Lifts of holonomy representations and the volume of a knot complement (Intelligence of Low-dimensional Topology)
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(2) 110. F_{n}\rightarrow G(K) the epimorphism from the free group associated with P to G(K) and by \overline{$\phi$} : \mathbb{Z}F_{n} \rightarrow \mathbb{Z}G(K) the ring homomorphism which is obtained from $\phi$ by extending linearly. Let $\alpha$ : G(K) \rightarrow H_{1}(E(K);\mathbb{Z}) \cong \mathbb{Z}= \langle t} be the abelianization homomorphism. It is given by $\alpha$(x_{1}) $\alpha$(x_{n}) =t since P is a Wirtinger presentation. By extending linearly, we have a homomorphism between group rings: \overline{ $\alpha$}:\mathbb{Z}G(K) \rightarrow \mathbb{Z}[t, t^{-1}] We denote by $\Phi$ the. We denote. by $\phi$. :. ,. =.. .. .. .. composed mapping. \overline{$\alpha$}\circ\overline{$\phi$}. that is,. ,. $\Phi$:\mathbb{Z}F_{n}\rightarrow \mathbb{Z}[t, t^{-1}].. \displayte\frac{prtial}{\prtialx_{j} \displayst le\frac{\parti lx_{i} \parti lx_{j}=$\delta$_{ij}. The map. by (1). \mathbb{Z}F_{n}\rightarrow \mathbb{Z}F_{n} is the linear extension of the map defined. :. free differential.. ,. (2). \displaystyle\frac{\partialx_{i}^{-1}{\partialx_{j}=-$\delta$_{ij}x_{i}^{-1}. We obtain. a. ,. (3). \displayst le(n-1) \frac{\partial(uv)}{\partialx_{j}=\frac{\partialu}{\partialx_{j}+u\frac{\partialv}{\partialx_{j}. matrix whose size is. .. on. the elements of. F_{n}. This is called the Foxs. \times n :. A= ( $\Phi$(\displaystyle \frac{\partial r_{i} {\partial x_{j} ) \in M(n-1, n;\mathb {Z}[t, t^{-1}]) the Foxs free differential to the relations r_{1} ,. by applying tion P and. composing. presentation We denote. matrix and. $\Phi$. .. by A_{j}. .. .. ,. r_{n-1} of the. Wirtinger presenta‐. The matrix A is called the Alexander matrix associated with the. P of the knot group. we. .. G(K). obtained from A. .. by deleting. define the Alexander polynomial of. the a. j. column of A. knot K. .. This becomes. a. square. by. $\Delta$_{K}(t)=\det A_{j} \in \mathbb{Z}[t, t^{-1}]. It is known that this becomes. Example. a. knot invariant up to. \pm t^{s}(s\in \mathbb{Z}). 2.1. The knot illustrated below is called the. hyperbolic. .. figure eight. knot and it is known. knot. The knot number is 4_{1}.. Its knot group has the next. presentation,. which is. a. Wirtinger presentation:. G(K)=\langle x, y|xy^{-1}x^{-1}yxy^{-1}xyx^{-1}y^{-1}\rangle.. as a.
