Quandle theory and optimistic limits of representations of knot groups

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1 Quandle theory and optimistic limits of representations of not groups arxiv: v4 [math.gt] 6 Oct 04 Jinseo Cho October 7, 04 Abstract Quandle of a lin diagram is very useful tool to describe the not group via Wirtinger presentation. Following the conjugation quandle proposed by Inoue and Kabaya, we see the octahedral triangulation is always non-degenerate for any boundary-parabolic representation of the not group. On the other hand, the same triangulation was also used in the optimistic limit. In this paper, we show the relationship between the quandle theory and the optimistic limit of a lin. Especially, we interpret the optimistic limit in the language of quandle, and generalize it to any boundary-parabolic representation of the not group. Furthermore, we show that the hyperbolicity equation, which determines the hyperbolic structure of the triangulation, always have a solution. This solution is explicitly constructed by the shadow-coloring of the quandle induced by the representation. Introduction Quandle of a lin diagram is an algebric object whose axioms are naturally induced by Reidemeister moves of lins. Due to the natural definition, it has fruitful structures and various applications to not theory, including classical nots, surface nots and virtual nots. A good guide of quandle theory is []. Especially, quandle is a very useful tool to describe the fundamental group of a lin via Wirtinger presentation. Inoue and Kabaya, in [7], used it for a boundary-parabolic representation ρ : π L PSL, C, where π L is the fundamental group of a lin L, which is called the not group of L, and boundary-parabolic means the image of the peripheral subgroup π S 3 \L is a parabolic subgroup of PSL, C. They defined shadowcoloring of the conjugation quandle consisting of parabolic elements induced by ρ, and suggested a natural way to determine the geometric shape of the octahedral triangulation of S 3 \L {two points}. Especially, they showed that any tetrahedron in the triangulation is non-degenerate. On the other hand, the same octahedral triangulation appears naturally from the lin diagram together with its dual graph see Section 3 of [4] for detail, and is used in many

2 areas including the optimistic limit in [3] and the cluster algebra in [6]. The purpose of this article is to show the relationship between quandle and optimistic limit, and to prove the existence of a solution of certain equations by constructing it explicitly from the shadowcoloring of the quandle. Especially, we will present new interpretation of the optimistic limit of [3] using the language of quandle in [7] and generalize it to any boundary-parabolic representation of a not group. The relationship with cluster algebra will be discussed in another article. The optimistic limit was first appeared in [8] when Kashaev proposed the volume conjecture, which relates certain limits of not invariants, called Kashaev invariants, with the hyperbolic volumes. The optimistic limit is the value of certain potential function evaluated at saddle point, where the function and the value are expected to be an analytic continuation of the Kashaev invariant and the limit of the invariant, respectively. As a matter of fact, physicists usually consider the optimistic limit as actual limit and develop their theories. Due to many wors on the optimistic limit in [0], [4] and [5], the volume conjecture will be solved if the coincidence of the optimistic limit and the actual limit is proved. On the other hand, the original definition of the optimistic limit was so complicated that applying it to other theories was really hard. Therefore, we improved the definition at [3] by using the octahedral triangulation of S 3 \L {two points} without any collapsing of tetrahedra. We claimed the revised definition is more natural because many technical assumptions are removed and the geometry of the triangulation wors more naturally. This article will show another example of such naturalness by the harmony with the quandle theory of [7]. The optimistic limit is defined by the potential function V z,..., z n, w j,.... Previously, in [3], this function was defined purely by the lin diagram, but here we modify it using the information of the representation ρ. The definition is in Section 3. We consider a solution of the following set { } V H := expz =, expw j V = j : degenerate, =,..., n, z w j which is a saddle-point of the potential function V. Then Proposition 3. will show that H becomes the hyperbolicity equations, which determines the hyperbolic structure of the octahedral triangulation. We assumed H has a solution in [3] because, if not, we cannot do anything with that potential function. This is because the chosen lin diagram fixes the triangulation and the function together. When there is no solution, we had to change the function by modifying the lin diagram and we conjecturally expected proper change will guarantee the existence of a solution. However, in this article, we will explicitly construct a solution z 0,..., z n 0, w j 0,... of H from the shadow-coloring induced by the representation ρ. The exact formulas are in Theorem 3.. Furthermore, after defining V 0 z,..., z n, w j,... := V z,..., z n, w j,... V z log z w j V z w j log w j, j:degenerate

3 we will show V 0 z 0,..., z 0 n, w j 0,... ivolρ + i csρ mod π at Theorem 3.3, where volρ and csρ are the volume and the Chern-Simons invariant of ρ defined in [6], respectively. The left-hand side of is called the optimistic limit of ρ, and volρ + i csρ in the right-hand side is called the complex volume of ρ. Note that the optimistic limit gives a very convenient method to calculate the complex volume of a given representation ρ. As a matter of fact, when we consider the following formal series of the asymptotic expansion of the Kashaev invariant in -3 of [5], Z M = exp S M,0 3 log + S M, + n S M,n, = πi N, n our potential function of the optimistic limit detects the complex volume term S M,0. The author expects there should be another function, which is also induced by the Kashaev invariant, that detects the Ray-Singer torsion term S M,. Actually, the results of [5] and [] support this conjecture. One of the advantage of the approach in this article is that we already now the right solution, so what should be done is to find the right function. This article consists of the following contents. In Section, we will summarize some results of [7]. Although the article [7] discussed general theory of quandle homology, we restrict our attention only to the conjugation quandle P,, which consists of parabolic elements of PSL,C with the operation induced by the conjugation. Also, we will discuss the triangulation of the lin complement of [3] and slightly modify it in the viewpoint of [7]. Section 3 will discuss the optimistic limit and Section 4 will contain two simple examples, the figure-eight not 4 and the trefoil not 3 with one degenerate crossing. Quandle In this section, we will survey the quandle theory of [7], which is an essential tool to new interpretation of the optimistic limit. We remar that all contents of this section come from [7] and series lectures of Ayumu Inoue given at Seoul National University during spring of 0.. Conjugation quandle of parabolic elements Definition.. A quandle is a set X with a binary operation satisfying the following three conditions:. a a = a for any a X,. the map b : X X a a b is bijective for any b X, 3. a b c = a c b c for any a, b, c X. 3

