Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links

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1 Optimistic imits of Kashaev invariants and compex voumes of hyperboic ins arxiv:30.69v [math.gt] Mar 03 Jinseo Cho, Hyu Kim and Seonhwa Kim December, 03 Abstract Yoota suggested an optimistic imit method of the Kashaev invariants of hyperboic nots and showed it determines the compex voumes of the nots. His method is very effective and gives amost combinatoria method of cacuating the compex voumes. However, to describe the trianguation of the not compement, he restricted his method to not diagrams with certain conditions. Athough these restrictions are genera enough for any hyperboic nots, we have to seect a good diagram of the not to appy his theory. In this artice, we suggest more combinatoria way to cacuate the compex voumes of hyperboic ins using the modified optimistic imit method. This new method wors for any in diagrams, and it is more intuitive, easy to hande and has natura geometric meaning. Introduction Kashaev conjectured the foowing reation in [4] : og L N π im N N = vol, where L is a hyperboic in, vol is the hyperboic voume of S 3 \L, and L N is the N-th Kashaev invariant of L. After that, the generaized conjecture was proposed in [9] that og L N πi im N N ivol + i csl mod π, where csl is the Chern-Simons invariant of S 3 \L defined moduo π in [7]. These are now caed Kashaev voume conjectures and vol + i csl is caed the compex voume of L. When Kashaev suggested the conjecture in [4], he cacuated a certain vaue using an anaytic function induced from the Kashaev invariant and showed numericay that the vaue coincides with the voume of the in for few cases. In [8], this function was named the potentia function, and the vaue was named the optimistic imit of the Kashaev invariant and

2 denoted by πi o-im N imit. Then Yoota proved og L N. See Section. of [] for the exact meaning of the optimistic N og K N πi o-im N N ivok + i csk mod π, for hyperboic nots K in [3] by introducing natura geometry corresponding to the optimistic imit. Eaborating on the geometry, he defined a trianguation of S 3 \K {two points} and transformed it into the trianguation of S 3 \K by coapsing certain tetrahedra. He defined the potentia function refecting this coapsing process and proved the derivation of this function gives the hyperboicity equations, i.e. Thurston s guing equations and the competeness conditions of the trianguation. See Section 3 for the definitions. His method is very effective and gives amost combinatoria method of cacuating the compex voumes of hyperboic nots. See [] for a brief survey. However, understanding Yoota s method in [3] is not easy for severa reasons. We thin a major difficuty ies on the coapsing process of the trianguation. To mae the coapsing wors we, he deformed the not diagram into certain,-tange diagram satisfying severa non-trivia conditions and restricted his method ony to nots. Furthermore, the coapsing process twists the natura trianguation to a compicate one. To overcome these difficuties, we wi deveop new version of Yoota theory without coapsing process here. Our method does not need to deform the diagram because it is appicabe to any in diagrams without restriction. In Section of this artice, we define the natura potentia function V z,..., z n of a hyperboic in L combinatoriay from the in diagram. Then we wi consider the foowing set of equations { } H := expz = z =,..., n. In Section 3, we introduce an idea trianguation of S 3 \L {two points}, and name it octahedra trianguation. It was the same one considered in [3] before the coapsing, and it aso appeared in [] as a natura trianguation of the in compement inside S [0, ]. On the other hand, Luo considered idea trianguations of cosed 3-manifods by removing vertices in [5] and considered their hyperboicity equations. Later, Luo, Timann and many others considered idea trianguations of any 3-manifods by removing non-idea vertices and found severa properties of their hyperboicity equations. See [0] for exampe. We consider the hyperboicity equations of the octahedra trianguation in this sense. Note that this idea trianguation of S 3 \L {two points} can be obtained by removing two non-idea points from the trianguation of S 3 \L, as in [0]. One of the most important properties of the potentia function V is the foowing proposition. Proposition.. For a hyperboic in L with a fixed diagram, consider the potentia function V z,..., z n defined in Section. Then the set H defined in becomes the hyperboicity equations of the octahedra trianguation of S 3 \L {two points}.