(3) 111. We. apply the Foxs free differential to the relation: r=xy^{-1}x^{-1}yxy^{-1}xyx^{-1}y^{-1}. ,. then. have:. we. \displaystyle \frac{\partial}{\partial x}r=\frac{\partial x}{\partial x}+x\frac{\partial}{\partial x}(y^{-1}x^{-1}yxy^{-1}xyx^{-1}y^{-1}) =1+x(\displaystyle \frac{\partial y^{-1} {\partial x}+y^{-1}\frac{\partial}{\partial x}(x^{-1}yxy^{-1}xyx^{-1}y^{-1}) =1+xy^{-1}(\displaystyle \frac{\partial x^{-1} {\partial x}+x^{-1}\frac{\partial}{\partial x}(yxy^{-1}xyx^{-1}y^{-1}) =1-xy^{-1}x^{-1}+xy^{-1}x^{-1}\displaystyle \frac{\partial}{\partial x}(yxy^{-1}xyx^{-1}y^{-1})=\cdots=. (2.1). =1-xy^{-1}x^{-1}+xy^{-1}x^{-1}y+xy^{-1}x^{-1}yxy^{-1}-xy^{-1}x^{-1}yxy^{-1}xyx^{-1}. Similarly,. \displaystyle \frac{\partial}{\partial y}r=-xy^{-1}+xy^{-1}x^{-1}-xy^{-1}x^{-1}yxy^{-1}+xy^{-1}x^{-1}yxy^{-1}x-xy^{-1}x^{-1}yxy^{-1}xyx^{-1}y^{-1} Since. $\alpha$(x)= $\alpha$(y)=t. ,. we. have:. $\Phi$(\displaystyle \frac{\partial r}{\partial x}) =1-t ^{-1}t^{-1}+t ^{-1}t^{-1}t+t ^{-1}t^{-1}t t^{-1}-t ^{-1}t^{-1}t t^{-1}t t^{-1} =1-\displaystyle \frac{1}{t}+1+1-t=-\frac{1}{t}+3-t,. $\Phi$(\displaystyle \frac{\partial r}{\partial y}) =-1+t^{-1}-1+t-1=\frac{1}{t}-3+t.. Thus the Alexander matrix is the matrix of 1\times 2 :. polynomial. of the. figure eight knot. K is:. (-\displaystyle \frac{1}{t}+3-t \frac{1}{t}-3+t). ,. and the Alexander. $\Delta$_{K}(t)=\displaystyle \det (- \frac{1}{t}+3-t) =-\frac{1}{t}+3-t (up. to. \pm t^{S}(s\in \mathbb{Z}. Originally element t. x. by. and y. the map. are $\alpha$. .. different generators in the knot group, but. 3. following section, improves. Twisted Alexander We. use. the. representation. same. are. It makes the calculation easy while this process. information included in knot groups. The twisted Alexander in the. they. nations. this. polynomial,. sent to the. same. reduce. some. might. which is introduced. point.. polynomials as. of a knot group. in the. G(K). .. previous. sections.. This map induces. Let $\rho$. naturally. \overline{ $\rho$}:\mathbb{Z}G(K)\rightarrow M(m;\mathbb{C}). ,. :. G(K). \rightarrow. \mathrm{S}\mathrm{L}(m, \mathbb{C}). the map between group. be. a. rings:.
(4) 112. moreover. by taking the tensor product with the map \overline{$\alpha$} induced in the previous section,. \overline{ $\rho$}\otimes\overline{ $\alpha$}:\mathbb{Z}G(K)\rightarrow M(m, \mathbb{C}[t, t^{-1}]) Set $\Phi$. we. have:. .. :. $\Phi$=(\overline{ $\rho$}\otimes\overline{ $\alpha$})\circ\overline{ $\phi$}:\mathbb{Z}F_{n}\rightar ow M(m;\mathbb{C}[t, t^{-1}]) \overline{$\phi$}. by composing the. (i,j). defined in the. element is the. m\times m. previous. section.. Suppose A_{ $\rho$}. (n-1). is the. \times n. matrix whose. matrix:. $\Phi$(\displaystyle \frac{\partial r_{i} {\partial x_{j} ) \in M((n-1)m\times nm;\mathb {C}[t, t^{-1}]). .. This matrix is called the twistedAlexander matrix associated with $\rho$ In order to make .. matrix, so. that. we we. corresponding A_{ $\rho$} \mathrm{a}(n-1)m\times(n-1)m matrix, which is denoted by A_{ $\rho$,k}. delete from. have. Alexander polynomial. Here. one column. to. a. generator x_{k} in the .. a. square. presentation P,. We define the twisted. as:. $\Delta$_{K, $\rho$}(t)=\displaystyle \frac{\det A_{ $\rho$,k} {\det $\Phi$(x_{k}-1)}.. we assume. Wada. \det $\Phi$(x_{k}-1)\neq 0.. proved the following. Theorem 3.1. of G(K). .. ([19]).. Let K be. a. [19].. knot and. G(K) the knot group. Suppose $\rho$ is a representation $\Delta$_{K, $\rho$}(t) is an invariant for the pair (G(K), $\rho$) up. The twisted Alexander polynomial. to\pm t^{s}(s\in \mathbb{Z}) Example. theorem in. .. 3.2. Let K be the. presentation. as. in. Example. figure eight knot,. then the knot group. G(K). has the. following. 