4 The inverse of b is notated by b. In other words, the equation a b = c is equivalent to c b = a. Definition.. Let G be a group and X be a subset of G satisfying Define the binary operation on X by gxg = X for any g G. a b = bab for any a, b X. Then X, becomes a quandle and is called the conjugation quandle. As an example, let P be the set of parabolic elements of PSL, C = Isom + H 3. Then gpg = P holds for any g PSL, C. Therefore, P, is a conjugation quandle, and this is the only quandle we are using in this article. To perform concrete calculations, explicit expression of P, was introduced in [7]. At first, note that p q p q + rs s = r s 0 r s r, rs p q for PSL, C. Therefore, we can identify C r s \{0}/± with P by + αβ β α β α, 3 αβ where ± means the equivalence relation α β α β. To use the left-side action of PSL, C on C \{0}/±, we consider the transpose of 3 by α + αβ α β β, αβ and define the operation by α γ + γδ γ := β δ δ γδ α β C \{0}/±, where the matrix multiplication on the right-hand side is the standard multiplication. Note that this definition coincides with the operation of the conjugation quandle P, by α γ + γδ γ α = β δ δ C \{0}/± γδ β + γδ γ + αβ α + γδ γ δ γδ β αβ δ γδ γ α γ = PSL, C. δ β δ 4

5 The inverse operation is given by α γ β δ γδ γ = δ + γδ α β. From now on, we use the notation P instead of C \{0}/±.. Knot group and shadow-coloring Consider a representation ρ : π L PSL, C of a hyperbolic lin L. We call ρ boundaryparabolic when the peripheral subgroup π S 3 \L of π L maps to a subgroup of PSL, C whose elements are all parabolic. For a fixed oriented lin diagram D of L, Wirtinger presentation gives an algorithmic expression of π L. For each arc α of D, we draw a small arrow labelled a as in Figure, which presents a loop. The details are in [3]. Here we are using the opposite orientation of a to be consistent with the operation of the conjugation quandle. This loop corresponds to one of the meridian curves of the boundary tori, so ρa is an element in P. Hence we call {ρa,..., ρa n } arc-coloring of D, whereas each ρa is assigned to the corresponding arc α. Figure : The figure-eight not 4 Wirtinger presentation shows that the not group is presented by π L =< a,..., a n ; r,..., r n >, We always assume the diagram does not contain a trivial not component which has only over-crossings or under-crossings or no crossing. If it happens, then we change the diagram of the trivial component slightly. For example, applying Reidemeister second move to mae different types of crossings or Reidemeister first move to add a in is good enough. This assumption is necessary to guarantee that the five-term triangulation becomes a topological triangulation of S 3 \L {two points} 5

6 where the relation r l is assigned to each crossing as in Figure. Note that r l coincides with, so we can write down relation of the arc-colors as in Figure 3. As a matter of fact, Figure 3 is usually the defining relation of arc-coloring in general quandle. Refer Section 4 of [7] for this. a r l : a l+ = a a l a b r l : a l = a a l+ a Figure : Relations at crossings ρa l ρa ρa l ρa Figure 3: Arc-coloring From now on, we always assume ρ : π L PSL, C is a given boundary-parabolic representation. Definition.3. The Hopf map h : P CP = C { } is defined by α α β β. To avoid redundant notations, arc-color is denoted by {a,..., a n } without ρ from now on. Choose an element s P corresponding to certain region of the diagram D and determine elements corresponding to the other regions using the relation in Figure 4. The assignment of elements of P to all regions using the relation in Figure 4 is called region-coloring. This assignment is well-defined because the two curves in Figure 5, which we call the cross-changing pair, determine the same region-coloring, and any pair of curves with the same starting and ending points can be transformed each other by finite sequence of cross-changing pairs. 6

7 a s s a Figure 4: Region-coloring a Positive crossing b Negative crossing Figure 5: Well-definedness of region-coloring An arc-coloring together with a region-coloring is called shadow-coloring. The following lemma shows important property of shadow-colorings, which is crucial for showing the existence of solutions of certain equations. Lemma.4. Let L be a lin and assume an arc-coloring is already given by the boundaryparabolic representation ρ : π L PSL, C. Then, for any triple a, s, s a of an arc-color a and its surrounding region-colors s, s a as in Figure 4, there exists a regioncoloring satisfying ha hs hs a ha. 4 Proof. We follow some part of the proof of Proposition in [7]. For the given arc-colors a,..., a n, we choose region-colors s,..., s m so that {hs,..., hs m } {ha,..., ha n } =. 5 This is always possible because, the number of hs satisfying hs {ha,..., ha n } is finite, and hs,..., hs m are uniquely determined by hs. Therefore, the number of hs satisfying {hs,..., hs m } {ha,..., ha n } is finite, but we have infinite freedom to choose hs CP. 7

8 Now consider the case of Figure 4 and assume hs a = hs. Then we obtain where â : CP CP is the Möbius transformation of a = α β that contradicts 5. hs a = â hs = hs, 6 â z = + α β z α β z + α β. Then 6 implies hs is the fixed point of â, which means ha = hs We remar that the condition 5 is stronger than what we actually need in 4. Even though some region-coloring does not satisfy 5, if it satisfies 4, then all results of this article can be applicable. Examples in Section 4 are this type. The arc-coloring induced by ρ together with the region-coloring satisfying Lemma.4 is called the shadow-coloring induced by ρ. This shadow-coloring will determine the exact coordinates of points of the octahedral triangulation in the next section..3 Octahedral triangulations of lin complements In this section, we describe the ideal triangulation of S 3 \L {two points} appeared in [3]. To obtain the triangulation, we consider the crossing j in Figure 6 and place an octahedron A j B j C j D j E j F j on each crossing j as in Figure 7a. Then we twist the octahedron by identifying edges B j F j to D j F j and A j E j to C j E j, respectively. The edges A j B j, B j C j, C j D j and D j A j are called horizontal edges and we sometimes express these edges in the diagram as arcs around the crossing as in Figure 6. a l a s a l D j C j s j s a l a A j B j a s a a l a a l s a l a D j A j a s C j j s a B j s a l a a l a a Positive crossing b Negative crossing Figure 6: Crossing j with shadow-coloring 8