3 The exact construction of the trianguation and the proof wi be in Section 3. We remar that this proposition aso hods for the potentia functions of the coapsed cases in [3] and [], but the proof of this artice is more natura and far easier than the previous ones. This is because the coapsing process distorts the natura geometry of the trianguation, so one has to eep trac of the changes carefuy. Let S = {z,..., z n } be the set of soutions of H in C n. In this artice, we aways assume S. Then, by Theorem of [0], a edges in the octahedra trianguation are essentia. Essentia edge roughy means it is not nu-homotopic. See [0] for the exact definition. Therefore, using Yoshida s construction in Section 4.5 of [6], for a soution z S, we can obtain the boundary-paraboic representation ρ z : π S 3 \L PSL, C. Note that the voume voρ z and the Chern-Simons invariant csρ z of ρ z were defined in [4]. We ca voρ z + i csρ z the compex voume of ρ z. For the soution set S, et S j be a path component of S satisfying S = j J S j for some index set J. We assume 0 J for notationa convenience. To obtain we-defined vaues of the potentia function V z,..., z n see Lemma., we sighty modify it to n V 0 z,..., z n := V z,..., z n z og z. 3 z Then the main resut of this artice is as foows: Theorem.. Let L be a hyperboic in with a fixed diagram and V z,..., z n be the potentia function of the diagram. Assume the soution set S = j J S j is not empty. Then, for any z S j, V 0 z is constant depends ony on j and = V 0 z i voρ z + i csρ z mod π, 4 where ρ z is the boundary-paraboic representation obtained in. Furthermore, there exists a path component S 0 of S satisfying for a z S 0. V 0 z i vol + i csl mod π, 5 The proof wi be given in Section 4. The main idea is to use Zicert s formua of the extended Boch group in [4], which was aready appeared in [3]. However our proof is simper because we do not consider any coapsing. Athough we restricted L to hyperboic ins, we remar that Proposition. and Theorem. sti hods for non-spitting diagrams of non-hyperboic ins except for the existence of We ony consider soutions satisfying the condition that, when the potentia function is expressed by V z,..., z n = ±Li za, the variabes inside the diogarithms satisfy za / {0,, }. Previousy, in [3] and [], these soutions were caed essentia soutions. The soution z S satisfies the competeness condition, so ρ z is boundary-paraboic. 3

4 S 0 and 5. The definitions of voρ z and csρ z are from [4]. That is because we do not use the hyperboic structure of L but the boundary-paraboic representation ρ z in, which can be non-discrete or non-faithfu. We ca the vaue V 0 z the optimistic imit of the Kashaev invariant. Note that it depends on the choice of the diagram and the path component S j. Finay, in Section 5, we appy our resuts to the twist nots and cacuate the compex voumes of representations. Potentia function V z,..., z n Consider a hyperboic in L and its diagram D. For simpicity, we aways assume D does not have any in by removing them as in Figure. Figure : Removing ins We define sides of D by the arcs connecting two adjacent crossing points. 3 For exampe, the diagram of the figure-eight not 4 in Figure has 8 sides. We assign compex variabes z,..., z n to each side of the diagram D. Using the diogarithm function Li z = z og t dt, we define the potentia function of a crossing as in 0 t Figure 3. Note that the potentia function in Figure 3 comes from the forma substitution of the R-matrix of the Kashaev invariant in []. See [] for the meaning of the forma substitution. In [], we defined the potentia function of the corner of a crossing from Figure 4. Foowing this definition, the potentia function of a crossing is then the summation of potentia functions of the four corners. The potentia function V z,..., z n of the diagram D is defined by the summation of a potentia functions of the crossings. For exampe, the potentia function of the figure-eight 3 Most peope use the word edge instead of side we are using here. However, in this paper, we want to eep the word edge for the edge of a tetrahedron. 4

5 z z 7 z 6 z z 5 z3 z 8 4 z Figure : The figure-eight not 4 Li Li + Li Li Figure 3: Potentia function of a crossing not 4 in Figure is { V z,..., z 8 = Li z 6 Li z 6 + Li z 7 Li z } 7 z z z z { + Li z 3 Li z 3 + Li z Li z } z 6 z 5 z 5 z { 6 + Li z 4 Li z 4 + Li z 5 Li z } 5 z 8 z 7 z 7 z { 8 + Li z Li z + Li z 8 Li z } 8. z 3 z 4 z 4 z 3 We define a modified potentia function V 0 z,..., z n as given in 3. Note that V 0 is anaytic since the diogarithm function Li z is anaytic and the term z z consists of ogarithms. This property wi be used impicity in Lemma. beow. Reca that H was defined in. Aso reca that we are considering the soutions z = 5

6 a Positive crossing zb Li π 6 b Negative crossing π 6 Li za Figure 4: Potentia function of a corner z,..., z n C n of H with the property that if the potentia function is expressed by V z,..., z n = ±Li za, then variabes inside the diogarithms satisfy za / {0,, }. This choice is reasonabe because, if za {0,, } for some soution, then at east one of the terms and of V 0 z,..., z n is not we-defined at that soution. In this artice, we aways assume the soution set S C n of H is nonempty. We cannot guarantee S for any in diagram. For exampe, the in diagrams containing Figure 5 aways satisfy S = because expz 4 z 4 = impies z = z 3. However, we can easiy remove this probem by reducing the redundant crossings in this case. We expect that if S = for a given in diagram, changing the diagram propery maes S. z 3 z 4 z z Figure 5: Diagram with S = Note that the functions Li z and og re muti-vaued functions. Therefore, to obtain we-defined vaues, we have to seect proper branch of the ogarithm by choosing arg nd arg z. The foowing emma shows why we consider the potentia function V 0 instead of V. Lemma.. Let z = z,..., z n S. For the potentia function V z,..., z n, the vaue of V 0 z is invariant under a choice of branch of the ogarithm moduo 4π. Proof. Let Li z and og e the functions with different og-branch corresponding to an anaytic continuation of Li z and og z respectivey. Aso et V z,..., z n = ±Li z z m. Then Li z Li z + aπi og z mod 4π z m z m z m 6

