑ˜(i,j,k)( italic_i , italic_j , italic_k ), that is a lamination with iݑ–iitalic_i cycles around the first hole, jݑ—jitalic_j cycles around the second and kݑ˜kitalic_k cycles around the third. Hence the image under the topological spectral map of GݐºGitalic_G is the lamination space of GݐºGitalic_G, i.e. the set of all (i,j,k)ݑ–ݑ—ݑ˜(i,j,k)( italic_i , italic_j , italic_k ) such that GݐºGitalic_G admits a lamination of type (i,j,k)ݑ–ݑ—ݑ˜(i,j,k)( italic_i , italic_j , italic_k ). The polytopes that arise in the image of the topological spectral map are exactly those that correspond to the lamination space of some graph on a pair of pants. The remainder of the paper will be formulated only in terms of laminations, no longer in terms of the determinant of the vector-bundle Laplacian, but the reader should keep in mind that the motivation behind this work comes from the spectral map associated with the vector-bundle Laplacian.

Organization of the paper

We introduce the relevant definitions and state our main results in Section 2. In Section 3 we describe an exploration process of a graph on a pair of pants and use it to realize the lamination space of that graph as a polytope. In passing we define three collections of special loops. In Section 4 we investigate the structure of the intersection of two collections of special loops. Deduce from this some necessary conditions for the polytopes arising as the lamination space of some graph. We show in Section 5 that these conditions are sufficient by constructing a class of graphs having as a lamination space a given polytope satisfying the aforementioned conditions.

2 Main results

We consider the three-holed sphere ΣΣ\Sigmaroman_Σ obtained by removing from the sphere ݕŠ2superscriptݕŠ2\mathbbS^2blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT three distinct points P1,P2subscriptݑƒ1subscriptݑƒ2P_1,P_2italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and P3subscriptݑƒ3P_3italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. Every simple closed curve Cݐ¶Citalic_C on ݕŠ2superscriptݕŠ2\mathbbS^2blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT which does not pass through the points Pisubscriptݑƒݑ–P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT separates ݕŠ2superscriptݕŠ2\mathbbS^2blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT into two hemispheres. which does not contain Pisubscriptݑƒݑ–P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT). A simple closed curve Cݐ¶Citalic_C is called of type iݑ–iitalic_i for 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3 if one of the hemispheres defined by Cݐ¶Citalic_C contains Pisubscriptݑƒݑ–P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and the other hemisphere contains the other two points, i.e.italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_C ) = italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_C ) = italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ( italic_C ) .
In the previous equalities, as well as in the remainder of this article, the indices 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3 should be considered modulo 3333.

Let GݐºGitalic_G be a connected nonempty graph embedded in ݕŠ2superscriptݕŠ2\mathbbS^2blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The connected components of ݕŠ2∖GsuperscriptݕŠ2ݐº\mathbbS^2\setminus Gblackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_G are topological disks, they are called the faces of GݐºGitalic_G and we denote by ℱℱ\mathcalFcaligraphic_F the set of faces of GݐºGitalic_G. We say that GݐºGitalic_G is a Σnormal-Σ\Sigmaroman_Σ-graph if there exist three distinct faces (F1,F2,F3)∈ℱsubscriptݐ¹1subscriptݐ¹2subscriptݐ¹3ℱ(F_1,F_2,F_3)\in\mathcalF( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ∈ caligraphic_F (called marked faces) such that Pisubscriptݑƒݑ–P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is in the interior of Fisubscriptݐ¹ݑ–F_iitalic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for all 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3. A ΣΣ\Sigmaroman_Σ-graph is more than just a graph embedded in ΣΣ\Sigmaroman_Σ because we require that the graph actually separates the three punctures. A lamination of the ΣΣ\Sigmaroman_Σ-graph GݐºGitalic_G is a collection Lݐ¿Litalic_L of pairwise disjoint simple loops on GݐºGitalic_G such that each loop in Lݐ¿Litalic_L is non-contractible on ݕŠ2∖P1,P2,P3superscriptݕŠ2subscriptݑƒ1subscriptݑƒ2subscriptݑƒ3\mathbbS^2\setminus\P_1,P_2,P_3\blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT . By disjoint we mean having no vertex in common. For any non-negative integers m1subscriptݑš1m_1italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, m2subscriptݑš2m_2italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and m3subscriptݑš3m_3italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, a lamination is said to be of type (m1,m2,m3)subscriptݑš1subscriptݑš2subscriptݑš3(m_1,m_2,m_3)( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) if for any 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3 it contains misubscriptݑšݑ–m_iitalic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT loops of type iݑ–iitalic_i. The lamination space ℒ(G)ℒݐº\mathcalL(G)caligraphic_L ( italic_G ) of a ΣΣ\Sigmaroman_Σ-graph GݐºGitalic_G is defined to be the set of all (m1,m2,m3)∈(ℤ+)3subscriptݑš1subscriptݑš2subscriptݑš3superscriptsubscriptℤ3(m_1,m_2,m_3)\in(\mathbbZ_{+})^3( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ∈ ( blackboard_Z start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT such that GݐºGitalic_G admit a lamination of type (m1,m2,m3)subscriptݑš1subscriptݑš2subscriptݑš3(m_1,m_2,m_3)( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ). Below we will describe the lamination space of a given ΣΣ\Sigmaroman_Σ-graph GݐºGitalic_G as the integer points of a lattice polytope defined in terms of some geometric characteristics of GݐºGitalic_G.