9 F j F j F j D j C j D j C j D j C j A j B j A j B j A j B j E j E j E j a b Figure 7: Octahedron on the crossing j c Then we glue faces of the octahedra following the lines of the lin diagram. Specifically, there are three gluing patterns as in Figure 8. In each cases a, b and c, we identify the faces A j B j E j C j B j E j to C j+ D j+ F j+ C j+ B j+ F j+, B j C j F j D j C j F j to D j+ C j+ F j+ B j+ C j+ F j+ and A j B j E j C j B j E j to C j+ B j+ E j+ A j+ B j+ E j+, respectively. C j B j+ D j B j+ C j A j+ A j B j C j+ D j+ B j C j C j+ D j+ A j B j B j+ C j+ a b c Figure 8: Three gluing patterns Note that this gluing process identifies vertices {A j, C j } to one point, denoted by, and {B j, D j } to another point, denoted by, and finally {E j, F j } to the other points, denoted by P t where t =,..., c and c is the number of the components of the lin L. The regular neighborhoods of and are two 3-balls and that of c t=p t is a tubular neighborhood of the lin L. Therefore, after removing all vertices of the gluing, we obtain an octahedral decomposition of S 3 \L {± }. The octahedral triangulation is obtained by subdividing all the octahedra of the decomposition using the arc-coloring from ρ as follows. If ha ha l in Figure 6, then we subdivide the octahedron into four tetrahedra by adding edge E j F j as in Figure 7b. Also, if ha = ha l, then we subdivide it by adding edge A j C j as in Figure 7c. The result is called the octahedral ideal triangulation of S 3 \L {± }. 9

10 Consider a shadow-coloring of a lin diagram D induced by ρ, and let {a, a,..., a n } be the arc-colors and {s, s,..., s m } be the region-colors. The number of these colors is finite, so can choose an element p P satisfying hp / {ha,..., ha n, hs,..., hs m }. The geometric shape of the triangulation is determined by the shadow-coloring induced by ρ in the following way. If ha ha l and the crossing j is positive, then let the signed coordinates of the tetrahedra E j F j C j D j, E j F j A j D j, E j F j A j B j, E j F j C j B j be a l, a, s a l, p, a l, a, s, p, a l a, a, s a, p, a l a, a, s a l a, p, 7 respectively. Here, the minus sign of the coordinate means the orientation of the tetrahedron does not coincide with the one induced by the vertex-ordering. Also, if the crossing j is negative, then let the signed coordinates of the tetrahedra E j F j C j D j, E j F j A j D j, E j F j A j B j, E j F j C j B j be a l, a, s, p, a l, a, s a l, p, a l a, a, s a l a, p, a l a, a, s a, p, 8 The tetrahedra in Figure 9 shows the signed coordinates of 7 and 8, and Figure 0 shows the relationship between Figure 7b and Figure 9. Here, gluing the pairs of faces E j F j D j and E j F jd j, E jf j B j and E jf jb j, E j F jc j and E jf jc j, E j F j A j and E jf j A j of Figure 0, respectively, induces Figure 7b. p p s *a l s a l a a l a s *a l *a s *a p s p p s *a l p a l *a a l *a a a p s *a p s *a l *a a Positive crossing b Negative crossing Figure 9: Cooridnates of tetrahedra when ha ha l On the other hand, if ha = ha l and j is positive, then let the signed coordinates of the tetrahedra F j A j C j D j, E j A j C j D j, E j A j C j B j, F j A j C j B j be a, s, s a l, p, a l, s, s a l, p, a l a, s a, s a l a, p, a, s a, s a l a, p, 9 0

11 D j E j C j F j D j A j C j B j F j A j E j B j Figure 0: Octahedron in Figure 7b after gluing pairs of faces respectively. If j is negative, then let the signed coordinates be a, s a l, s, p, a l, s a l, s, p, a l a, s a l a, s a, p, a, s a l a, s a, p, 0 respectively. The tetrahedra in Figure shows the signed coordinates of 9 and 0. Note that the orientations of 7 0 are different from [7] and match with [3]. Definition.5. Let v 0, v, v, v 3 CP = C { } = H 3. The hyperbolic ideal tetrahedron with signed coordinate σv 0, v, v, v 3 with σ {±} is called degenerate when some of the vertices v 0, v, v, v 3 have the same coordinate, and non-degenerate when all the vertices have different coordinates. The cross-ratio [v 0, v, v, v 3 ] σ of the non-degenerate signed coordinate σv 0, v, v, v 3 is defined by σ [v 0, v, v, v 3 ] σ v3 v 0 v v = C\{0, }. v v 0 v 3 v The tetrahedra in 7 0 have elements of the coordinates in P. Therefore, we need to send them to points in the boundary of the hyperbolic 3-space H 3 so as to obtain hyperbolic ideal tetrahedra. The Hopf map h, defined in Definition.3, plays the role. Lemma.6. The images of 7 0 under the Hopf map h are non-degenerate tetrahedra. Specifically, if ha ha l and the crossing j is positive, then ha l, ha, hs a l, hp, ha l, ha, hs, hp, ha l a, ha, hs a, hp, ha l a, ha, hs a l a, hp,

12 a Positive crossing b Negative crossing Figure : Cooridnates of tetrahedra when ha = ha l and, if ha ha l and the crossing j is negative, then ha l, ha, hs, hp, ha l, ha, hs a l, hp, ha l a, ha, hs a l a, hp, ha l a, ha, hs a, hp, are non-degenerate hyperbolic ideal tetrahedra. If ha = ha l and the crossing j is positive, then ha l, hs, hs a l, hp, ha, hs, hs a l, hp, 3 ha, hs a, hs a l a, hp, ha l a, hs a, hs a l a, hp, and, if ha = ha l and the crossing j is negative, then ha l, hs a l, hs, hp, ha, hs a l, hs, hp, 4 ha, hs a l a, hs a, hp, ha l a, hs a l a, hs a, hp, are non-degenerate hyperbolic ideal tetrahedra.

13 Proof. The shadow-coloring we are considering satisfies Lemma.4, and all endpoints of edges are adjacent, as a, s, s a in Figure 4, or one of them is p, expect the edges a l, a, a l a, a in the case of ha ha l. Therefore, it is enough to show that ha ha l implies ha l a ha, which is trivial because ha l a = ha a implies ha l = ha. Note that, when ha = ha l, the first two tetrahedra in 3 share the same coordinate with different signs and the other two do the same. Therefore, all tetrahedra cancel each other out and we can remove the octahedron of the crossing. Also, the same holds for 4. When a geometric shape of a triangulation is given, it determines another representation ρ : π L PSL, C by the Yoshida s construction in Section 4.5 of [9]. This construction determines the representation ρ up to conjugate, and we now ρ is conjugate with ρ by the following observation. The Poincaré polyhedron theorem claims the fundamental group π L is generated by the face identifications with certain relations. The face identifications in Figure 9, after applying the Hopf map h, are the Möbius transformations â,..., â n and the identity map. Note that a,..., a n are also generators in Wirtinger presentation of π L in Section.. Hence, two representations ρ and ρ map generators to the same elements, which implies they are the same..4 Complex volume of ρ Consider an ideal tetrahedron with vertices v 0, v, v, v 3, where v CP. For each edge v v l, we assign g l and ĝ l CP, and call them long-edge parameter and edge parameter, respectively. See Figure. Later, we will distinguish them by considering g l is assigned to the edge of a triangulation and ĝ l to the edge of a tetrahedron. v 3 g 3 g 03 g 3 v g 0 g v 0 g 0 v Figure : Long-edge parameter Definition.7. For the long-edge parameter g l of an ideal tetrahedron, Ptolemy identity is the following equation: g 0 g 3 = g 0 g 3 + g 03 g. For example, if we define long-edge parameter g l := v l v, then direct calculation shows v v 0 v 3 v = v v 0 v 3 v + v 3 v 0 v v, 5 3

