Conneted sum of representtions of knot groups Jinseok Cho rxiv:141.6970v4 [mth.gt] 3 Mr 016 November 4, 017 Abstrct When two boundry-prbolic representtions of knot groups re given, we introduce the connected sum of these representtions nd show severl nturl properties including the unique fctoriztion property. Furthermore, the complex volume of the connected sum is the sum of ech complex volumes modulo iπ nd the twisted Alexnder polynomil of the connected sum is the product of ech polynomils with normliztion. 1 Introduction For ny oriented knots K 1 nd K, the connected sum K 1 #K is well-defined nd hs mny nturl properties. For exmple, ny knot cn be uniquely decomposed into prime knots. Also, the simplicil volumes vol(k 1, vol(k nd vol(k 1 #K of K 1, K nd K 1 #K, respectively, stisfy vol(k 1 #K = vol(k 1 +vol(k. Furthermore, for the Alexnder polynomils K1, K nd K1 #K of K 1, K nd K 1 #K, respectively, we hve K1 #K = K1 K. On the other hnd, mny importnt invrints re defined for boundry-prbolic representtion ρ : π 1 (K PSL(, C nd its lift ρ : π 1 (K SL(, C of the knot group π 1 (K, where the knot group is the fundmentl group of the knot complement S 3 \K nd the boundry-prbolic 1 mens ny meridin loop of the boundry-torus mps to prbolic element in PSL(, C under ρ. For exmple, the complex volume vol(ρ + i cs(ρ nd the twisted Alexnder polynomil K, ρ re some of the importnt invrints. For two boundry-prbolic representtions ρ 1 : π 1 (K 1 PSL(, C nd ρ : π 1 (K PSL(, C, we will define the connected sum of ρ 1 nd ρ ρ 1 #ρ : π 1 (K 1 #K PSL(, C in Section. (Note: After the publiction of this rticle, serious errors were found. The uthor wnted to preserve the content of the publiction, so he dded the errt in the ppendix. Then this definition stisfies the unique fctoriztion property; for ny oriented 1 Boundry-prbolic representtion is lso clled prbolic representtion in mny other texts. 1
knot K = K 1 #... #K g nd ny boundry-prbolic representtion ρ : π 1 (K PSL(, C, there exist unique boundry-prbolic representtions ρ j : π 1 (K j PSL(, C (j = 1,..., g stisfying ρ = ρ 1 #... #ρ g up to conjugte. (If two sme knots K j nd K k pper in K, then the indices of ρ j nd ρ k cn be exchnged. Using this definition, we will show the following dditivity of complex volumes vol(ρ 1 #ρ + i cs(ρ 1 #ρ (vol(ρ 1 + i cs(ρ 1 + (vol(ρ + i cs(ρ (mod iπ, (1 in Section 3. The uthor believes (1 ws lredy known to some experts becuse the knot complement S 3 \(K 1 #K {two points} is obtined by gluing S 3 \(K 1 {two points} nd S 3 \(K {two points} long T \{two points}, torus minus two points. However, the proof in Section 3 will be combintoril nd very simple. Furthermore, while proving (1, we will show the solutions of the hyperbolicity equtions I 1 nd I, which correspond to the five-term tringultions of S 3 \(K 1 {two points} nd S 3 \(K {two points}, respectively, re determined by the solution of I, which corresponds to the tringultion of S 3 \(K 1 #K {two points}. (See Lemm 3.4. This is not usul sitution becuse, in generl, if we glue two mnifolds, then the set of the hyperbolicity equtions chnges, nd even smll chnge on the equtions induces rdicl chnge on the solutions. Therefore, the solution of the glued mnifold usully cnnot detect the solutions of the originl two mnifolds. However, it works for our cse in Section 3 becuse we will use combintoril method. In Section 4, we will show the twisted Alexnder polynomil K1 #K, ρ 1 # ρ is the product of K1, ρ 1 nd K, ρ with normliztion. Finlly, Section 5 will discuss n exmple ρ 1 #ρ : π 1 (3 1 #4 1 PSL(, C nd its lift ρ 1 # ρ : π 1 (3 1 #4 1 SL(, C. Although we restrict our ttention to boundry-prbolic representtions for simplicity, under certin condition, ll results in Section nd 4 re still true for generl representtions. It will be discussed briefly lter. Note tht ll representtions in this rticle re defined up to conjugte. We follow the definition of the complex volume of representtion ρ in [7] nd tht of the twisted Alexnder polynomil in Section of [6]. Definition nd the unique fctoriztion.1 Writinger presenttion nd rc-coloring For fixed oriented knot digrm 3 D of knot K, let α 1,..., α n be the rcs of D. These rcs cn be regrded s the meridin loops of the boundry-torus, which is expressed by smll rrows in Figure 1. Then Wirtinger presenttion gives presenttion of the knot group π 1 (K =< α 1,..., α n ; r 1,..., r n 1 >, ( The gluing mp here is topologiclly unique becuse it is obtined by gluing two pirs of two vertexoriented idel tringles. 3 We ssume D hs t lest one crossing.