2.1:. G(K)=\{x, y|xy^{-1}x^{-1}yxy^{-1}xyx^{-1}y^{-1}\rangle. (3.1). .. Define. Then. we can. $\rho$(x)= \left(\begin{ar ay}{l } 1 & 1\ 0 & 1 \end{ar ay}\right) , $\rho$(y)= (_{\frac{-1+\sqrt{-3}1 {2} 01). confirm that $\rho$ becomes. representation. a. from. G(K). (3.2) to. \mathrm{S}\mathrm{L}(2, \mathbb{C}). .. Set. $\rho$(x). =. $\Phi$(\displaystle\frac{\parti lr}{\parti lx}) I-\displaystyle \frac{1}{t}XY^{-1}X^{-1}+XY^{-1}X^{-1}Y+XY^{-1}X^{-1}YXY^{-1}-tXY^{-1}X^{-1}YXY^{-1}XYX^{-1},. X, $\rho$(y)=Y. ,. where I is the. then. we. have. :. =. identity matrix of size 2\times 2. x\rightarrow X, y\rightarrow Y. and 1\rightarrow I with t to the. .. Note that this. appropriate. can. be obtained from. power. Calculate these. (2.1) by changing matrices, then. have:. $\Delta$_{K, $\rho$}(t)=\displaystyle \frac{\det $\Phi$(\frac{\partial r}{\partial x})}{\det $\Phi$(y-1)}=\frac{1/t^{2}(t-1)^{2}(t^{2}-4t+1)}{(t-1)^{2} =t^{2}-4t+1.. we.
(5) 113. This. to make up for the lack of the information caused. seems. However the twisted Alexander not be called. knot invariant,. a. polynomial depends namely,. Furthermore, is is. not easy to find. thinkable ways to. apply might be (1). to consider the restricted. (2). or. determine in. non‐fibered knot. a. [6] which. unimodular the. holonomy representation. to find. [10].. 4. On. of. a. only G(K). for. distinguishing. knot group in. a. two. it. might. given. knots.. ,. so. Therefore the. general.. I think. example. an. of the former. representation, i.e.,. polynomials. we. of fibered knots. case. is to. gave the theorem. are. monic for any. for the researches which followed this theorem.. [3, 15]. hyperbolic. $\alpha$.. property of a knot satisfied for any representation. will consider the twisted Alexander. of. but also $\rho$. the map. knot, which. conceptions. polynomial. corresponds. on. to. the. In. associated with the. case. (2) above.. the twisted Alexander. polynomial,. see. for its recent researches.. [4, 9]. hyperbolic knots. We refer We. not. use. any unimodular. For the details of basic notations and. See. a. representation.. by using See. representation. we. representation. twisted Alexander. states that the. following sections,. a. on. it is hard to. by going through. [11, 18] for the former half in this section.. regard the. upper half space model. \mathbb{H}^{3}. as a. subspace. of the. quatemion. field and set. \mathbb{H}^{3}=\{(x+yi)+tj \in \mathbb{C}+\mathbb{R}j|t>0\} where. 1, i, j. are. the part of. basis, i. =. \sqrt{-1}. ,. and. we. suppose. \partial \mathbb{H}^{3}. =. \mathbb{C}\cup\{\infty\}. .. We. give. the. metric. ds^{2}=\displaystyle \frac{1}{t^{2} (dx^{2}+dy^{2}+dt^{2}) to. \mathbb{H}^{3} then. preserving. we. call this \mathbb{H}^{3} the 3‐dimensional. hyperbolic. space. It is known that the orientation. isometric transformation group of \mathbb{H}^{3} is:. \mathrm{P}\mathrm{S}\mathrm{L}(2,\mathb {C})=\{\left(\begin{ar ay}{l} a&b\ c&d \end{ar ay}\right)|a,b,c d\in\mathb {C},ad-bc=1\}/\{ pm\left(\begin{ar ay}{l} 1&0\ 0&1 \end{ar ay}\right)\}. Here the action. on. \mathbb{H}^{3} of \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathb {C}) is given by. \left(\begin{ar ay}{l } a & b\ c & d \end{ar ay}\right)w=(aw+b)(cw+d)^{-1} (w\in \mathb {H}^{3}) We calculate the. tion of \mathbb{H}^{3} is. right‐hand. side. as. elements of the. quatemion. .. field. The isometric transforma‐. \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathb {C}) on \mathbb{H}^{3} is transitive. Moreover its stabilizer of a point is \mathrm{P}\mathrm{S}\mathrm{U}(2, \mathbb{C})(\cong \mathrm{S}\mathrm{O}(3) that is, if f(p) and the mapping between tan‐ gent spaces T_{p}\mathbb{H}^{3}\rightar ow T_{f(p)}\mathbb{H}^{3} are given for f\in \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathbb{C}) and a point p\in \mathbb{H}^{3} then f may be a. conformal. mapping,. and the action of ,. ,.