14 which is the Ptolemy identity. Furthermore, these long-edge parameters satisfy [v 0, v, v, v 3 ] = g 03g g 0 g 3. 6 The Ptolemy identity and 6 are the most important properties, so this definition of longedge parameters loos good enough now. However, to apply the results of [6] and [6], the value of g l should depend only on the edge of the triangulation. In other words, if two edges are glued together in the triangulation and they have the long-edge parameters g l and g ab respectively, then we need g l = g ab. We call this condition the coincidence of longedge parameter. We also need extra condition that the orientations of the two glued edges induced by the vertex-orientations of each tetrahedra should coincide. However, the vertexorientation in 4 always satisfy it. Unfortunately, g l := v l v does not satisfy this condition, so we will modify them using [7] as follows. α β At first, consider two elements a =, b = in P. We define determinant deta, b by α α β deta, b := ± det α β β = ±α β β α. Note that the determinant is defined up to sign due to the choice of the representative α α a = = P. To remove this ambiguity, we fix representatives α α of arccolors in C \{0} once and for all. Then we fix a representative of one region-color, which uniquely determines the representatives of all the other region-colors by the arc-coloring. This is due to the fact that s ±a = s a for any s, a C \{0}. After fixing all the representatives of shadow-colors, we obtain a well-defined determinant α β deta, b = det = α α β β β α. 7 Lemma.8. For a, b, c C \{0}, the determinant satisfies Proof. Let a = α α, b = β deta c, b c = detc deta c, b c = deta, b. γ + γ γ, c =, and C = γ γ γ. Then γ γ α β, C = det C deta, b = deta, b. β β α The difference with [7] is that they chose a sign of the determinant once and for all. Their choice is good enough to define long-edge parameter g j, but not for edge parameter ĝ j. 4

15 Consider the shadow-coloring and the coordinates of tetrahedra in Figures 9 and. We define the edge parameter ĝ l using those coordinates. Specifically, when the signed coordinate of the tetrahedron is σa 0, a, a, a 3 with σ {±} and a C \{0}, we define the edge parameter by ĝ l = deta, a l. 8 For example, the edge parameters of the tetrahedron a l, a, s, p in the left-hand or the right-hand side of Figures 9 are defined by ĝ 0 = deta l, a, ĝ 0 = deta l, s, ĝ 03 = deta l, p, ĝ = deta, s, ĝ 3 = deta, p, ĝ 3 = dets, p. Lemma.9. The edge parameter ĝ l of the tetrahedron σa 0, a, a, a 3 defined in 8 satisfies the Ptolemy identity and Proof. From 7, we obtain where x = Let a = x x α [ha 0, ha, ha, ha 3 ] = ĝ03ĝ ĝ 0 ĝ 3. 9 hx hy = x y = x y y and y =. y β detx, y x y, 0 for = 0,..., 3, and let v = ha = α β. Then 5 and 0 imply deta 0, a β 0 β deta, a 3 β β 3 = deta 0, a β 0 β deta, a 3 β β 3 + deta 0, a 3 β 0 β 3 deta, a β β, which is equivalent to the Ptolemy identity ĝ 0 ĝ 3 = ĝ 0 ĝ 3 + ĝ 03 ĝ. Also, using 0, we obtain [ha 0, ha, ha, ha 3 ] = deta 0,a 3 β 0 β 3 deta,a 3 β β 3 deta,a β β deta 0,a = ĝ03ĝ. ĝ 0 ĝ 3 β 0 β Note that, by the same calculation of the proof above, we obtain [ha 0, ha 3, ha, ha ] = ĝ0ĝ 3 ĝ 0 ĝ 3, [ha 0, ha, ha 3, ha ] = ĝ0ĝ 3 ĝ 03 ĝ. If we put z σ = [ha 0, ha, ha, ha 3 ], using Ptolemy identity, the above equations are expressed by z σ = ĝ03ĝ, ĝ 0 ĝ 3 z = ĝ0ĝ 3, σ ĝ 0 ĝ 3 z = ĝ0ĝ 3. σ ĝ 03 ĝ 5

16 The edge parameter ĝ j defined above satisfies all needed properties of the long-edge parameter g j except the coincidence of long-edge parameter, which ĝ j satisfies up to sign. To see this phenomenon, consider the two edges of Figure 9a as in Figure 3, which are glued in the triangulation. Assume the chosen representative of a m in Figure 3 satisfies a m = a l a C \{0}. This actually happens often. For example, the minus signs of 46 and 47 in Section 4 show this situation. Then the edge parameters satisfy ĝ 0 = deta l, a = deta l a, a = deta m, a = ĝ 0. a l ĝ 0 a a ĝ 0 a m = a l a Figure 3: Example of the inconsistency of edge parameter To define the long-edge parameter g j, we assign certain signs to the edge parameters g j = ±ĝ j, so that the consistency of the long-edge parameter holds. Due to Lemma 6 of [7], any choice of values of g j determines the same complex volume. Actually, in Section 3, we do not consider the exact values of g j, but use the existence of them. The relations of the edge parameters become z σ = ± g 03g g 0 g 3, z σ = ±g 0g 3 g 0 g 3, z σ = ±g 0g 3 g 03 g. Using, we define integers p and q by { pπi = log z σ + log g 03 + log g log g 0 log g 3, qπi = log z σ + log g 0 + log g 3 log g 0 log g 3. 3 Now we consider the tetrahedron with the signed coordinate σa 0, a, a, a 3 and the signed triples σ[z σ ; p, q] PC. The extended pre-bloch group is denoted by PC here. For the definition, see Definition.6 of [6]. To consider all signed triples corresponding to all tetrahedra in the triangulation, we denote the triple by σ t [zt σt ; p t, q t ], where t is the index of tetrahedra. We define a function L : PC C/π Z by [z; p, q] Li z + πi π log z log z + q log z + p log z 6, 4 6