In order to simplify the inequalities defining the lamination space, we have allowed a lamination to be empty, in which case its topological type is (0,0,0)000(0,0,0)( 0 , 0 , 0 ). Note however that the polynomials P(X,Y,Z)ݑƒݑ‹ݑŒݑP(X,Y,Z)italic_P ( italic_X , italic_Y , italic_Z ) arising in the image of the spectral map have no constant term, so the only difference between the image of a graph GݐºGitalic_G under the topological spectral map and the lamination space of GݐºGitalic_G will be the presence or absence of the point (0,0,0)000(0,0,0)( 0 , 0 , 0 ).

We define a distance function dGsubscriptݑ‘ݐºd_Gitalic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT on ℱℱ\mathcalFcaligraphic_F such that any two faces sharing a vertex are at distance 1111 for dGsubscriptݑ‘ݐºd_Gitalic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT. Let G*superscriptݐºG^{*}italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT be the dual graph of GݐºGitalic_G (seen as a graph in ݕŠ2superscriptݕŠ2\mathbbS^2blackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT). Construct G*~~superscriptݐº\widetildeG^{*}over~ start_ARG italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_ARG by adding to G*superscriptݐºG^{*}italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT a dual edge between any two dual vertices such that the corresponding two primal faces share a primal vertex. The distance dGsubscriptݑ‘ݐºd_Gitalic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT is defined to be the usual graph distance on the vertex set of G*~~superscriptݐº\widetildeG^{*}over~ start_ARG italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_ARG, which is canonically in bijection with ℱℱ\mathcalFcaligraphic_F. In the special case when all the vertices of GݐºGitalic_G have degree 3333, then G*~=G*~superscriptݐºsuperscriptݐº\widetildeG^{*}=G^{*}over~ start_ARG italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_ARG = italic_G start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT and dGsubscriptݑ‘ݐºd_Gitalic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT is the classical distance between two faces corresponding to the graph distance on the dual graph. From now on, whenever we mention the distance between two faces of GݐºGitalic_G, the distance function will implicitly be dGsubscriptݑ‘ݐºd_Gitalic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT.

Define d1(G):=dG(F2,F3)assignsubscriptݑ‘1ݐºsubscriptݑ‘ݐºsubscriptݐ¹2subscriptݐ¹3d_1(G):=d_G(F_2,F_3)italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_G ) := italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ), d2(G):=dG(F1,F3)assignsubscriptݑ‘2ݐºsubscriptݑ‘ݐºsubscriptݐ¹1subscriptݐ¹3d_2(G):=d_G(F_1,F_3)italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_G ) := italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) and d3(G):=dG(F1,F2)assignsubscriptݑ‘3ݐºsubscriptݑ‘ݐºsubscriptݐ¹1subscriptݐ¹2d_3(G):=d_G(F_1,F_2)italic_d start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_G ) := italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). Also, for any 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3, define Mi(G)subscriptݑ€ݑ–ݐºM_i(G)italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_G ) to be the maximal number of pairwise disjoint simple loops of type iݑ–iitalic_i one can simultaneously draw on GݐºGitalic_G. Given a ΣΣ\Sigmaroman_Σ-graph GݐºGitalic_G, we define the sextuple

s(G):=(M1(G),M2(G),M3(G),d1(G),d2(G),d3(G))∈(ℤ+)3×ℕ3.assignݑ ݐºsubscriptݑ€1ݐºsubscriptݑ€2ݐºsubscriptݑ€3ݐºsubscriptݑ‘1ݐºsubscriptݑ‘2ݐºsubscriptݑ‘3ݐºsuperscriptsubscriptℤ3superscriptℕ3s(G):=(M_1(G),M_2(G),M_3(G),d_1(G),d_2(G),d_3(G))\in(\mathbbZ_{+% })^3\times\mathbbN^3.italic_s ( italic_G ) := ( italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_G ) , italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_G ) , italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_G ) , italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_G ) , italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_G ) , italic_d start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_G ) ) ∈ ( blackboard_Z start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × blackboard_N start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT .
See Figure 1 for an example.