(6) 114. determined. formation is there exists. of each. element of. an. by determining (It does. group of. a. such that. the. in M'. \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathb {C}). hyperbolic. homomorphism. $\rho$. the. we. call M. as. points. path.. \overline{$\gam a$}. local coordinate such that. a. we. take. a. a. 3‐manifold M'. $\pi$_{1}(M) \rightar ow \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathbb{C}). the. a. from the base. path. image. of p. corresponding. Let $\gamma$ be. lift of $\gamma$ to. a. local coordinate of. a. We may have the. path.). a. 3‐manifold. This is. hyperbolic. a. follows. Give. of the coordinate functions. 3‐manifold M and :. f. simply‐connected hyperbolic. a. the way to take. on. ,. point p. For any. .. along. image. depend. not. \mathbb{H}^{3} uniquely. Further,. transforms any 3. \mathbb{H}^{3} and the coordinate transfor‐. to an open set in. map from M' to \mathbb{H}^{3}. developing point. the isometric trans‐. point by. a. to the whole space. \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathb {C}). point is homeomorphic. sequence of local coordinates. the. f. \in. of. points pí, p_{2}',p_{3}'\in\partial \mathbb{H}^{3}.. into any 3. borhood of the base. order.. neighborhood. mapping. concept defined in Section 1. For. same. map. extend the. one can. be written in. may define the. a. of the. 3‐dimensional differentiable manifold. If M has. a. neighborhood the. image. the transformation. uniquely. Let M be. can. then. given,. p_{1},p_{2},p_{3}\in\partial \mathbb{H}^{3}. mation. if the. uniquely. Thus,. by. point the. ,. we. neigh‐. a. to p. and. developing. to the sequence in. element of the fundamental. an. universal. covering \overline{M} of M. holonomy representation. We call. .. $\rho$( $\gamma$). of M if. is the. (\in \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathbb{C}) which maps the image by the developing map of the neighborhood of the base point of \overline{$\gam a$} to that of the neighborhood of the end point of \overline{$\gam a$} Let $\Gamma$ be the image $\rho$($\pi$_{1}(M)) element. .. for the. urally. holonomy representation and M is. 3‐manifolds is may think the. is shown. by. homeomorphic. a. complete hyperbolic. \mathb {H}^{3}/ $\Gamma$. to. equivalent essentially. 3‐manifold M then $\Gamma$ acts \mathbb{H}^{3} nat‐ ,. Therefore the classification of. .. to that of. a. kind of discrete. holonomy representation. a. [2] it is proved that the lift has. be. a. structure of M Then .. a. we. a. submanifold of. hyperbolic. a. 3‐manifold is. obtain from M. we. by getting rid of a neighborhood of the cusp. with finite volume is. resp.). is said to be. 3‐manifold with. a. a. closed 3‐manifold. hyperbolic. cusp. (cusps. if. .. \mathrm{S}\mathrm{L}(2, \mathbb{C}) representation,. structure of M Let $\eta$ .. a. a. boundary. a. of the. particular,. a. structure of the. torus knot. nor a. is. a. a. torus. 3‐manifold. complete hyperbolic. 3‐manifold with cusps. In. A knot which is neither. product. cusp. A 3‐manifold M with. compact 3‐manifold whose. It is known that. or a. a. to the direct. S^{3}-K ( S^{3}-L resp.) admits the. resp. and. .. homeomorphic. 2‐dimensional torus and the half‐line, the submanifld is called cusp is non‐compact, and. so we. ,. following homomorphism:. \mathrm{H}\mathrm{o}1_{(M, $\eta$)} : $\pi$_{1}(M, $\eta$)\rightar ow \mathrm{S}\mathrm{L}(2, \mathbb{C}) If. of \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathb {C}). 3‐manifold is included in $\Gamma$ It. be lift to. correspondence to the spin. one‐to‐one. have the. can. complete hyperbolic. subgroups. geometrical information of a complete hyperbolic. Thurston that. in. spin. $\rho$ of. knot. K(L. hyperbolic. satellite knot is. hyperbolic. In the. case. of. a. knot in S^{3}. ,. we. let. A_{1}. ,. .. .. .. ,. A_{n}. be the. images. of generators a_{1} ,. .. .. .. ,. a_{n} of. a. Wirtinger presentation of G(K) by the holonomy representation $\rho$ then their lifts to \mathrm{S}\mathrm{L}(2, \mathbb{C}) are A_{1} A_{n} or -A_{1} -A_{n} (Corollary 2.3 in [14]). We denote by $\rho$^{\pm}(a_{i}) \pm A_{i}(\in ,. ,. .. .. .. ,. ,. .. .. .. ,. =.