17 where Li z = z log t dt is the dilogarithm function. Well-definedness of L was proved 0 t in []. Recall that, for a boundary-parabolic representation ρ, the volume volρ and the Chern-Simons invariant csρ was already defined in [6]. We call volρ + i csρ the complex volume of ρ. The following theorem is one of the main result of [7]. Theorem.0 [6], [7]. For a given boundary-parabolic representation ρ and the shadowcoloring induced by ρ, the complex volume of ρ is calculated by σ σ t L[z t t ; p t, q t ] ivolρ + i csρ mod π, t where t is over all tetrahedra of the triangulation defined in Section.3. Proof. See Theorem 5 of [7]. Note that the removal of the tetrahedra in 3 and 4 does not have any effect on the complex volume. For example, if we put [z; p, q] and [z ; p, q ] the corresponding triples of the tetrahedron ha l, hs, hs a l, hp and ha, hs, hs a l, hp in 3, respectively, and put {g l }, {g l } the sets of long-edge parameters of the two tetrahedra, respectively. Then, from ha l = ha, we obtain z = z. Furthermore, we can choose long-edge parameters so that g l = g l holds for all pairs of edges sharing the same coordinate, which induces p = p, q = q and L[z; p, q] L[z ; p, q ] = 0. 3 Optimistic limit In this Section, we will use the result of Section to redefine the optimistic limit of [3] and prove the existence of solutions of H. At first, we consider a given boundary-parabolic representation ρ and its shadow-coloring of a lin diagram D. For the diagram, define sides of the diagram by the lines connecting two adjacent crossings. The word edge is more common than side here. However, we want to eep the word edge for the edges of a triangulation. For example, the diagram in Figure 4 has eight sides. We assign z,..., z n to sides of D as in Figure 4 and call them side variables. For the crossing j in Figure 5, let z e, z f, z g, z h be side variables and let a l, a be the arc-colors. If ha ha l, then we define the potential function V j of the crossing j by V j z e, z f, z g, z h = Li z f z e Li z f z g + Li z h z g Li z h z e. 5 On the other hand, if ha l = ha in Figure 5, then we introduce new variables w j e, w j f, wj g of the crossing j and define V j z e, z f, z g, z h, w j e, w j f, wj g 6 = log w j e log z e + log w j f log z f log w j g log z g + log wj ew j g w j f log z h. 7

18 Figure 4: Sides of a lin diagram a l a D j C j z h z g j z e z f A j B j Figure 5: A crossing j with arc-colors and side variables For notational convenience, we put w j h := wj ew j g/w j f. We can choose any three of wj e, w j f, wj g, w j h as free variables in 6. We call the crossing j in Figure 5 degenerate when ha l = ha holds. In particular, when the degenerate crossing forms a in, as in Figure 6, we put V j z e, z f, z g, w j e, w j f = log w j e log z e + log w j f log z f log w j f log z f + log wj ew j f w j f = log w j e log z e + log w j e log z g. log z g Consider the crossing j in Figure 5 and place the octahedron A j B j C j D j E j F j as in Figure 7. When the crossing j is non-degenerate, in other words ha ha l, we consider Figure 7b and assign shape parameters z f z z e, g z z f, h zg and ze z h to the horizontal edges A j B j, B j C j, C j D j, D j A j, respectively. On the other hand, if the crossing j is degenerate, in other words ha = ha l, then we consider Figure 7c and assign shape parameters we, j w j f, wj g and w j h to the edges A j F j, B j E j, C j F j and D j E j, respectively. 3 3 Note that, when ha = ha l, by adding one more edge B j D j to Figure 7c, we obtain another 8

19 Figure 6: Kin The potential function V z,..., z n, w j,... of the lin diagram D is defined by V z,..., z n, w j,... = j V j, where j is over all crossings. For example, if ha ha in Figure 4, then a 4 = a a implies 4 ha 4 ha, a = a a 3 does 5 ha ha 3 ha, a = a 3 a 4 does ha 4 ha 3, a 4 = a 3 a does ha 4 ha, and the potential function becomes { V z,..., z 8 = Li z 5 Li z 5 + Li z 4 Li z } 4 7 z 7 z 8 z 8 z { 7 + Li z Li z + Li z 8 Li z } 8 z 3 z 4 z 4 z { 3 + Li z 3 Li z 3 + Li z Li z } z 6 z 5 z 5 z { 6 + Li z 6 Li z 6 + Li z 7 Li z } 7. z z z z Note that, if ha l ha for any crossing j in Figure 5, then the definition of the potential function above coincides with the definition in Section of [3]. Therefore, the above definition is a slight modification of the previous one. On the other hand, if ha = ha in Figure 4, then a a = a. This equation and subdivision of the octahedron with five tetrahedra. This subdivision was already used in []. Focusing on the middle tetrahedron that contains all horizontal edges, we obtain wew j g j = w j f wj h. Furthermore, the shape-parameters assigned to D j F j and B j F j are /wj e w j g and /wj g we j, respectively. 4 If ha 4 = ha, then ha a = ha = ha 4 = ha a induces ha = ha, which is contradiction. 5 If ha = ha 3, then ha 3 a 3 = ha 3 = ha = ha a 3 induces ha = ha 3 = ha, which is contradiction. Liewise, if ha = ha 3, then ha = ha a 3 = ha is contradiction. 9