Given a sextuple of integers τ=(a,b,c,d,e,f)∈(ℤ+)3×ℕ3ݜݑŽݑݑݑ‘ݑ’ݑ“superscriptsubscriptℤ3superscriptℕ3\tau=(a,b,c,d,e,f)\in(\mathbbZ_{+})^3\times\mathbbN^3italic_τ = ( italic_a , italic_b , italic_c , italic_d , italic_e , italic_f ) ∈ ( blackboard_Z start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × blackboard_N start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT, we define the convex lattice polytope ݒ«τsubscriptݒ«ݜ\mathcalP_\taucaligraphic_P start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT by

ݒ«τ:=(x,y,z)∈(ℤ+)3.assignsubscriptݒ«ݜconditional-setݑ¥ݑ¦ݑ§superscriptsubscriptℤ3formulae-sequenceݑ¥ݑŽformulae-sequenceݑ¦ݑformulae-sequenceݑ§ݑformulae-sequenceݑ¦ݑ§ݑ‘formulae-sequenceݑ¥ݑ§ݑ’ݑ¥ݑ¦ݑ“\mathcalP_\tau:=\left\x\leq a,\ y\leq b,\ % z\leq c,\ y+z\leq d,\ x+z\leq e,\ x+y\leq f\right\.caligraphic_P start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT := italic_x ≤ italic_a , italic_y ≤ italic_b , italic_z ≤ italic_c , italic_y + italic_z ≤ italic_d , italic_x + italic_z ≤ italic_e , italic_x + italic_y ≤ italic_f .

For any Σnormal-Σ\Sigmaroman_Σ-graph GݐºGitalic_G, its lamination space ℒ(G)ℒݐº\mathcalL(G)caligraphic_L ( italic_G ) is the polytope ݒ«s(G)subscriptݒ«ݑ ݐº\mathcalP_s(G)caligraphic_P start_POSTSUBSCRIPT italic_s ( italic_G ) end_POSTSUBSCRIPT.

Proposition 2.1 is proved in Section 3.

The inequalities mi≤Mi(G)subscriptݑšݑ–subscriptݑ€ݑ–ݐºm_i\leq M_i(G)italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_G ) are not redundant with the inequalities mi+mi+1≤di+2(G)subscriptݑšݑ–subscriptݑšݑ–1subscriptݑ‘ݑ–2ݐºm_i+m_i+1\leq d_i+2(G)italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ≤ italic_d start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ( italic_G ), as illustrated by Figure 2. On that picture, d1(G)=d2(G)=d3(G)=2subscriptݑ‘1ݐºsubscriptݑ‘2ݐºsubscriptݑ‘3ݐº2d_1(G)=d_2(G)=d_3(G)=2italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_G ) = italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_G ) = italic_d start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_G ) = 2 and M1(G)=M2(G)=M3(G)=1subscriptݑ€1ݐºsubscriptݑ€2ݐºsubscriptݑ€3ݐº1M_1(G)=M_2(G)=M_3(G)=1italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_G ) = italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_G ) = italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_G ) = 1. The triple (m1,m2,m3)=(2,0,0)subscriptݑš1subscriptݑš2subscriptݑš3200(m_1,m_2,m_3)=(2,0,0)( italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = ( 2 , 0 , 0 ) verifies the inequalities mi+mi+1≤di+2subscriptݑšݑ–subscriptݑšݑ–1subscriptݑ‘ݑ–2m_i+m_i+1\leq d_i+2italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ≤ italic_d start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT, but that graph has no lamination of type (2,0,0)200(2,0,0)( 2 , 0 , 0 ). This proposition corrects a statement made in [7], where the inequalities mi≤Mi(G)subscriptݑšݑ–subscriptݑ€ݑ–ݐºm_i\leq M_i(G)italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_G ) were missing.

We can now characterize all the convex lattice polytopes that arise as the lamination space of some ΣΣ\Sigmaroman_Σ-graph. By the previous proposition, it suffices to characterize the sextuples τݜ\tauitalic_τ that arise as some s(G)ݑ ݐºs(G)italic_s ( italic_G ).