7 for a certain integer a, og z og z, og z m og z m, og and, because of z = z,..., z n S, we have z z m og z z m mod πi, Therefore, ± { Li z ± ± ± ±Li z z z m ±Li z z z m m z z m z { m Li z Li z /z m z m z m { Li z z m Li z /z m Li z z /z m z + aπi og z z m z m og z Li z z /z m m z m z Li z /z m z og z m + aπi og z m } + aπi og z z m z m } og z m + aπi og z m z m Li z /z m z z { Li z z m 0 mod πi. og z m } og z aπi og z Li z /z m z og z aπi og z z } Li z /z m og z z m og z m z m mod 4π. The potentia function V 0 is the summation of the above terms, so the proof foows. Lemma.. Let S = j J S j C n be the soution set of H with S j being a path component. Assume S. Then, for any z = z,..., z n S j, V 0 z C j mod 4π, where C j is a compex constant depending ony on j J. Proof. Note that z z is continuous on S j and expz z = for any z S. Therefore, z z = r j, πi, 6 on S j for an integer constant r j, depending on j and. The integer r j, can be changed when z passes through the branch cut of the ogarithm. In this case, we change the branch cut so that r j, is ocay constant. The goba invariance of V 0 is obtained by the oca invariance discussed beow and Lemma.. For any path at = α t,..., α n t : [0, ] S j, 7

8 using 6 and the Chain rue, we have dv 0 at dt = n dv dt at d dt = = n atα z t = This impies V 0 is constant on S j. = n = r j, πi α t α t n = r j, πi og α t n = r j, πi α t α t r j, πi α t α t = 0. Athough we are considering the soution set S in C n, it is more natura to consider S as a subset of the compex projective space CP n. This fact is not used in this artice, but we show the foowing emma for reference. Coroary.3. If z = z,..., z n S j, then λz := λz,..., λz n S j compex number λ. Furthermore, for any nonzero V 0 z V 0 λz mod 4π. Proof. The equations in H are products of the foowing terms Li z /z m exp z = z Li z /z m and exp z m = z, z z m z m z m which are represented ony with ratios of the variabes. This proves the first statement. The second statement comes from Lemma. by choosing a path from z to λz. 3 Octahedra trianguation of S 3 \L {two points} In this section, we describe an idea trianguation of S 3 \L {two points}. We remar that this trianguation was aready appeared in many different paces because it naturay came from the in diagram. For exampe, see Section 3 of []. It was aso appeared in Section. of [] and we named it uncoapsed Yoota trianguation. To obtain the trianguation, we pace an octahedron A B D E F on each crossing as in Figure 6 and twist it by identifying edges B F to D F and A E to E respectivey. The edges A B, B, D and D A are caed horizonta edges and we sometimes express these edges in the diagram as arcs around the crossing in the eft hand side of Figure 6. Then we gue faces of the octahedra foowing the sides of the diagram. Specificay, there are three guing patterns as in Figure 7. In each cases a, b and c, we identify the faces A B E B E to + D + F + + B + F +, B F D F to 8

9 B D F D D B A A B D A E B A C Figure 6: Octahedron on the crossing B + D B + A + A B + D + B + D + A B B + + a b c Figure 7: Three guing patterns D + + F + B + + F + and A B E B E to + B + E + A + B + E + respectivey. Note that this guing process identifies vertices {A, } to one point, denoted by, and {B, D } to another point, denoted by, and finay {E, F } to the other points, denoted by P j where j =,..., s and s is the number of the components of the in L. The reguar neighborhoods of and are 3-bas and that of s j=p j is a tubuar neighborhood of the in L. Therefore, if we remove the vertices P,..., P s from the guing, then we obtain a trianguation of S 3 \L, denoted by T. On the other hand, if we remove a the vertices of the guing, the resut becomes an idea trianguation of S 3 \L {± }. We ca this idea trianguation octahedra trianguation and denote it by T. Let M = S 3 \L and M = S 3 \L {± }. Then there exists a continuous deformation of the deveoping maps from M H 3 to M H 3, caed Thurston s spinning construction. Section 3 of [6] expains this construction for cosed manifods, but it can be appied to our trianguation T by fixing idea points P,..., P s and sending points ± to H 3 = CP. Therefore, the parameter space of T in [6] determines the compex voume of M. We wi 9