20 the relations at crossings induce 6 a = a = a 3 = a 4, and the potential function becomes V z,..., z 8, w 8, w 4, w 7, w 4, w 8, w 3, w 3 6, w 3 3, w 3 5, w 4, w 4 7, w 4 = log w 8 log z 8 + log w 4 log z 4 log w 7 log z 7 + log w 5 log z 5 log w 4 log z 4 + log w 8 log z 8 log w 3 log z 3 + log w log z log w 3 6 log z 6 + log w 3 3 log z 3 log w 3 5 log z 5 + log w 3 log z log w 4 log z + log w 4 7 log z 7 log w 4 log z + log w 4 6 log z 6, where w5 = w8w 7/w 4, w = w4w 3/w 8, w 3 = w6w 3 5/w and w6 4 = ww 4 /w For the potential function V z,..., z n, w j,..., let H be the set of equations { } V H := expz =, expw j V = =,..., n, j : degenerate, 8 z w j and S = {z,..., z n, w j,...} be the solution set of H. Here, solutions are assumed to satisfy the properties that z 0 for all =,..., n and z f z e, zg z f, z h zg, ze z h, zg z e, z h zf in Figure 5 for any non-degenerate crossing, and w j 0 for any degenerate crossing j and the index. All these assumptions are essential to avoid singularity of the equations in H and log 0 in the formula V 0 defined in 3. Even though we allow w j = here, the value we are interested in always satisfies w j. Proposition 3.. For the arc-coloring of a lin diagram D induced by ρ and the potential function V z,..., z n, w j,..., the set H induces the whole set of hyperbolicity equations of the octahedral triangulation defined in Section.3. The hyperbolicity equations consist of the Thurston s gluing equations of edges and the completeness condition. Proof of Proposition 3.. When no crossing is degenerate, this proposition was already proved in Section 3 of [3]. To see the main idea, chec Figures 0 3 and equations of [3]. Equation 3. is a completeness condition along a meridian of certain annulus, and are gluing equations of certain edges. These three types of equations induce all the other gluing equations. Therefore, we consider the case when the crossing j in Figure 5 is degenerate. Then, the following three equations expwe j V w j e = z h z e =, expw j f V w j f = z f =, expw j V g z h w j g = z h z g = 9 induce z e = z f = z g = z h. This guarantees the gluing equations of horizontal edges trivially by the assigning rule of shape parameters. Note that the shape parameters assigned to the horizontal edges of the octahedron at a degenerate crossing are always. 0

21 z f z f C j z B j+ D j z A j+ A j B j C j+ a D j+ z e z f B j C j B j+ b C j+ z e z f D j z B j+ C j z A j+ B j C j C j+ c D j+ z e A j B j B j+ d C j+ z e Figure 7: Four cases of gluing pattern There are four possible cases of gluing pattern as in Figure 7, and we assume the crossing j is degenerate and j + is non-degenerate. The case when both of j and j + are degenerate can be proved similarly. The part of the potential function V containing z in Figure 7a is and V a = log w j log z + Li ze z V V exp z = exp z a = w j z z is equivalent with the following completeness condition w j z e z z f z Li zf z, z e z f = z z = along a meridian m in Figure 8a. Compare it with Figure of [3]. Here, a j, b j, c j, b j+, c j+, d j+ in Figure 8a are the points of the cusp diagram, which lie on the edges A j E j, B j E j, C j E j, B j+ F j+, C j+ F j+, D j+ F j+ of Figure 7a, respectively. The part of the potential function V containing z in Figure 7b is V b = log w j log z Li z z e + Li z z f and V V exp z = exp z b = z z w j z z = z e z f is equivalent with the following completeness condition z z = z f z e w j 6 The relation a 4 = a a induces a 4 = a, a 4 = a 3 a does a 4 = a 3, and a = a 3 a 4 does a = a 4.,

22 b j+ a j = c j+ m m b j = d j+ c j a b b j+ c j c d Figure 8: Four cusp diagrams from Figure 7 along a meridian m in Figure 8b. Here, b j, c j, d j, a j+, b j+, c j+ in Figure 8b are the points of the cusp diagram, which lie on the edges B j F j, C j F j, D j F j, A j+ E j+, B j+ E j+, C j+ E j+ of Figure 7a, respectively. To simplify the cusp diagram in Figure 8b, we subdivided the polygon A j B j C j D j F j in Figure 7c into three tetrahedra by adding the edge B j D j. The part of the potential function V containing z in Figure 7c is V c = log w j log z ze + Li z Li zf z, and V V exp z = exp z c = z z w j z e z f = z z

23 is equivalent with the following gluing equation w j z e z f = z z of c j = c j+ in Figure 8c. Compare it with Figure of [3]. Here, b j, c j, d j, b j+, c j+, d j+ in Figure 8c are the points of the cusp diagram, which lie on the edges B j F j, C j F j, D j F j, B j+ F j+, C j+ F j+, D j+ F j+ of Figure 7a, respectively, and the edges d j c j and b j c j are identified to b j+ c j+ and d j+ c j+, respectively. To simplify the cusp diagram in Figure 8c, we subdivided the polygon A j B j C j D j F j in Figure 7c into three tetrahedra by adding the edge B j D j. The part of the potential function V containing z in Figure 7d is and V d = log w j log z Li z z e V V exp z = exp z d = w j z z + Li z z f is equivalent with the following gluing equation w j z z = z e z f, z z = z e z f of b j = b j+ in Figure 8d. Compare it with Figure 3 of [3]. Here, a j, b j, c j, a j+, b j+, c j+ in Figure 8d are the points of the cusp diagram, which lie on the edges A j E j, B j E j, C j E j, A j+ E j+, B j+ E j+, C j+ E j+ of Figure 7a, respectively, and the edges a j b j and c j b j are identified to c j+ b j+ and a j+ b j+, respectively. Note that the case when both of the crossings j and j + in Figure 7 are degenerate can be proved by the same way. On the other hand, it was already shown in [3] that all hyperbolicity equations are induced by these types of equations see the discussion that follows Lemma 3. of [3], so the proof is done. In [3], we could not prove the existence of a solution of H, in other words S, so we assumed it. However, the following theorem proves the existence by directly constructing one solution from the given boundary-parabolic representation ρ together with the shadowcoloring. Theorem 3.. Consider a shadow-coloring of a lin diagram D induced by ρ which satisfies Lemma.4 and the potential function V z,..., z n, w j,... from D. For each side of D with the side variable z, arc-color a l and the region-color s, as in Figure 9, we define z 0 := deta l, p deta l, s. 30 3

24 Also, if the positive crossing j in Figure 0a is degenerate, then we define we j 0 dets, p := dets a, p, wj f 0 := dets a l a, p, dets a, p wg j 0 := dets a l a, p, w j dets, p h dets a l, p 0 := dets a l, p, and, if the negative crossing j in Figure 0b is degenerate, then we define w j e 0 := dets a l, p dets a l a, p, wj f 0 := w j g 0 := dets a, p dets, p dets a, p dets a l a, p,, w j h 0 := dets a l, p. dets, p Then z 0 0,,, w j 0 0, for all possible j,, and z 0,..., z 0 n, w j 0,... S. a l z s s a l Figure 9: Region-coloring a l a s a l z h z g s s a l a j z e z f s a a ±a l a a Positive crossing a l a s z h z g s a l s a j z e z f a s a l a ±a l a b Negative crossing Figure 0: Crossings with shadow-colors and side-variables Note that the ± signs in the arc-colors of Figure 0 appears due to the representatives of the colors in C \{0}. However, ± does not change the value of z 0 because det±a l, p det±a l, s = deta l, p deta l, s = z0. Liewise, the value of w j 0 does not depend on the choice of ± because the representatives of region-colors are uniquely determined from the fact s ±a = s a for any s, a C \{0}. 4