(7) 115. \mathrm{S}\mathrm{L}(2, \mathbb{C}). for the lifts of the. Irreducible. 5. holonomy representation $\rho$(a_{i})=A_{i}(\in \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathb {C}. \mathrm{S}\mathrm{L}(m, \mathbb{C}) ‐representations of \mathrm{S}\mathrm{L}(2, \mathbb{C}). We review irreducible. representations of \mathrm{S}\mathrm{L}(2, \mathbb{C}) briefly.. symmetric product \mathrm{S}\mathrm{y}\mathrm{m}^{m-1}(\mathb {C}^{2}). action of \mathrm{S}\mathrm{L}(2, \mathbb{C}) It is known that the .. by \mathrm{S}\mathrm{L}(2, \mathbb{C}) give. an m ‐dimensional. representation. with the vector space of homogeneous. polynomials. V_{m}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}_{\mathbb{C} \langle x^{m-1}, x^{m-2}y The action of. A\in \mathrm{S}\mathrm{L}(2, \mathbb{C}). is. The vector space \mathb {C} has the standard. expressed. ,. \mathrm{S}\mathrm{L}(2, \mathbb{C}). of on. .. .. .. We. .. can. and the induced action. identify. \mathrm{S}\mathrm{y}\mathrm{m}^{m-1}(\mathb {C}^{2}). \mathb {C}^{2} with degree m-1 namely, ,. ,. xy^{m-2}, y^{m-1}\rangle.. as. A\cdotp\left(\begin{ar y}{l x\ y \end{ar y}\right)=p(A^{-1}\left(\begin{ar y}{l x\ y \end{ar y}\right) where mined. p\left(bgin{ar y}{l x\ y \end{ar y}\right). a. homogeneous polynomial. the action of A^{-1}. by. (V_{m}, $\sigma$_{m}). is. the. the column vector. on. into. GL(V_{m}). .. matrix. as a. above action of. representation given by the. morphism from \mathrm{S}\mathrm{L}(2, \mathbb{C}). and the variables in the. It is known that. right‐hand. multiplication.. \mathrm{S}\mathrm{L}(2, \mathbb{C}). side. deter‐. are. We denote. by. where $\sigma$_{m} denotes the homo‐. (1) each representation (V_{m}, $\sigma$_{m}). tums. \mathrm{S}\mathrm{L}(m, \mathbb{C}) ‐representation and (2) every irreducible m ‐dimensional represen‐ \mathrm{S}\mathrm{L}(2, \mathbb{C}) equivalent to (V_{m}, $\sigma$_{m}) Let M be a complete hyperbolic 3‐manifold and \mathrm{H}\mathrm{o}1_{(M, $\eta$)} the homomorphism defined in Sec‐ tion 4. By composing \mathrm{H}\mathrm{o}1_{(M, $\eta$)} and $\sigma$_{m} we have the representation: into. an. irreducible. tation of. is. .. ,. $\rho$_{m}:$\pi$_{1}(M)\rightarrow \mathrm{S}\mathrm{L}(m, \mathbb{C}) Example 5.1.. It is known that the map. tation of the. figure eight knot. instead of. and y in the group. x. Note that these. are. xy—y2, y^{2}=y^{2}. ,. .. reduplication,. presentation (3.1). \left(bgin{ar y}{l 1& \ 0&1 \end{ar y}\ight). the elements in. we. given by (3.2) in Example. K To avoid the. $\rho$(a)=. .. \mathrm{S}\mathrm{L}(2, \mathbb{C}). have the next matrix. .. a. holonomy represen‐. and b be generators of. G(K). and set:. $\rho$(b)=. ;. let. 3.2 is the. Since. by taking. (_{\frac{-1+\sqrt{-3}1 {2} 01) (x-y)^{2}. =. x^{2}-2xy+y^{2}, (x-y)y. the coefficients:. $\rho$_{3}(a)=\left(\begin{ar y}{l 1-2&1\ 0&1- \ 0&01 \end{ar y}\right). .. =.