where the reltions r 1,..., r n re defined in Figure. (We cn remove one reltion in {r 1,..., r n } becuse it cn be obtined by ll the others. Figure 1: Knot digrm with rcs α 1,..., α n nd rc-colors 1,..., n ( r l : α l+1 = α k α l α 1 k (b r l : α l = α k α l+1 α 1 k Figure : Reltions t crossings Let P be the set of prbolic elements in PSL(, C. For boundry-prbolic representtion ρ : π 1 (K PSL(, C, put k = ρ(α k P nd cll k the rc-color of α k (induced by ρ. Note tht, due to the Wirtinger presenttion, the rc-coloring determines the representtion ρ uniquely (up to conjugte. Therefore, from now on, we express the representtion ρ by using the rc-coloring of digrm D. For, b P, we define the opertion by b = bb 1 PSL(, C. (3 3
Then the rc-colors of crossing stisfy the reltion in Figure 3. Furthermore, the opertion b : b is bijective nd stisfies = nd ( b c = ( c (b c, for ny, b, c P, which implies (P, is qundle. (See [4] or [3] for detils. We define the inverse opertion 1 by 1 c = b = b c. l k l k Figure 3: Arc-coloring One trivil, but importnt fct is tht the rc-coloring uniquely chnges under the Reidemeister moves. (This is trivil becuse rc-coloring is uniquely determined by the representtion ρ. Another wy to see this fct is to consider the reltionship between the Reidemeister moves nd the xioms of qundle. See Figure 4. Definition.1. Let K 1 nd K be oriented knots with digrms nd, respectively. For j = 1,, let ρ j : π 1 (K j PSL(, C be boundry-prbolic representtion. For the rc-colorings of nd (induced by ρ 1 nd ρ, respectively, we mke one rc-color of nd nother rc-color of coincided by conjugtion. We denote the coincided rc-color by P. Then we define the rc-coloring of # following Figure 5. The boundry-prbolic representtion induced by this rc-coloring is denoted by ρ 1 #ρ : π 1 (K 1 #K PSL(, C nd is clled the connected sum of ρ 1 nd ρ. Theorem.. The connected sum ρ 1 #ρ is well-defined up to conjugte. Proof. At first, note tht the well-definedness of K 1 #K (up to isotopy is lredy proved in stndrd textbooks. Let P be the coincided rc-color in the definition. For nother rc-color b P of, there exists unique c P such tht b c =. We will show the rc-coloring of the right-hnd side of Figure 6 is conjugte with tht of Figure 5. (In Figure 6, c mens the rc-coloring of obtined by cting c to ll rc-colors. 4
b b RI RII b b ( = (b Opertion b is bijective b c b c RIII c (c ( b c = ( c (b c c Figure 4: Reidemeister moves nd the xioms of qundle To show the coincidence, we need the observtion on the chnges of rc-colors in Figure 7. The observtion shows tht the rc-colors outside D or D x does not chnge by moving D or D x cross the crossing. Also note tht the rc-colors of the two open rcs of D or D x re lwys the sme. Moving the digrm of the right-hnd side of Figure 6 (or the left-hnd side of Figure 8 inside c, we obtin the middle picture of Figure 8. (The chnged rc-color of is determined by the rc-color c of the two rcs. By cting 1 c to ll rc-colors, we obtin the right-hnd side of Figure 5, nd the coincidence of the rc-colors is proved. On the other hnd, the rc-colorings chnged by pplying Reidemeister moves to the digrms nd re uniquely determined. (See Figure 4. Therefore, chnging digrms does not hve ny impct on the definition of ρ 1 #ρ. Proposition.3. For boundry-prbolic representtion ρ : π 1 (K 1 #K PSL(, C, there exist unique ρ 1 : π 1 (K 1 PSL(, C nd ρ : π 1 (K PSL(, C stisfying ρ = ρ 1 #ρ up to conjugte. (If K 1 = K, then the decomposition is not unique but ρ 1 #ρ = ρ #ρ 1 = ρ up to conjugte. Proof. Choose digrm # of K 1 #K s in Figure 9(. Then the rc-colors, b P 5