10 appy Zicert s formua of [4] to T for cacuating the compex voumes of M. See Section 4 for detais. To describe the parameter space of the octahedra trianguation T, we divide each idea octahedron A B D E F into four idea tetrahedra A B E F, B E F, D E F and D A E F. When,, and are assigned to the sides around the octahedron as in Figure 8, we parametrize each tetrahedra by assigning shape parameters, zc, zc and za to the horizonta edges A B, B, D and D A respectivey. D zd A zc C z c B Figure 8: Parametrizing tetrahedra Note that if we assign a shape parameter u C\{0, } to an edge of an idea tetrahedron, then the other edges are aso parametrized by u, u := and u u := as in Figure 9. u u u u u u u Figure 9: Parametrization of an idea tetrahedron with a parameter u For a given idea trianguation of S 3 \L or S 3 \L {± }, we require two conditions to obtain the compete hyperboic structure; the product of shape parameters on an edge is one for a edges, and the hoonomies of meridian and ongitude act as transations on the cusp. The former are caed Thurston s guing equations and the atter competeness conditions. Note that these conditions are expressed as equations of shape parameters. The whoe set of these equations are caed the hyperboicity equations. The wors of Luo, Timann and others in [6] and [0] use ony Thurston s guing equations, but, in this artice, we aso require competeness conditions. Therefore, if z is a soution of the hyperboicity equations, then the induced representation ρ z : π S 3 \L PSL, C is boundary-paraboic. The rest of this section is devoted to the proof of Proposition.. Note that Proposition. was aready appeared and proved in [3] in a sighty different way. 0

11 Proof of Proposition.. For each octahedron in Figure 6 of the octahedra trianguation, et A be the set of horizonta edges A B, B, D and D A of a crossings. Let B be the set of edges B F, D F, A E, E of a crossings and other edges gued to them. Let C be the set of edges E F of a crossings and et D be set of the other edges in the trianguation. Note that if the diagram D is aternating, then D =. The rue of assigning shape parameters to horizonta edges maes the edge conditions of A and C hod triviay. Lemma 3.. The set of equations H consists of the competeness conditions aong the meridian and Thurston s guing equations of the eements in D. Proof. Consider the foowing three cases in Figure 0. We ca the case a aternating guing and the other cases b and c non-aternating guings. Note that eements of D appear ony in non-aternating guings. Specificay F = + F + D in the case b and B E = B + E + D in the case c. z B + D z B + z A + A B a + D + B b + D + A B c B + + Figure 0: Three cases of guings The variabes,,, and z are assigned to each sides in Figure 0. The potentia function V a of the four corners in Figure 0a is defined by V a z zb zc z = Li + Li Li Li, z z and it induces the foowing equation exp z = exp z a z z = z z z c z = H. 7 z On the other hand, the cusp aong the side z in Figure 0a can be visuaized by the annuus in Figure. In Figure, a, b, c, b +, c +, d + are the points of the cusp, which ie on the edges A E, B E, E, B + F +, + F +, D + F + respectivey, and m is the meridian of the cusp. The competeness condition aong m in Figure becomes { zd z } z zb z z = z z z b z c =, z z

12 F F + b + b + B + c + d + a = c + z A c B + D + z z m z b = d + a b c E E + Figure : Cusp diagram of Figure 0a which is equivaent to 7. The potentia function V b of the four corners in Figure 0b is defined by V b za zb zc zd = Li + Li Li + Li, z z z z and it induces the foowing equation exp z = exp z b z z = z z z c z d = H. 8 z z On the other hand, the cusp aong the side z in Figure 0b can be visuaized by Figure. In Figure, b, c, d, b +, c +, d + are the points of the cusp, which ie on the edges B F, F, D F, B + F +, + F +, D + F + respectivey, and the edges c d and c b are identified to c + b + and c + d + respectivey. Thurston s guing equation of the edge F = + F + D around c = c + in Figure becomes z zb z zd = z a z b z c z d =, z z z z z z which is equivaent to 8. The potentia function V c of the four corners in Figure 0c is defined by V c z z z z = Li Li + Li Li,

13 F F + d b + B b c D B + + c + d + D + d b + z z c = c + z z b d + E E + Figure : Cusp diagram of Figure 0b and it induces the foowing equation exp z = exp z c z z = z z z z = H. 9 On the other hand, the cusp aong the side z in Figure 0c can be visuaized by Figure 3. In Figure 3, a, b, c, a +, b +, c + are the points of the cusp, which ie on the edges A E, B E, E, A + E +, B + E +, + E + respectivey, and the edges b a and b c are identified to b + c + and b + a + respectivey. F F + a c + A + z z b = b + z z A B B + + c a + c a + a b b + c + E E + Figure 3: Cusp diagram of Figure 0c 3