(8) 116. By setting. $\rho$_{3}(b)=. u=\displaystyle \frac{-1+\sqrt{-3} {2}. \left(bgin{ar y}{l 1&0 \ u&1 0\ u^{2}& u 1 \end{ar y}\right),. and. calculating similarly,. obtain:. \left(bgin{ar y}{l 1-3& -1\ 0&1-2 \ 0& 1-\ 0& 01 \end{ar y}\ight), \left(bgin{ary}l 1&0 \ u&1 0 \ u^{2}&u 1&0\ u^{3}&u^{2}&3u 1 \end{ary}\ight). $\rho$_{4}(a)=. $\rho$_{4}(b)=. (\cdot)^{T} means the transposed matrix.. Here. Main Theorem and the outline of the. ó. we. Let K be. hyperbolic knot,. a. holonomy representation. of. and $\rho$_{m} the. proof. \mathrm{S}\mathrm{L}(m, \mathbb{C}) ‐representation which is. G(K) by the method described in Sections 4 and 5.. \displaystyle\mathcal{A}_{K,2k}(t)=\frac{$\Delta$_{K,$\rho$_{2k}(t)}{$\Delta$_{K,$\rho$_{2}(t)} \displaystyle\mathcal{A}_{K,2k+1}(t)=\frac{$\Delta$_{K,$\rho$_{2k+1}(t)}{$\Delta$_{K,$\rho$\mathrm{s}(t)} ;. Our main result is the Theorem ó.l. obtained from the Set:. (6.1). .. following:. ([5]).. \displaystyle\lim_{k\rightar ow\infty}\frac{\log|\mathcal{A}_{K,2k}(1)|}{(2k)^{2} =\lim_{k\rightar ow\infty}\frac{\log|\mathcal{A}_{K,2k+1}(1)|}{(2k+1)^{2} =\frac{\mathrm{V}\mathrm{o}1(K)}{4$\pi$}. As in the. case. \bullet. \bullet. (6.1), \mathcal{A}_{K,$\rho$_{m} of. m even.. is defined. by dividing. We may describe. as. the. but it is inessential,. principal part,. follows if there is. no. \displaystyle\lim_{k\rightar ow\infty}\frac{\log|$\Delta$_{K,2k}(1)|}{(2k)^{2}=\frac{\mathrm{V}\mathrm{o}1(K)}{4$\pi$} \displaystyle \lim_{k\rightar ow\infty}\frac{1}{(2k+1)^{2} (\log(\lim_{t\rightar ow 1}|\frac{$\Delta$_{K,2k+1}(t)}{t-1}|) =\displayst le\frac{\mathrm{V}\mathrm{o}1(K)}{4$\pi$}.. volume of. a. The crucial. section,. knot. points. demeister torsion. we. give sample. complement are are. the. same. perbolic. using. combing them,. 3‐manifold. can. calculations of the. be. using. essentially the. one. a. kind of. of them states the. for unimodular. analytic. we are. figure eight knot.. approximated using. the results of Müller:. the volume formula. manifold. Thus,. torsion. able to have. ([17]). a. the Reidemeister torsion.. a. complete hyperbolic. 3‐manifold with cusps in [14]. torsion and the Rei‐. representations ([16]) for. a. closed. Applying. a. closed. the Thurstons. Theorem. 6.4),. a. review. some. results of Menal‐Ferrer and Porti. Let M be. bolic 3‐manifold whose. boundary. is. one. torus cusp,. i.e.,. we. an. 3‐. complete hy‐. hyperbolic. Dehn. volume formula for. so we. have. clear the relation between the Reidemeister torsion and the twisted Alexander us. and the other. complete hyperbolic. volume formula for. (see. As shown there the. combinatorial method.. analytic. surgery theorem to these Müllers works, Menal‐Ferrer and Porti gave. Let. in. ;. In the next. gives. especially. corrections:. oriented. only. a. to make. polynomial. complete hyper‐ \partial\overline{M}=T^{2}.. will consider M with.