b b Figure 5: Arc-coloring of # Figure 6: Arc-coloring of # obtined by connecting different rcs should stisfy = b becuse the corresponding meridin loops re homotopic. Hence we cn define ρ 1 nd ρ using the rc-colorings in Figure 9(b. To show the uniqueness, ssume ρ 1#ρ = ρ = ρ 1 #ρ up to conjugte. Then ρ 1#ρ lso induces n rc-coloring of #, which should be conjugte with the rc-coloring induced by ρ. Therefore, ρ 1 = ρ 1 nd ρ = ρ up to conjugte. The generl cse of K = K 1 #... #K g in Section 1 cn be proved by Proposition.3 nd the induction on g. Remrk tht ll discussions in this section cn be esily generlized to ny representtion ρ j : π 1 (K j GL(k, C. One obstruction is tht, for ρ 1 nd ρ, ρ 1 #ρ is defined only when ρ 1 (α is conjugte with ρ (β for some meridin loops α π 1 (K 1 nd β π 1 (K. Also, generliztion to links is possible if we specify which components re connected by the connected sum. 3 Complex volume of ρ To clculte the complex volume of ρ 1 #ρ explicitly, we briefly review the shdow-coloring of [3] nd the min result of []. We identify C \{0}/± with P by ( α β ( 1 + αβ α β 1 αβ. (4 Then the opertion defined in (3 is given by ( ( ( α γ 1 + γδ γ = β δ δ 1 γδ ( α β P, 6
y D y y x x ( Moving under the crossing y D y y y y D x x (b Moving over the crossing y Figure 7: Chnges of rc-colors b Figure 8: Coincidence of the rc-coloring where the opertion on the right-hnd side is the usul mtrix multipliction. The inverse opertion 1 is given by ( ( ( ( α γ 1 γδ γ 1 α = β δ δ P. 1 + γδ β The Hopf mp h : P CP 1 = C { } is defined by ( α α β β. For the given rc-coloring of the digrm D with rc-colors 1,..., n, we ssign regioncolors s 1,..., s m P to regions of D stisfying the rule in Figure 10. Note tht, if n rc-coloring is fixed, then choice of one region-color determines ll the other region-colors. Lemm 3.1. Consider the rc-coloring induced by the boundry-prbolic representtion ρ : π 1 (K PSL(, C. Then, for ny triple ( k, s, s k of n rc-color k nd its surrounding region-colors s, s k s in Figure 10, there exists region-coloring stisfying Proof. See Proof of Lemm.4 in [3]. h( k h(s h(s k h( k. 7
b ( (b Figure 9: Arc-colors of nd induced by the rc-color of # k s s k Figure 10: Region-coloring The rc-coloring induced by ρ together with the region-coloring stisfying Lemm 3.1 is clled the shdow-coloring induced by ρ. We choose p P so tht h(p / {h( 1,..., h( n, h(s 1,..., h(s m }. (5 From now on, we fix the representtives of shdow-colors in C \{0}, not in P. Note tht this my cuse inconsistency of some signs of rc-colors under the opertion. (In other words, for rc-colors j, k, l P with j = k l, we llow j = ± k ( l C \{0}. α1 As discussed in [3], this inconsistency does not mke ny problem. For = nd ( α β1 b = in C \{0}, we define the determinnt det(, b by β ( α1 β det(, b := det 1 α β = α 1 β β 1 α. For the knot digrm D, we ssign vribles w 1,..., w m to the regions with region-colors s 1,..., s m, respectively, nd define potentil function of crossing j s in Figure 11, where Li (z = z log(1 t dt is the dilogrithm function. 0 t Then the potentil function of D is defined by W (w 1,..., w m := W j, nd we modify it to j : crossings W 0 (w 1,..., w m := W (w 1,..., w m 8 m k=1 ( W w k log w k. w k