25 Proof of Theorem 3.. At first, when the crossing j in Figure 0 is degenerate, we will show z e 0 = z 0 f = z g 0 = z 0 h, 3 α c α which satisfies 9. Using ha = ha l, we put a = and a β l = = c a c β for some constant c C\{0}. Then we obtain a l a = a l and, if j is positive crossing, then z 0 e = c deta, p c deta, s = deta l, p deta l, s = z0 h, z 0 f = det±a l a, p det±a l a, s a = deta l a, p deta l a, s a = deta l, p deta l, s = z0 h, z 0 g = c deta, p c deta, s a l = deta l, p deta l, s a l = z0 h. If j is negative crossing, then by exchanging the indices e g in the above calculation, we obtain the same result. Note that Lemma.4 and the definition of p in Section.3 guarantee z 0 0,, and w j 0 0,, so we will concentrate on proving z 0,..., z n 0, w j 0,... S. Consider the positive crossing j in Figure 0a and assume it is non-degenerate. Also consider the tetrahedra in Figures 9a and 0, and assign variables z e, z f, z g, z h to sides of the lin diagram as in Figure 0a. Then, using 9 and 30, the shape parameters assigned to the horizontal edges A j B j and D j A j are [hs a, hp, h±a l a, ha ] = dets, a detp, ±a l a dets a, ±a l a detp, a = z0 f z e 0 [hs, hp, ha, ha l ] = dets, a l detp, a dets, a detp, a l = z0 e respectively. Liewise, the shape parameters assigned to B j C j and C j D j are z0 g and z0 z 0 h f z g 0 respectively. Furthermore, for any a, b C \{0}, we can easily show that ha b a = hb. If zg z e = deta,s we obtain zg z e =, then ha deta,s a l = hs a l s = ha l, which is contradiction. Therefore,, and z h zf can be obtained similarly. We can verify the same holds for non-degenerate negative crossing j by the same way. Now consider the case when the positive crossing j in Figure 0a is degenerate. See Figures 7c and a. Then, using 9 and 3, the shape parameters assigned to the, z 0 h, 5

26 edges F j A j, E j B j, F j C j and E j D j in Figure 7c are [ha, hs, hp, hs a l ][ha, hs a, hs a l a, hp] dets, p = dets a, p = wj e 0, [h±a l a, hp, hs a l a, hs a ] = detp, s a l a = w j f detp, s a 0, [ha, hs a l a, hp, hs a ][ha, hs a l, hs, hp] = dets a l a, p = w j dets a l, p g 0, [ha l, hp, hs, hs a l ] detp, s = detp, s a l = wj h 0, respectively. We can verify the same holds for degenerate negative crossing j by the same way. Therefore z 0,..., z n 0, w j 0,... satisfies the hyperbolicity equations of octahedral triangulation defined in Section.3 and, from Proposition 3., we obtain z 0,..., z n 0, w j 0,... is a solution of H. By the definition of S, we obtain z 0,..., z n 0, w j 0,... S. We remar that, in the viewpoint of [3], Theorem 3. can be interpreted that a sufficient condition for S of the solution set S defined in [3] is that the parabolic elements corresponding to meridians of two arcs, in other words a l and a in Figure 0, of any crossing have distinct fixed points. To obtain the complex volume of ρ from the potential function V z,..., z n, w j,..., we modify it to V 0 z,..., z n, w j,... := V z,..., z n, w j,... 3 V z log z w j V z w j log w j. j:degenerate This modification guarantees the invariance of the value under the choices of branches of z 0 and w j 0. See Lemma. of [3]. Note that V 0 z 0,..., z n 0, w j 0,... means evaluation of V 0 z,..., z n, w j,... at z0,..., z n 0, w j 0,.... Theorem 3.3. Consider a hyperbolic lin L, the shadow-coloring induced by ρ, the potential function V z,..., z n, w j,... and the solution z0,..., z n 0, w j 0,... S defined in Theorem 3.. Then, V 0 z 0,..., z 0 n, w j 0,... ivolρ + i csρ mod π. 33 6

27 Proof. When the crossing j is degenerate, direct calculation shows that the potential function V j of the crossing in 6 satisfies V j 0 z, z, z, z, w, w, w 3 = 0, 34 for any nonzero values of z, w, w, w 3. To simplify the potential function, we rearrange the side variables z,..., z n to z,..., z r, z r+, zr+, zr+, zr+, 3..., z t,..., zt 3 so that the endpoints of sides with variables z,..., z r are non-degenerate crossings and the degenerate crossings induce z 0 r+ = zr+ 0 = zr+ 0 = zr+ 3 0,..., z 0 t =... = zt 3 0. Then we define simplified potential function V by V z,..., z t := V j z,..., z r, z r+, z r+, z r+, z r+,..., z t, z t, z t, z t. j:non-degenerate Note that V is obtained from V by removing the potential functions 6 of the degenerate crossings and substituting the side variables z f, z g, z h with z e. From 34, we have V 0 z 0,..., z 0 t = V 0 z 0,..., z 0 n, w j 0,..., which suggests V is just a simplification of V with the same value. Therefore, from now on, we use V instead of V and substitute the side variables zr+, zr+, zr+ 3 to z r+ and zt,..., zt 3 to z t, etc. Also, we remove octahedra 3 of degenerate crossings because they do not have any effect on the complex volume. See the comment in the proof of Theorem.0. Now we will follow ideas of the proof of Theorem. in [3]. However, due to the degenerate crossings, we will improve the proof to cover more general cases. At first, we define r by r πi = z V z z =z 0,...,zt=z0 t, 35 for =,..., t, where z means the evaluation of the equation at z 0 =z 0,...,zt=z0,..., z 0 t. t Unlie [3], we cannot guarantee r is an even integer yet, so we need the following lemma. Lemma 3.4. For the value z 0 defined in Theorem 3., z 0,..., z 0 t is a solution of the following set of equations { Ĥ = expz V } = z =,..., t. Proof. Note that, for the degenerate crossing j in Figure 5, we have Also, for r + t, 3 implies w j f 0 w j h 0 we j 0 wg j =. 0 z 0 = z 0 =... = z