14 Thurston s guing equation of the edge B E = B + E + D around b = b + in Figure 3 becomes z zb z zd = z z z z =, z z which is equivaent to 9. It competes the proof of Lemma 3.. We remar that the cusp diagram of aternating guing becomes an annuus, but that of non-aternating guing eventuay becomes a part of annuus. This comes from the cusp diagrams of the two cases in Figure 4 and Figure 5. C 0 A B 0 D C A D B + C A + A 0 B B C +... a = d = = d 0 a... b = a 0... c = a c = c + 0 = =... = + c b b =b a + b Figure 4: First non-aternating guing and its cusp diagram Due to the competeness conditions in H, the edges d c and b c are identified to b + a + and b + c + respectivey in Figure 4b, and the edges a b and c b are identified to c + d + and c + b + respectivey in Figure 5b. These identifications mae the cusp diagrams topoogica annui. Furthermore, due to the Thurston s guing equations of the edges in C D, the annui have Eucidean structures. 4

15 B 0 D 0 D C 0 A C B D A C C + B B + D +... d = a = = a 0 a 0 = d b = d b = b + c = =... = + b c c =c b d + Figure 5: Second non-aternating guing and its cusp diagram To compete the proof of Proposition., we show the competeness conditions in H and Thurston s guing equations of the the edges in A C D induce the other guing equations of the edges in B. Consider the crossing in Figure 6. The crossing is the previous under-crossing of and the crossing m is the next under-crossing. Thurston s guing equation of B F = D F B foows from the guing equation around b = d of the cusp diagram in Figure 7 since c d and d a m are parae to a b and b c m respectivey. The proof of the case of A E = E B is aso obtained by considering Figure 8 and foowing the same argument as before. As a concusion, we showed H induces Thurston s guing equations of a the edges. The competeness conditions aong the meridian in H and a the guing equations together induce the competeness condition aong the ongitude, so H induces the whoe hyperboicity equations. 5

16 C B A m m A B D A B m C m Figure 6: The case of B F = D F B b a = c a = c m d c a m Figure 7: The cusp diagram of Figure 6 D... A... B m B C m... B D... C m D m Figure 8: The case of A E = E B 6

17 4 Proof of Theorem. In this section, we aways assume z = z,..., z n is a soution in S j and drop the index j of r j, in 6. The main technique of the proof of Theorem. is the extended Boch group theory in [4]. To appy it, we first define the vertex ordering of the octahedra trianguation. In Figure 6, we assign 0 and to the vertices E and F respectivey, to the vertices A and, and 3 to the vertices B and D. This assignment induces the vertex orderings of the four tetrahedra. Note that the vertex ordering of each tetrahedron induces the orientations of the edges and the tetrahedron. The induced orientation of the tetrahedron can be different from the origina orientation induced by the trianguation. For exampe, the tetrahedra E F B and E F A D in Figure 6 are the cases. If the two orientations are the same, we define the sign of the tetrahedron σ =, and if they are different, then σ =. One important property of this vertex orientation is that when two edges are gued together in the trianguation, the orientations of the two edges induced by each vertex orderings coincide. We ca this condition edge-orientation consistency. Because of this property, we can appy the formua in [4]. The trianguation we are using is an idea trianguation, so we aready parametrized a idea tetrahedra of the trianguation by assigning shape parameters to horizonta edges in Section 3. For each tetrahedron with the vertex-orientation, we define an eement of the extended pre-boch group σ[u σ ; p, q] PC, where σ is the sign of the tetrahedron, u is the shape parameter assigned to the edge connecting the 0th and st vertices, and p, q are certain integers. Zicert suggested a way to determine p and q from the deveoping map of the representation ρ : π M PSL, C of a hyperboic manifod M in [4], and showed that L σ[u σ ; p, q] ivoρ + i csρ mod π, 0 where the summation is over a tetrahedra and L[u; p, q] = Li u π 6 + qπi og u + og uog u + pπi is a compex vaued function defined on PC. Athough our idea trianguation T is that of S 3 \L {± }, the formua of [4] is sti vaid because of Thurston s spinning construction. Theorem 4. of [4] aready considered our case and the deveoping map of the representation is the one obtained by Thurston s spinning construction. The map sends ± to idea points corresponding to the trivia ends. To determine p, q of σ[u σ ; p, q] of a tetrahedron with vertex orientation, we assign certain compex numbers g j to the edge connecting the jth and th vertices, where j, {0,,, 3} and j <. We assume g j satisfies the property that if two edges are gued together in the trianguation, then the assigned g j s of the edges coincide. We do not use the exact vaues 7