w c w d w b W j := Li ( wc w b Li ( wc w d + Li ( wwc w b w d + Li ( w b w + Li ( w d w π w + log w b 6 w log w d j w ( Positive crossing w c w d w b W j := Li ( wc w b + Li ( wc w d Li ( wwc w b w d Li ( w b w Li ( w d w + π w log w b 6 w log w d j w (b Negtive crossing Figure 11: Potentil function of the crossing j Also, from the potentil function W (w 1,..., w m, we define set of equtions { ( } W I := exp w k = 1 w k k = 1,..., m. Then, from Proposition 1.1 of [1], I becomes the set of hyperbolicity equtions of the fiveterm tringultion of S 3 \(K {two points}. Here, hyperbolicity equtions re the equtions tht determine the complete hyperbolic structure of the tringultion, which consist of gluing equtions of edges nd completeness condition. According to Yoshid s construction in Section 4.5 of [5], solution w = (w 1,..., w m of I determines the boundry-prbolic representtion up to conjugte. ρ w : π 1 (S 3 \(K {two points} = π 1 (S 3 \K PSL(, C, Theorem 3. ([]. For ny boundry-prbolic representtion ρ : π 1 (K PSL(, C nd ny knot digrm D of K, there exists the solution w (0 of I stisfying ρ w (0 = ρ, up to conjugte. Furthermore, W 0 (w (0 i(vol(ρ + i cs(ρ (mod π. (6 The vlue vol(ρ + i cs(ρ is clled the complex volume of ρ. The explicit formul of w (0 = (w (0 1,..., w m (0 is very simple. For region of D with region-color s k stisfying Lemm 3.1 nd region-vrible w k, the vlue w (0 k of the regionvrible is defined by w (0 k := det(p, s k. (7 9
Corollry 3.3. For boundry-prbolic representtion ρ 1 #ρ : π 1 (K 1 #K PSL(, C, we hve vol(ρ 1 #ρ + i cs(ρ 1 #ρ (vol(ρ 1 + i cs(ρ 1 + (vol(ρ + i cs(ρ (mod i π. (8 Proof. For the connected sum K 1 #K, consider digrm # nd its shdow-coloring induced by ρ 1 #ρ. (Remrk tht the shdow-coloring stisfies Lemm 3.1. By rerrnging the indices, we ssume {s 1,..., s l, s l+1 } nd {s l, s l+1,..., s m } re the region-colors of nd, respectively, nd s l is the region-color ssigned to the unbounded region of #. (See Figure 1(. s l s l+1 s l s l+1 s l s l+1 s l ( # (b (c Figure 1: Region-colorings of digrms Let W 1 (w 1,..., w l, w l+1 nd W (w l, w l+1,..., w m be the potentil functions of the digrms nd in Figures 1(b nd (c, respectively. Then holds trivilly. W (w 1,..., w m = W 1 (w 1,..., w l, w l+1 + W (w l, w l+1,..., w m Lemm 3.4. For the solution w (0 = (w (0 1,..., w (0 l, w (0 l+1,..., w(0 m of I defined by (7, let w (0 1 := (w (0 1,..., w (0 l, w (0 l+1 nd w(0 := (w (0 l, w (0 l+1,..., w(0 m { ( }. Then w (0 1 nd w (0 re solu- W tions of I 1 := exp w 1 k w k = 1 k = 1,..., l, l + 1 respectively. Furthermore, = ρ j up to conjugte, nd for j = 1,. ρ w (0 j nd I := { exp ( w k W } w k = 1 k = l, l + 1,..., m, (W j 0 (w (0 j i(vol(ρ j + i cs(ρ j (mod π (9 Proof. Note tht the rc-colorings of nd induce the representtions ρ 1 nd ρ, respectively. Both of the region-colorings {s 1,..., s l, s l+1 } nd {s l, s l+1,..., s m } of nd in Figures 1(b nd (c, respectively, stisfy Lemm 3.1. Therefore, by pplying Theorem 3. to Figures 1(b nd (c, we obtin the results of this lemm. The reltion (8 is directly obtined by (6, (9 nd which complete the proof of Corollry 3.3. W 0 (w (0 = (W 1 0 (w (0 1 + (W 0 (w (0, 10