28 Therefore, assuming wj f wj h w j ew j g = for any degenerate crossing j, we obtain expz V V = expz expz z z V expz 3 V, z =z z 3 =z for r + t. Therefore, Theorem 3. induces the statement of this lemma. z z 3 To avoid redundant complicate indices, we use z instead of z 0 on. Using the even integer r, we can denote V 0 z,..., z t by in this proof from now V 0 z,..., z t = V z,..., z t t r πi log z. 36 Now we use variables α m, β m, γ l, δ j for the long-edge parameters in 8. We assign α m and β m to non-horizontal edges as in Figure, where m = a, b, c, d. We also assign γ l to horizontal edges, where l is over all regions, and δ j to the edge E j F j inside the octahedron. Although we have α a = α c and β b = β d, we use α a for the tetrahedron E j F j A j B j and E j F j A j D j, α c for E j F j C j B j and E j F j C j D j, β b for E j F j A j B j and E j F j C j B j, β d for E j F j C j D j and E j F j A j D j, respectively. Note that the labeling is consistent even when some crossing is degenerate because, when the crossing j in Figure is degenerate, we obtain z a = z b = z c = z d and, after removing the octahedron of the crossing, the long-edge parameters satisfy α a = α b = α c = α d and β a = β b = β c = β d. = F j z d z c D j C j j A j B j z a z b D j A j C j B j E j Figure : Long-edge parameters of non-horizontal edges 8

29 z b z z a a z b z z a b Figure : Two cases with respect to z Now consider a side with variable z and two possible cases in Figure. We consider the case when the crossing is non-degenerate, or equivalently, z a z z b. If it is degenerate, we assume there is no octahedron at the crossing. For m = a, b, let σ m {±} be the sign of the tetrahedron 7 between the sides z and z m, and u m be the shape parameter of the tetrahedron assigned to the horizontal edge. We put τ m = when z is the numerator of u m σm and τ m = otherwise. We also define pm and qm by 3 so that σm [um σm ; p m, q m] becomes the element of PC corresponding to the tetrahedron. Then,m σm [um σm ; p m, q m] is the element 8 of BC corresponding to the octahedral triangulation in Section.3, and σ m L[u m σm ; p m, q m ] ivolρ + i csρ mod π, 37,m from Theorem.0. By definition, we now In the case of Figure a, we have u a = z z a, u b = z b z. 38 σ a =, σ b = and τ a = τ b =. Using the equation 3 and Figure 3a, we decide p m and qm as follows: { log z za + p a πi = log α log β log α a log β a, log z zb + p b πi = log α log β log α b log β b, 39 { log z za + q aπi = log β + log α a log γ log δ, log z zb + q bπi = log β + log α b log γ log δ. In the case of Figure b, we have 40 σ a =, σ b = and τ a = τ b =. 7 Sign of a tetrahedron is the sign of the coordinate in or. 8 The coefficient appears because the same tetrahedron is counted twice in the summation. 9

30 a b Figure 3: Tetrahedra of Figure Using the equation 3 and Figure 3b, we decide p m and qm as follows: { log z a z + p a πi = log α a log β a log α log β, log z b z + p b πi = log α b log β b log α log β, 4 { log z a z + q aπi = log β a + log α log γ log δ, log z b z + q bπi = log β b + log α log γ log δ. The equations 39 and 4 holds for all possible non-degenerate crossings, so we get the following observation. Observation 3.5. We have log α log β log z + A mod πi, for all =,..., t, where A is a complex constant number independent of. Note that the potential function V is expressed by V z,..., z t = σ m Li u m σm, 43,m where is over all sides and the range of m = a, b,... is determined by. From now on, we put the range of m by m = a,..., d. 9 Recall that r was defined in 35. Direct calculation 9 This does not mean m = a, b, c, d in general. If z is connected to a degenerate crossing, then m can have bigger range. 30 4

31 shows r πi = m=a,...,d σ m τ m log u m σm. Combining 40 and 4, we obtain { σ m τ m log u m σm + q m πi } = log γ + log γ, m=a,b for both cases in Figure. Note that α a = α b in 40 and β a = β b in 4. Therefore, we obtain { σ m τ m log u m σm + q m πi } = 0, and m=a,...,d r πi = m=a,...,d Lemma 3.6. For all possible and m, we have σ m τ m q m πi. 44 t σ m q m πi logu m σm r πi log z mod π. 45,m = Proof. Note that, by definition, σ m = σ m, τ m = τ m and τ m u m z σm = = z τ m zm τ m. z m Using the above and 44, we can directly calculate t t σ m q m πi logu m σm = m=a,...,d = = m=a,...,d t r πi log z. = σ m τ m q m πi log z mod π Lemma 3.7. For all possible and m, we have,m σ m log u m σm logu m σm + p m πi t r πi log z l mod π. = Proof. From 39 and 4, we have logu m σm + p m πi = τ m log α log β + τ mlog α m log β m. 3

32 Therefore,,m = σ m log u m σm logu m σm + p m πi t = = m=a,...,d σ m τ m log u m σm log α log β t r πilog α log β. = Note that t r πi = = t = z V z = 0 because V is expressed by the summation of certain forms of Li za z b and z a Li z a /z b z a + z b Li z a /z b z b = log z a z b + log z a z b = 0. By using Observation 3.5, the above and the fact that r is even, we have t r πilog α log β = t r πilog z + A = = t r πi log z mod π. = Combining 37, 43, Lemma 3.6 and Lemma 3.7, we complete the proof of Theorem 3.3 as follows: ivolρ + i csρ σ m L[u m σm ; p m, q m ],m = σ m Li u m π σm + σ m q m πi log u m σm 6 4,m,m + σ m log u m σm log u m σm + p m πi 4,m V z,..., z n t r πi log z = V 0 z,..., z n mod π. = 3

33 s s s 3 s 4 s 6 s 5 Figure 4: Figure-eight not 4 with parameters 4 Examples 4. Figure-eight not 4 For the figure-eight not diagram in Figure 4, let the elements of P corresponding to the arcs be 0 t t a =, a t =, a 0 3 =, a + t 4 =, t where t is a solution of t + t + = 0. These elements satisfy a a = a 4, a 3 a 4 = a, a a 3 = a, a 3 a = a 4, 46 where the identities are expressed in C \{0}, not in P = C \{0}/±. Let ρ : π 4 PSL, C be the boundary-parabolic representation determined by a,..., a 4. We define the shadow-coloring of Figure 4 induced by ρ by letting s =, s = 0 s 5 =, s 3 = t t + 4 t t, s t + 4 =, t + 3, s 6 =, p =. t + Direct calculation shows this shadow-coloring satisfies 4 in Lemma.4. However, this does not satisfy 5. All values of ha,..., ha 4 are different, hence the potential function V z,..., z 8 of 33