(9) 117. ó.2. Proposition (2) If m. ([13]).. odd, then. is. ó.3. fundamental. group of T. a. ([14]). Suppose as a. m. subgroup of $\pi$_{1}(M). .. Choose. by $\rho$_{m}(G) If i .. :. a. A basis for. (2). Let. H_{1}(M;$\rho$_{m}). [T] \in H_{2}(T;\mathbb{Z}) i_{*}([v\otimes T. is. be. the above notations,. Here \mathrm{T}\mathrm{o}\mathrm{r}. means. Theorem 6.4. a case. a. fundamental. we. H_{1}(T;\mathbb{Z}). \in. ,. class. of T.. A basis. the. evaluating. given by. is. ;. ([14]).. \displaystyle \lim_{k\rightar ow\infty}\frac{\log|T_{2k+1}(M)|}{(2k+1)^{2} =\lim_{k\rightar ow\infty}\frac{\log|T_{2k}(M)|}{(2k)^{2} =\frac{\mathrm{V}\mathrm{o}1(M)}{4 $\pi$}. 6.2. in the. (1), the twisted homology vanishes. corresponding. chain. is said to be. complex. t=1 in this case, that. can. acyclic. complement E(K). get the. even case. representation. representation. same as. a. so. ([5]).. following equation. that. we. Let $\lambda$ be. .. In such. even.. to discuss. proved by. Kitano. polynomial by. .. main result via Theorem 6.4. $\rho$_{3} in. action of the. adjoint. our. setting essentially.. ([20, 21]). have this a. is. relatively. knot K It is. \mathrm{S}\mathrm{L}(2, \mathbb{C}) ‐representation The next. which treats the. proposition. adjoint. fundamental group. We restrict the base $\theta$ in. $\lambda$ and handle it well, ó.5. a. m. is,. theorem. Yamaguchis of. our. that. be obtained from the twisted Alexander. obtained from the. damental group is the ization of the. of. case. and it is easy. of. \mathrm{T}\mathrm{o}\mathrm{r}(M;$\rho$_{2k})=$\Delta$_{K,$\rho$_{2k}}(1). Proposition. for H_{2}(M;$\rho$_{m}). the Reidemeister torsion.. Proposition. we. cycle $\theta$. set:. ([8]) that the Reidemeister torsion. The. fixed realization of the. some. non‐trivial. given by i_{*}([v\otimes $\theta$. the Reidemeister torsion. Let M be the. Thus. 2.. T\rightarrow M denotes the inclusion, then the. T_{2k+1}(M)=\displayst le\frac{\mathrm{T}\mathrm{o}\mathrm{r}(M;$\rho$_{2k+1};$\theta$)}{\mathrm{T}\mathrm{o}\mathrm{r}(M;$\rho$_{3};$\theta$)} T_{2k}(M)=\displayst le\frac{\mathrm{T}\mathrm{o}\mathrm{r}(M;$\rho$_{2k}){\mathrm{T}\mathrm{o}\mathrm{r}(M;$\rho$_{2}).. As in. be. ,. assertions hold.. (1). Using. G<$\pi$_{1}(M). is odd and let. non‐trivial vector v\in V_{m} fixed. following. \dim_{\mathbb{C}}H_{i}(M;$\rho$_{m})=0 for any i.. \dim_{\mathbb{C}}H_{0}(M;$\rho$_{m})=0 and \dim_{\mathbb{C}}H_{i}(M;$\rho$_{m})=1 for i=1. Proposition and. is even, then. (1) If m. is. action of the. Proposition. 6.3 to. a. of. a. a. fun‐. general‐. \mathrm{S}\mathrm{L}(2, \mathbb{C})longitude. proposition:. longitude of a. knot K and M the. holds:. |\displaystyle \mathrm{T}\mathrm{o}\mathrm{r}(M;$\rho$_{2k+1}; $\lambda$)|=\lim_{t\rightar ow 1}\frac{|$\Delta$_{K,$\rho$_{2k+1} (t)|}{t-1}.. complement of K. ,. then the.