4 Twisted Alexnder polynomil of ρ To clculte (Wd s twisted Alexnder polynomil, we briefly summrize the clcultion method in Section of [6]. At first, we lift the boundry-prbolic representtion ρ : π 1 (K PSL(, C to ρ : π 1 (K SL(, C by ssuming ll rc-colors hve trce two. As mtter of fct, this ssumption ws lredy reflected in the right-hnd side of (4. Under this lifting, we cn trivilly obtin ρ 1 #ρ = ρ 1 # ρ. Therefore, we will use ρ 1 # ρ insted of ρ 1 #ρ from now on. Consider the Wirtinger presenttion of π 1 (K in (. Let γ : π 1 (K Z =< t > be the beliniztion homomorphism given by γ(α 1 =... = γ(α n = t. We define the tensor product of ρ nd γ by ( ρ γ(x = ρ(xγ(x, for x π 1 (K. From the mps ρ nd γ, we obtin nturl ring homomorphisms ρ : Z[π 1 (K] M(, C nd γ : Z[π 1 (K] Z[t, t 1 ], where Z[π 1 (K] is the group ring of π 1 (K nd M(, C is the mtrix lgebr consisting of mtrices over C. Combining them, we obtin ring homomorphism ρ γ : Z[π 1 (K] M(, C[t, t 1 ]. Let F n =< α 1,..., α n > be the free group nd ψ : Z[F n ] Z[π 1 (K] be the nturl surjective homomorphism. Define Φ : Z[F n ] M(, C[t, t 1 ] by Φ = ( ρ γ ψ. Consider the (n 1 n mtrix M ρ whose (k, j-component is the mtrix ( rk Φ M(, C[t, t 1 ], α j where α j denotes the Fox clculus. We cll M ρ the Alexnder mtrix ssocited to ρ. We denote by M ρ, j the (n 1 (n 1 mtrix obtined from M ρ by removing the jth column for ny j = 1,..., n. Then the twisted Alexnder polynomil of K ssocited to ρ is defined by K, ρ (t = det M ρ, j det Φ(1 α j, (10 nd it is well-defined up to t p (p Z. 11
If we concentrte on boundry-prbolic representtion ρ nd its lift ρ, then det Φ(1 α j in (10 is lwys (1( t independent of the choice of j by the following clcultion: fter 1 0 putting ρ(α j = P P 1 1 1 for certin invertible mtrix P, det Φ(1 α j = det(p P 1 t P ( 1 0 1 1 ( 1 0 P 1 = det(1 t 1 1 = (1 t. (Even when we consider non-boundry-prbolic representtion, the vlue of det Φ(1 α j in (10 is still independent of j becuse ll rc-colors of the knot digrm re conjugte ech other. Now we pply this clcultion method to the cse of K 1 #K ssocited to ρ 1 # ρ. For Figure 13(, consider the Wirtinger presenttion of π 1 (K 1 nd π 1 (K by nd π 1 (K 1 =< α 1,..., α l r 1,..., r l 1, r l >=< α 1,..., α l r 1,..., r l 1 > π 1 (K =< α l,..., α n r l, r l+1, r l+,..., r n >=< α l,..., α n r l+1, r l+,..., r n >, respectively. (In Figure 13, nd re the digrms of K 1 nd K, respectively. ( Knot digrms nd (b Knot digrm # Figure 13: Knot digrms with some rcs Lemm 4.1. In the bove Wirtinger presenttion of π 1 (K 1 nd π 1 (K, we cn present π 1 (K 1 #K by π 1 (K 1 #K =< α 1,..., α n r 1,..., r l 1, r l+1,..., r n >. Proof. In Figure 13(b, the meridin loop corresponding to α l is homotopic to tht of α l. Therefore, fter writing down the Wirtinger presenttion of π 1 (K 1 #K nd substituting α l to α l in ll the reltions, the resulting presenttion is π 1 (K 1 #K =< α 1,..., α n r 1,..., r l 1, r l, r l, r l+1,..., r n >. (11 From the fct tht α l is homotopic to α l, two reltions in (11 re redundnt, one from nd nother from. After removing r l nd r l, we complete the proof. 1