18 of g j in this artice, but remar that there is an expicit method in [4] for cacuating these numbers using the deveoping map. With the given numbers g j, we can cacuate p, q using the foowing equations, which appeared as equation 3.5 in [4]: { pπi = og u σ + og g 03 + og g og g 0 og g 3, qπi = og u σ + og g 0 + og g 3 og g 3 og g 0. To avoid confusion, we use variabes α m, β m, γ m, δ m instead of g j. We assign α m and β m to non-horizonta edges as in Figure 9, where m = a, b, c, d. We aso assign γ to horizonta edges and δ to the edge E F inside the octahedron. Athough we have α a = α c and β b = β d, we use α a for the tetrahedron E F A B and E F A D, α c for E F B and E F D, β b for E F A B and E F B, β d for E F D and E F A D. We assign vertex orderings of the tetrahedra in Figure 9 by assigning 0 to E, to F, to A and, and 3 to B and D. Then the orientation of the octahedra trianguation induced by this ordering satisfies the edge-orientation consistency. F D A B D A α a=α c β a α d β b=β d α a=α c β b=β d α β c b B E Figure 9: Labeings of non-horizonta edges Observation 4.. For a fixed in diagram with the octahedra trianguation, we have og α og β og z + A mod πi, for a =,..., n, where n is the number of sides of the diagram and A is a compex constant number independent of. Proof. Appying the definition of pπi in to the tetrahedra E F A B and E F B in Figure 9, we have og og og α b og β b og α a og β a mod πi, og og og α b og β b og α c og β c mod πi. 8

19 Note that these equations hod for a tetrahedra in the trianguation. Therefore, by etting A = og α a og β a og, we compete the proof. Now we consider the three cases in Figure 0. For m = a, b, c, d, et σ m be the sign of the tetrahedron between the sides z and z m, and u m be the shape parameter of the tetrahedron assigned to the horizonta edge. We put τ m = when z is the numerator of u m σm and τ m = otherwise. We aso define p m and q m so that σ m [u m σm ; p m, q m ] becomes the eement of PC corresponding to the tetrahedron. By definition, we now In the case a of Figure 0, we have u a = z, u b = z, u c = z, u d = z. σ a =, σ b =, σ c =, σ d = and τ a =, τ b =, τ c Using the equation and Figure 0, we decide p m and q m as foows: =, τ d =. β d 3 β a α = α a d δ γ γ 3 z β β α β α 0 γ 3 γ α δ 3 β = β b c α c α b 0 Figure 0: Case a of Figure 0 9

20 og z a + p a πi = og α + og β a og β og α a, og z + p b πi = og α b + og β og β b og α, og zc z + p c πi = og α c + og β og β c og α, og z + p d πi = og α + og β d og β og α d, og z a + q aπi = og β + og α a og γ og δ, og z + q bπi = og β b + og α og γ og δ, og zc z + q cπi = og β c + og α og γ og δ, og z + q dπi = og β + og α d og γ og δ. In the case b of Figure 0, we have σ a =, σ b =, σ c =, σ d = and τ a = τ b = τ c Using the equation and Figure, we decide p m and q m as foows: og za z + p a πi = og α a + og β og β a og α, og z + p b πi = og α b + og β og β b og α, og zc z + p c πi = og α c + og β og β c og α, og z + p d πi = og α d + og β og β d og α, og za z + q aπi = og β a + og α og γ og δ, og z + q bπi = og β b + og α og γ og δ, og zc z + q cπi = og β c + og α og γ og δ, og z + q dπi = og β d + og α og γ og δ. In the case c of Figure 0, we have σ a =, σ b =, σ c =, σ d = and τ a = τ b = τ c Using the equation and Figure, we decide p m and q m as foows: og z a + p a πi = og α + og β a og β og α a, og z + p b πi = og α + og β b og β og α b, og z c + p c πi = og α + og β c og β og α c, og z + p d πi = og α + og β d og β og α d, og z a + q aπi = og β + og α a og γ og δ, og z + q bπi = og β + og α b og γ og δ, og z c + q cπi = og β + og α c og γ og δ, og z + q dπi = og β + og α d og γ og δ. = τ d =. = τ d =

21 α 3 α β 3 γ δ γ z α a α d β = β a d β α 0 γ 3 α γ δ 3 β = β b c α c α b 0 Figure : Case b of Figure 0 Note that σ m = σm, τ m = τm, u m = u m, p m = p m, q m = qm and σ m [u m σm ; p m, q m ] = σm[u m σ m ; p m, qm] PC. If we put the eement 4,m σm [u m σm ; p m, q m ] PC corresponding to the trianguation of S 3 \L {± }, the potentia function defined in Section can be expressed by the foowing way: V z,..., z n = σ m Li u m σm. By direct cacuation, we obtain for a =,..., n. z z = m=a,...,d,m σ m τ m og u m σm 9 4 The eement has the coefficient because a tetrahedra appear twice in the summation.

22 β d 3 β a α = α a d γ δ γ 3 z β 0 β α 3 α = α b c γ β c γ β b δ 3 α β β 0 Figure : Case c of Figure 0 Lemma 4.. For a =,..., n, we have r πi = m=a,...,d σ m τ m q m πi. Proof. In the case a of Figure 0, using 6, 4, 9, α a = α d and β b = β c, we can directy cacuate the foowing: r πi = z z = m=a,...,d = q a πi + q b πi q c πi + q d πi. σ m τ m og u m σm 0 The cases b and c of Figure 0 aso can be proved by the direct cacuation using 6 and 8.