Fix τ=(μ1,μ2,μ3,δ1,δ2,δ3)∈(ℤ+)3×ℕ3ݜsubscriptݜ‡1subscriptݜ‡2subscriptݜ‡3subscriptݛ¿1subscriptݛ¿2subscriptݛ¿3superscriptsubscriptℤ3superscriptℕ3\tau=(\mu_1,\mu_2,\mu_3,\delta_1,\delta_2,\delta_3)\in(\mathbbZ_% {+})^3\times\mathbbN^3italic_τ = ( italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_δ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ∈ ( blackboard_Z start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × blackboard_N start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. There exists a Σnormal-Σ\Sigmaroman_Σ-graph GݐºGitalic_G such that s(G)=τݑ ݐºݜs(G)=\tauitalic_s ( italic_G ) = italic_τ if and only if the following inequalities hold for all 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3:

(T1subscriptݑ‡1T_1italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT)
max(μi+1,μi+2)≤δi≤μi+1+μi+2subscriptݜ‡ݑ–1subscriptݜ‡ݑ–2subscriptݛ¿ݑ–subscriptݜ‡ݑ–1subscriptݜ‡ݑ–2\max(\mu_i+1,\mu_i+2)\leq\delta_i\leq\mu_i+1+\mu_i+2roman_max ( italic_μ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT ) ≤ italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_μ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT + italic_μ start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT;

(T2subscriptݑ‡2T_2italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT)
min(δi+1-μi+2,δi+2-μi+1)+⌊μi+1+μi+2-δi2⌋≤μisubscriptݛ¿ݑ–1subscriptݜ‡ݑ–2subscriptݛ¿ݑ–2subscriptݜ‡ݑ–1subscriptݜ‡ݑ–1subscriptݜ‡ݑ–2subscriptݛ¿ݑ–2subscriptݜ‡ݑ–\min(\delta_i+1-\mu_i+2,\delta_i+2-\mu_i+1)+\left\lfloor\frac\mu_i+% 1+\mu_i+2-\delta_i2\right\rfloor\leq\mu_iroman_min ( italic_δ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT - italic_μ start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT , italic_δ start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT - italic_μ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ) + ⌊ divide start_ARG italic_μ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT + italic_μ start_POSTSUBSCRIPT italic_i + 2 end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ⌋ ≤ italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.

The fact that conditions (T1)subscriptݑ‡1(T_1)( italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and (T2)subscriptݑ‡2(T_2)( italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) are necessary is proved in Section 4, while the fact that they are sufficient is proved in Section 5 by explicitly constructing a ΣΣ\Sigmaroman_Σ-graph GτsubscriptݐºݜG_\tauitalic_G start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT such that s(Gτ)=τݑ subscriptݐºݜݜs(G_\tau)=\tauitalic_s ( italic_G start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT ) = italic_τ whenever τݜ\tauitalic_τ satisfies the two conditions.

The second set of inequalities seems to suggest that the image of sݑ sitalic_s in (ℤ+)3×ℕ3superscriptsubscriptℤ3superscriptℕ3(\mathbbZ_{+})^3\times\mathbbN^3( blackboard_Z start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT × blackboard_N start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT may be a non-convex set. This is indeed the case, since τ1=(9,6,3,9,9,12)subscriptݜ19639912\tau_1=(9,6,3,9,9,12)italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( 9 , 6 , 3 , 9 , 9 , 12 ) and τ2=(9,4,9,9,9,12)subscriptݜ29499912\tau_2=(9,4,9,9,9,12)italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = ( 9 , 4 , 9 , 9 , 9 , 12 ) are both in the image of sݑ sitalic_s, but τ1+τ22subscriptݜ1subscriptݜ22\frac\tau_1+\tau_22divide start_ARG italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG is not.

In order to prove Proposition 2.1 and the necessity of the conditions (T1)subscriptݑ‡1(T_1)( italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and (T2)subscriptݑ‡2(T_2)( italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) in Theorem 2.3, we will explore any ΣΣ\Sigmaroman_Σ-graph GݐºGitalic_G starting from the face F1subscriptݐ¹1F_1italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, then discover a first layer consisting of the faces at distance 1111 from F1subscriptݐ¹1F_1italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, then a second layer consisting of the faces at distance 2222, etc. We will perform the same exploration starting from the faces F2subscriptݐ¹2F_2italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and F3subscriptݐ¹3F_3italic_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and understand how the boundaries of the layers arising in each of these three explorations interact with each other. In the case of simple triangulations, our construction is very similar to the layer decomposition of Krikun [13]. More generally, this construction resembles the peeling process for planar maps (see for example [2]). The difference is that here we use a distance which differs slightly from the graph distance on the dual graph. Instead of peeling an edge by discovering the face on the other side of the edge, we are peeling a vertex, by discovering all the unknown faces containing a vertex which is on the boundary of what we have already explored.