Corollry 4.. For the boundry-prbolic representtions ρ 1, ρ nd their lifts ρ 1, ρ, the twisted Alexnder polynomils stisfy K1 #K, ρ 1 # ρ = (1 t K1, ρ 1 K, ρ. (1 Proof. Consider the Wirtinger presenttions of π 1 (K 1, π 1 (K nd π 1 (K 1 #K bove. Let M 1 be the (l 1 (l 1 mtrix whose (k, j component is ( rk Φ (k, j = 1,..., l 1, α j nd M be the (n l (n l mtrix whose (k, j component is ( rk Φ (k, j = l + 1,..., n. α j Then ( M1 0 det 0 M K1 #K, ρ 1 # ρ = det Φ(1 α j det(m 1 det(m = det Φ(1 α j det Φ(1 α j det Φ(1 α j = (1 t K1, ρ 1 K, ρ. Remrk tht the nturl generliztion of the Alexnder polynomil K (t is to define the twisted Alexnder polynomil K, ρ (t, using different normliztion from (10, by Then the product formul (1 chnges to K, ρ (t := det M ρ, j = (1 t K, ρ (t. K 1 #K, ρ 1 # ρ = K 1, ρ 1 K, ρ, which is nturl generliztion of K1 #K = K1 K. Note tht, for non-boundry-prbolic representtions of oriented knots, Corollry 4. still holds with slight modifiction. The term (1 t in (1 should be chnged to det Φ(1 α j, where α j is the rc connecting two digrms. However, s shown before, choosing ny rc α k insted of the connecting rc α j gives the sme eqution det Φ(1 α k = det Φ(1 α j. 5 Exmple For the trefoil knot 3 1 in the left-hnd side nd the figure-eight knot 4 1 in the right-hnd side of Figure 14, we put the boundry-prbolic representtion ρ : π 1 (3 1 #4 1 PSL(, C 13
s 4 s 6 1 3 s7 s s 3 s 5 6 5 s 1 s 8 s 9 4 3 determined by the rc-colors ( 1 1 =, 1 = ( x + 1 4 = x Figure 14: 3 1 #4 1 ( 1 0, 5 =, 3 = ( x x ( 0 1, 6 = = 3, ( x, 0 where x = 1± 3 i is solution of x +x+1 = 0. (We consider ech rc-color k is ssigned to the rc α k. Let ρ = ρ 1 #ρ for ρ j : π 1 (K j PSL(, C with j = 1,, nd ρ 1, ρ, ρ = ρ 1 # ρ be their lifts to SL(, C. If we put s 1 = by ( 1 ( ( s 1 =, s 1 =, s 3 3 = ( ( 4x + 3 4x + 3 s 6 =, s 4x + 7 7 = 4, then ll region-colors re uniquely determined ( 1 1, s 8 = Note tht this region-coloring stisfies Lemm 3.1. If we put ( 1 p =, ( ( 1 1, s 4 =, s 3 5 =, 4 ( ( 4x 3x, s x 1 9 =. x 1 then it stisfies (5. Let W 1 (w 1,..., w 5 nd W (w 4,..., w 9 be the potentil functions of 3 1 nd 4 1 from Figure 14, respectively. Then { W 1 = Li ( w Li ( w + Li ( w w 3 + Li ( w 1 + Li ( w 4 + log w 1 log w } 4 w 1 w 4 w 1 w 4 w 3 w 3 w 3 { Li ( w 3 w 1 Li ( w 3 w 4 + Li ( w 3w 5 w 1 w 4 + Li ( w 1 w 5 + Li ( w 4 w 5 + log w 1 w 5 log w 4 + { + Li ( w 5 Li ( w 5 + Li ( w w 5 + Li ( w 1 + Li ( w 4 + log w 1 log w 4 w 1 w 4 w 1 w 4 w w w w 14 w 3 w 5 } } π,
{ W = Li ( w 4 + Li ( w 4 Li ( w 4w 7 Li ( w 5 Li ( w 6 log w 5 log w } 6 w 5 w 6 w 5 w 6 w 7 w 7 w 7 w { 7 + Li ( w 7 + Li ( w 7 Li ( w 4w 7 Li ( w 6 Li ( w 9 log w 6 log w } 9 w 6 w 9 w 6 w 9 w 4 w 4 w 4 w { 4 } + + Li ( w 5 w 7 Li ( w 5 w 8 + Li ( w 5w 9 w 7 w 8 + Li ( w 7 w 9 + Li ( w 8 w 9 + log w 7 w 9 log w 8 { Li ( w 9 w 4 Li ( w 9 w 8 + Li ( w 5w 9 w 4 w 8 + Li ( w 4 w 5 + Li ( w 8 w 5 + log w 4 w 5 log w 8 w 5 nd the potentil function W (w 1,..., w 9 of 3 1 #4 1 from Figure 14 is W (w 1,..., w 9 = W 1 (w 1,..., w 5 + W (w 4,..., w 9. w 9 }, Let { ( } W I := exp w k = 1 w k k = 1,..., 9, { ( } W 