23 Coroary 4.3. For a possibe and m, we have σ m q m πi ogu m σm,m n r πi og z mod π. = Proof. Note that q m is an integer. Using and Lemma 4., we can directy cacuate n n σ m q m πi ogu m σm σ m τ m q m πi og z mod π = m=a,...,d = = m=a,...,d n r πi og z. = Lemma 4.4. For a possibe and m, we have,m σ m og u m σm ogu m σm + p m πi n r πi og z mod π. = Proof. Substituting the term ogu m σm + p m πi to the summation of ±og α og β terms by appying 3 or 5 or 7 and 0, we can verify,m = σ m og u m σm n = = m=a,...,d ogu m σm + p m πi σ m τ m og u m σm n r πiog α og β. = Note that r is an even integer and z j Li z j /z z j + z Li z j /z z og α og β = og z j z + og z j z = 0 impies n r πi = 0. By using Observation 4. and the above property, we have = n n n r πiog α og β r πiog z + A = r πi og z mod π. = = = 3

24 Combining 0, Coroary 4.3 and Lemma 4.4, we prove 4 as foows: ivoρ z + i csρ z L σ m [u m σm ; p m, q m ],m = σ m Li u m π σm + σ m q m πi og u m σm 6 4,m,m + σ m og u m σm og u m σm + p m πi 4,m V z,..., z n n r πi og z = V 0 z mod π. = The existence of a soution z satisfying voρ z = vol foows from Theorem. of [6]. Athough this theorem was proved for cosed manifods, it is sti true for our case because Thurston s spinning construction and a the other steps of the proof are vaid. From Thurston-Gromov-Godman rigidity Theorem 7. in [3] and 4, we now ρ z is the discrete and faithfu representation and 5 hods. One minor remar is that Theorem. of [6] considered parameter space of Thurston s guing equations without competeness condition, so z ies in the parameter space. However, because ρ z is discrete and faithfu, it is boundary-paraboic and z aso satisfies the competeness condition. Therefore z S and the path component of S containing z is S 0. 5 Exampes of the twist nots Let T n n be the twist not with n + 3 crossings in Figure 3. For exampe, T is the figure-eight not 4 and T is the 5 not. In this section, we show an appication of Theorem. to the twist not T n and severa numerica resuts. We assign variabes a, b, x 0,..., x n+, y 0,..., y n+ to sides of Figure 3. Then the potentia function becomes V T n ; a, b, x 0,..., x n+, y 0,..., y n+ { = Li y 0 b Li y 0 a + Li Li a } y n+ y n+ b { + Li b Li b x 0 a + Li x n+ a Li x } n+ x 0 n { + Li y + Li y + + Li x Li x }. x + y x + =0 We abbreviate the notation of this function to V T n. Finding the whoe soution set of the hyperboicity equations { HT n := expz T } n = z z = a, b, x 0,..., x n+, y 0,..., y n+ 4 y

25 a b y 0 x 0 x y x y... x n- y n- x n y n x n+ y n+ Figure 3: Twist nots T n is neither easy nor usefu. Instead, we can obtain enough soutions by fixing certain numbers, Tn say a =, b = and y n+ =. Then, from expa =, we find x a n+ = 3. If we denote x 0 = t, then we can express a the other variabes using t as foows: Tn from expb =, we find y b 0 = +. Aso, from expx Tn T t 0 x 0 = and expy n 0 y 0 =, we find x = tt+ and y t 4t+8 = 4. t x - y - x y x + y + Figure 4: Crossings of the twist not =,..., n T For =,..., n, the equations expx n T x = and expy n y = of Figure 4 induce the foowing recursive formuas x + = x y x + x + y, y + = x + y x y y. They enabe us to express a the variabes in rationa poynomias of t. Tabe shows x and y in t for = 0,..., 5. 5

26 x 0 t + t y 0 t t + t x 8 4t + t 4 y t 4t x 6 + 6t 7t + t 3 y 3 6t + t + t 3 8t 4t + t 3 x 3 3t 6t + t 3 + t 4 8 9t + 8t 40t 3 + 5t 4 y t 4t + t 4 6t + 6t 7t 3 + t 4 x 4 64t + 64t 4t 3 + t t 464t + 4t 3 57t 4 + 6t 5 y t + 480t t 3 6t 4 + 5t 5 8t 9t + 8t 3 40t 4 + 5t 5 x 5 5t 768t + 480t 3 t 4 6t 5 + 5t t t 3840t t 4 36t 5 + 9t 6 y t 79t + 768t 3 4t 4 6t 5 + 6t 6 56t + 5t 464t 3 + 4t 4 57t 5 + 6t 6 Tabe : x and y for = 0,..., 6 Furthermore, expy n+ T n y n+ = gives a simpe reation y n = 3t 3t 4, which determines the defining equation of t. Tabe shows these equations for n =,..., 5. n Defining equation of t 6 t + 3t = t 40t + 7t 3 = t + 336t 0t 3 + 7t 4 = t 4608t + 464t 3 696t 4 + 8t 5 = t t 9984t t 4 980t t 6 = 0 Tabe : Defining equation of t for n =,..., 5 We checed a the soutions t of the defining equation in Tabe satisfy the equations T expx n T n x n =, expy n T n y n = and expx n n+ x n+ =. 5 Therefore, a the soutions 5 As a matter of fact, checing ony two of them is enough. This is because, from the fact T n...,z,... z z = 0, one of the equations of HT n can be deduced from the others. 6