3 A collection of special loops around a puncture

In this section we first describe an exploration process of a ΣΣ\Sigmaroman_Σ-graph GݐºGitalic_G starting from a marked face, which will trace a collection of special loops on GݐºGitalic_G centered around a marked face. Then we will use this exploration to prove Proposition 2.1.

3.1 Some special loops around a puncture

We start by an elementary observation, which we will be using several times. Let GݐºGitalic_G be a connected planar graph and G~~ݐº\widetildeGover~ start_ARG italic_G end_ARG be a subgraph of GݐºGitalic_G. One defines the distance function dG~subscriptݑ‘~ݐºd_\widetildeGitalic_d start_POSTSUBSCRIPT over~ start_ARG italic_G end_ARG end_POSTSUBSCRIPT on the set of the connected components of ݕŠ2∖G~superscriptݕŠ2~ݐº\mathbbS^2\setminus\widetildeGblackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ over~ start_ARG italic_G end_ARG in exactly the same way as the distance dGsubscriptݑ‘ݐºd_Gitalic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT was defined on the faces of GݐºGitalic_G. Note that the connected components of ݕŠ2∖G~superscriptݕŠ2~ݐº\mathbbS^2\setminus\widetildeGblackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ over~ start_ARG italic_G end_ARG do not have to be topological disks, they may be disks with multiple punctures or even the whole sphere is G~~ݐº\widetildeGover~ start_ARG italic_G end_ARG is empty. Then we have the following result, the proof of which is easy and omitted.

Let GݐºGitalic_G be a connected planar graph and G~normal-~ݐº\widetildeGover~ start_ARG italic_G end_ARG be a subgraph of GݐºGitalic_G. Let Fݐ¹Fitalic_F and F′superscriptݐ¹normal-′F^\primeitalic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT be two faces of GݐºGitalic_G and let F~normal-~ݐ¹\widetildeFover~ start_ARG italic_F end_ARG and F′~normal-~superscriptݐ¹normal-′\widetildeF^\primeover~ start_ARG italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG be the two connected components of ݕŠ2∖G~superscriptݕŠ2normal-~ݐº\mathbbS^2\setminus\widetildeGblackboard_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∖ over~ start_ARG italic_G end_ARG containing respectively Fݐ¹Fitalic_F and F′superscriptݐ¹normal-′F^\primeitalic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Then dG~(F~,F′~)≤dG(F,F′)subscriptݑ‘normal-~ݐºnormal-~ݐ¹normal-~superscriptݐ¹normal-′subscriptݑ‘ݐºݐ¹superscriptݐ¹normal-′d_\widetildeG(\widetildeF,\widetildeF^\prime)\leq d_G(F,F^\prime)italic_d start_POSTSUBSCRIPT over~ start_ARG italic_G end_ARG end_POSTSUBSCRIPT ( over~ start_ARG italic_F end_ARG , over~ start_ARG italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ) ≤ italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_F , italic_F start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ).

Let GݐºGitalic_G be a ΣΣ\Sigmaroman_Σ-graph. For any k≥0ݑ˜0k\geq 0italic_k ≥ 0 and 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3, define

Aik=dG(F,Fi)=k.subscriptsuperscriptݐ´ݑ˜ݑ–conditional-setݐ¹ℱsubscriptݑ‘ݐºݐ¹subscriptݐ¹ݑ–ݑ˜A^k_i=\left\F\in\mathcalF.italic_A start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_F , italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_k . (3.1)
For any k≥1ݑ˜1k\geq 1italic_k ≥ 1 and 1≤i≤31ݑ–31\leq i\leq 31 ≤ italic_i ≤ 3 such that Aiksubscriptsuperscriptݐ´ݑ˜ݑ–A^k_iitalic_A start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is nonempty, define Biksubscriptsuperscriptݐµݑ˜ݑ–B^k_iitalic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to be the boundary of the set ⋃j=0k-1Aijsuperscriptsubscriptݑ—0ݑ˜1subscriptsuperscriptݐ´ݑ—ݑ–\bigcup_j=0^k-1A^j_i⋃ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_A start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of faces that are at distance less than kݑ˜kitalic_k to Fisubscriptݐ¹ݑ–F_iitalic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Each Biksubscriptsuperscriptݐµݑ˜ݑ–B^k_iitalic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the union of simple loops that are not necessarily pairwise disjoint. The case when Biksubscriptsuperscriptݐµݑ˜ݑ–B^k_iitalic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT consists in the union of several loops corresponds to a branching event in the peeling terminology, see e.g. [

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