1 I 1 := exp w k = 1 w k k = 1,..., 5, { ( } W I := exp w k = 1 w k k = 4,..., 9, nd define w (0 := (w (0 1,..., w (0 9 using the formul (7 s follows: w (0 1 = 3, w (0 = 1, w (0 3 = 1, w (0 4 = 5, w (0 5 = 6, w (0 6 = 4x + 1, w (0 7 = 8x, w (0 8 = 9x + 3, w (0 9 = 7x + 3. We put w (0 1 = (w (0 1,..., w (0 5 nd w (0 = (w (0 4,..., w (0 9. Then w (0 1, w (0 nd w (0 re solutions of I 1, I nd I, respectively. Furthermore, numericl clcultion shows i(vol(ρ 1 + i cs(ρ 1 (W 1 0 (w (0 1 i(0 + 1.6449...i (mod π, { i(vol(ρ + i cs(ρ (W 0 (w (0 i(.099... + 0 i i(.099... + 0 i if x = 1 3 i if x = 1+ 3 i (mod π, nd i(vol(ρ 1 #ρ + i cs(ρ 1 #ρ W 0 (w (0 { i(.099... + 1.6449... i i(.099... + 1.6449... i if x = 1 3 i if x = 1+ 3 i (W 1 0 (w (0 1 + (W 0 (w (0 i(vol(ρ 1 + i cs(ρ 1 + i(vol(ρ + i cs(ρ (mod π, 15
which confirms the dditivity of the complex volume in Corollry 3.3. To clculte the twisted Alexnder polynomils, we put the Wirtinger presenttions of 3 1, 4 1 nd 3 1 #4 1 from Figure 14 by respectively. simply by π 1 (3 1 = < α 1, α, α 3 α 1 α α1 1 α3 1, α α 3 α 1 α1 1, α 3 α 1 α3 1 α 1 > = < α 1, α, α 3 α 1 α α1 1 α3 1, α α 3 α 1 α1 1 >, π 1 (4 1 = < α 3, α 4, α 5, α 6 α 3 α 6 α3 1 α5 1, α 5 α 4 α5 1 α3 1, α 6 α 4 α6 1 α5 1 >, π 1 (3 1 #4 1 = < α 1, α, α 3, α 3, α 4, α 5, α 6 α 1 α α1 1 α3 1, α α 3α 1 α1 1, α 3α 1 (α 3 1 α 1, α 3 α 6 α3 1 α5 1, α 5 α 4 α5 1 α3 1, α 6 α 4 α6 1 α5 1 >, (If we use Lemm 4.1, the fundmentl group π 1 (3 1 #4 1 cn be expressed π 1 (3 1 #4 1 = < α 1, α, α 3, α 4, α 5, α 6 α 1 α α1 1 α3 1, α α 3 α 1 α1 1, α 3 α 6 α3 1 α5 1, α 5 α 4 α5 1 α3 1, α 6 α 4 α6 1 α5 1 >. This presenttion shows (1 trivilly, so we re using the Wirtinger presenttion of π 1 (3 1 #4 1 insted. The Alexnder mtrices ssocited to ρ 1, ρ nd ρ 1 # ρ obtined by the bove Wirtinger presenttions re 1 t 0 0 t 1 0 M ρ1 = t 1 t t t 0 1 1 0 1 t t t, 0 1 t 1 t 0 t M ρ = nd M ρ1 # ρ = 1 + xt (x + 1t 0 0 1 0 t 0 (x + 1t 1 (x + t 0 0 0 1 t t 1 0 xt (x + 1t 1 t 0 0 0 0 1 (x + 1t (x + t t 1 t 0 0 0 0 t (x + 1t 1 0 1 + xt (x + 1t 0 0 0 t 0 1 (x + 1t 1 (x + t 1 t 0 0 t 1 0 0 0 0 0 0 0 0 0 t 1 t t t 0 1 0 0 0 0 0 0 0 0 1 0 1 t 0 0 t t 0 0 0 0 0 0 0 1 t 1 t 0 0 0 t 0 0 0 0 0 0 t 0 1 0 0 0 1 t t 0 0 0 0 0 0 t t 0 1 0 0 0 1 t 0 0 0 0 0 0 0 0 0 0 1 + xt (x + 1t 0 0 0 0 1 0 t 0 0 0 0 0 (x + 1t 1 (x + t 0 0 0 0 0 1 t t 0 0 0 0 1 0 0 0 xt (x + 1t 1 t 0 0 0 0 0 0 0 0 1 0 0 (x + 1t (x + t t 1 t 0 0 0 0 0 0 0 0 0 0 t (x + 1t 1 0 1 + xt (x + 1t 0 0 0 0 0 0 0 0 0 t 0 1 (x + 1t 1 (x + t respectively. The corresponding twisted Alexnder polynomils obtined by (10 re 31, ρ 1 (t = 1 + t, 41, ρ (t = t (1 4t + t, 31 #4 1, ρ 1 # ρ (t = (1 t (1 + t t (1 4t + t, 16,,
respectively. 4 Therefore, we obtin which confirms Corollry 4.. 31 #4 1, ρ 1 # ρ (t = (1 t 31, ρ 1 (t 41, ρ (t, (13 A Errt This ppendix is the errt of this rticle. (The uthor pprecites Seonhw Kim for pointing out the error. The uthor found the errors fter the publiction, so he wrote this errt nd submitted it to the sme journl gin. He sincerely pologizes to the reders for confusing them. For given boundry-prbolic representtions ρ j : π 1 (K j PSL(, C (j = 1,, the connected sum ρ 1 #ρ : π 1 (K 1 #K PSL(, C ws defined t this rticle. He proved tht ρ 1 #ρ is well-defined up to conjugtion t Theorem., but the sttement nd the proof re not correct. As n counterexmple of Theorem., consider the exmple of Fig 14 in Section 5. We put ( 1 1 =, 1 = ( x + 1 4 = x ( 1 0, 5 =, 3 = ( x x ( 0 1, 6 = = 3, (14 ( x, 0 but we cn conjugte the figure-eight knot prt by the mp 3 : P P. The chnged rc-colors re ( ( ( ( ( 1 1 0 0 0 1 =, 1 =, 0 3 = = = 1 1 1 3, (15 ( ( ( ( ( ( x + 1 0 x + 1 x 0 x 4 = =, x 1 x + 1 5 = =, x 1 x ( ( ( x 0 x 6 = =. 