27 t of the defining equation determine the soutions of HT n. We denote the corresponding representation of t by ρt n t : π S 3 \T n PSL, C. Then Tabe 3 shows the vaues of t and the corresponding compex voumes of ρt n t for n =,..., 5. Note that, when n =, the vaues in Tabe 3 coincide with the resut of the 5 not in Exampe 6.6 of [4]. n t V 0 T n t ivoρt n t + i csρt n t t = i i i t = i i i t = i i i t = i i i t = i i 3 t = i i i t = i i i t = i i i t = i i i 4 t = i i i t = i i i t = i i i t = i i i t = i i 5 t = i i i t = i i i t = i i i t = i i i t = i i i t = i i i Tabe 3: Compex voumes of ρt n t for n =,..., 5 Note that, from the we-nown property see Proposition 4.8 of [6] for exampe, the vaue V 0 T n t with the maxima imaginary part is the compex voume ivot n + i cst n of the hyperboic not T n. We paced them at the top in Tabe 3. We finay remar that the cacuation method in this section aso wors for n > 5 and finding compete soutions of HT n for sma n and probaby for a n is possibe. However, a the vaues of V 0 T n evauated at the compete soutions ie in Tabe 3 for n 5 and probaby do for any n > 5 too. Therefore, our restricted soutions are genera enough for cacuating compex voumes of twist nots. Acnowedgments The authors appreciate Yunhi Cho and Jun Muraami for discussions and suggestions on this wor. The first author is supported by POSCO TJ Par foundation. 7

28 References [] J. Cho. Yoota theory, the invariant trace fieds of hyperboic nots and the Bore reguator map [] J. Cho and J. Muraami. Optimistic imits of the coored Jones poynomias [3] S. Francavigia. Hyperboic voume of representations of fundamenta groups of cusped 3-manifods. Int. Math. Res. Not., 9:45 459, 004. [4] R. M. Kashaev. The hyperboic voume of nots from the quantum diogarithm. Lett. Math. Phys., 393:69 75, 997. [5] F. Luo. Voume optimization, norma surfaces and thurston s equation on trianguated 3-manifods [6] F. Luo, S. Timann, and T. Yang. Thurston s spinning construction and soutions to the hyperboic guing equations for cosed hyperboic 3-manifods [7] R. Meyerhoff. Density of the Chern-Simons invariant for hyperboic 3-manifods. In Low-dimensiona topoogy and Keinian groups Coventry/Durham, 984, voume of London Math. Soc. Lecture Note Ser., pages Cambridge Univ. Press, Cambridge, 986. [8] H. Muraami. Optimistic cacuations about the Witten-Reshetihin-Turaev invariants of cosed three-manifods obtained from the figure-eight not by integra Dehn surgeries. Sūriaiseienyūsho Kōyūrou, 7:70 79, 000. Recent progress towards the voume conjecture Japanese Kyoto, 000. [9] H. Muraami, J. Muraami, M. Oamoto, T. Taata, and Y. Yoota. Kashaev s conjecture and the Chern-Simons invariants of nots and ins. Experiment. Math., 3:47 435, 00. [0] H. Segerman and S. Timann. Pseudo-deveoping maps for idea trianguations I: essentia edges and generaised hyperboic guing equations. In Topoogy and geometry in dimension three, voume 560 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, 0. [] J. Wees. Computation of hyperboic structures in not theory. In Handboo of not theory, pages Esevier B. V., Amsterdam, 005. [] Y. Yoota. On the voume conjecture for hyperboic nots. [3] Y. Yoota. On the compex voume of hyperboic nots. J. Knot Theory Ramifications, 07: , 0. 8

29 [4] C. K. Zicert. The voume and Chern-Simons invariant of a representation. Due Math. J., 503:489 53, 009. Department of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seou 30-7, Repubic of Korea Department of Mathematica Sciences, Seou Nationa University, Gwana-ro, Gwana-Gu, Seou 5-747, Repubic of Korea Department of Mathematica Sciences, Seou Nationa University E-mai: do045@gmai.com hyuim@snu.ac.r ryeona7@gmai.com 9