0 1 x The representtions defined by (14 nd (15 cnnot be conjugte, so the connected sum cnnot be well-defined. The error lies in the third sentence of the proof of Theorem.: For ny b P, there exists unique c P such tht b c =. The uthor confused tht the bijectiveness of the mp c implies this sttement. This sttement is wrong, so ll of the proof is wrong. (For exmple, in Fig 8, the second digrm is wrong. We cnnot gurntee the rc-color of the smll box becomes c. Therefore, Theorem. is wrong nd Definition.1 should be modified. 4 To clculte the determinnts, finl two columns of ll three mtrices re removed. 17
One wy to solve these errors is to consider the connected sum ρ 1 #ρ not s definition, but method to construct boundry-prbolic representtions. This construction does not define the unique representtion, but it defines mny representtions. Under this construction, Proposition.3 should be chnged s follows. Proposition A.1 (New version of Proposition.3. For boundry-prbolic representtion ρ : π 1 (K 1 #K PSL(, C, there exist ρ 1 : π 1 (K 1 PSL(, C nd ρ : π 1 (K PSL(, C, which re unique up to conjugtion, such tht one of ρ 1 #ρ becomes ρ. Proof. The existence is trivil from Fig 9. The uniqueness follows from Fig 7 becuse the rc-color of D is invrint under the moves up to conjugtion. Interestingly, Section 3 4 re still true under this construction. This implies tht ny representtion obtined by ρ 1 #ρ hs the sme complex volume nd the sme twisted Alexnder polynomil (vol(ρ 1 + i cs(ρ 1 + (vol(ρ + i cs(ρ K 1, ρ 1 K, ρ. Acknowledgments The uthor pprecites Teruki Kitno for giving very nice introductory lectures on twisted Alexnder polynomil t Seoul Ntionl University in November, 014. Section 4 of this rticle is motived by his tlk. Also, discussions with Sungwoon Kim, Yuichi Kby nd Hyuk Kim helped the uthor lot for prepring this rticle. References [1] J. Cho. Optimistic limits of colored Jones polynomils nd complex volumes of hyperbolic links. rxiv:1303.3701, 03 013. [] J. Cho. Optimistic limit of the colored Jones polynomil nd the existence of solution. rxiv:1410.055, 10 014. [3] J. Cho. Qundle theory nd optimistic limits of representtions of knot groups. rxiv:1409.1764, 09 014. [4] A. Inoue nd Y. Kby. Qundle homology nd complex volume. Geom. Dedict, 171:65 9, 014. [5] F. Luo, S. Tillmnn, nd T. Yng. Thurston s spinning construction nd solutions to the hyperbolic gluing equtions for closed hyperbolic 3-mnifolds. Proc. Amer. Mth. Soc., 141(1:335 350, 013. [6] T. Morifuji. Twisted Alexnder polynomils of twist knots for nonbelin representtions. Bull. Sci. Mth., 13(5:439 453, 008. 18
[7] C. K. Zickert. The volume nd Chern-Simons invrint of representtion. Duke Mth. J., 150(3:489 53, 009. Pohng Mthemtics Institute (PMI, Pohng 37673, Republic of Kore E-mil: dol045@gmil.com 19