(10) 118. The odd. case. in. our. main result follows from the. proposition.. Some calculations. 7. Here. we. give. some. complement of K We. use. the lifts. proceed the. is. calculations. equal to. the. figure eight knot K. .. It is known that the volume of the. 2. 0298832\cdots.. \left(bgin{ar y}{l 1& \ 0&1 \end{ar y}\ight). $\rho$^{+}(a)=. calculation in. on. ,. $\rho$^{+}(b)=. Example 3.2,. then. (_{\frac{-1+\sqrt{-3}1 {2} 01). we. ,. stated in. Example 5.1,. and. we. have:. $\Delta$_{K,$\rho$_{2}^{+} (t)=\displaystyle \frac{1}{t^{2} (t^{2}-4t+1) $\Delta$_{K,$\rho$_{3}^{+} (t)=-\displaystyle \frac{1}{t^{3} (t-1)(t^{2}-5t+1) $\Delta$_{K,$\rho$_{4}^{+} (t)=\displaystyle \frac{1}{t^{4} (t^{2}-4t+1)^{2}. ,. In the. same. way,. we can. ,. have:. $\Delta$_{K,$\rho$_{5}^{+} (t)=-\displaystyle \frac{1}{t^{5} (t-1)(t^{4}-9t^{3}+44t^{2}-9t+1) We denote. by. \mathcal{A}_{K,m}^{+} the corresponding \mathcal{A}_{K,m} with $\rho$^{+}. ,. so we. .. obtain:. \displaystyle \frac{4 $\pi$\log|\mathcal{A}_{K,4}^{+}(t)|}{4^{2} =\frac{ $\pi$\log|t^{2}-4t+1|}{4}\rightar ow\frac{ $\pi$\log 2}{4}\ap rox 0.54 397t=1\ldots. ;. \displaystyle \frac{4 $\pi$\log|\mathcal{A}_{K,5}^{+}(t)|}{5^{2} =\frac{ $\pi$\log|\frac{t^{4}-9t^{3}+4 t^{2}-9t+1}{t^{2}-5t+1}| {5^{2} \rightar ow\frac{4 $\pi$\log\frac{28}{3} {5^{2} \ap rox 1.12 73t=1\ldots The the. following. is the results. holonomy representation. using by. a. computer. The symbol. \mathcal{A}_{K,m}^{-} corresponds to the lift of. of K :. $\rho$^{-}(a)=-\left(\begin{ar y}{l 1&1\ 0&1 \end{ar y}\right) $\rho$^{-}(b)=-(_{\frac{-1+\sqrt{-3}1}{2} 01) ;. Note that. \mathcal{A}_{K,m}^{+}(t) =\mathcal{A}_{K,m}^{-}(t). these and. we. when. m. is odd. Mr.. Tetsuya. Takahashi. .. helped. me. to calculate. used the softwares Wolfram Mathematica and MathWorks Matlab. It took about. 4\sim 5 hours to compute in the. degree. 33. case..
(11) 119. Acknowldgements This work. was. supported by JSPS. KAKENHI Grant Numbers \mathrm{J}\mathrm{P}15\mathrm{K}04868.. References. [1] Alexander, J.W., Topological invariants of knots and links. Trans. Amer. Math. Soc., 30. (1928),. no.. 2, 275‐306.. [2] Culler, M., Lifting representations. to. covering. groups, Adv. in Math., 59. (1986), 64‐70.. arXiv:0906. 1500v4.. [3] Dunfield, N.M., Friedl, S., Jackson, N., Twisted Alexander Polynomials of Hyperbolic knots.. Exp. Math.. ,. 21. (2012),. no.. 4, 329‐352.. [4] Friedl, S., Vidussi. S., A survey of twisted Alexander polynomials, The Mathematics of Knots:. Theory. ences), (2010), [5] Goda,. and. Application (Contributions. Twisted. H.,. in Mathematical and. Computational. Sci‐. 45‐94. Alexander. invariants. and. Hyperbolic. volume,. preprint,. arXiv: 1604.07490.. [6] Goda, H., Kitano, T., and Morifuji, T., Reidemeister torsion, twisted Alexander polynomial and fibered knots, Comment. Math. Helv. 80. (2005),. no.. 1, 51‐61.. [7] Kirk, P. and Livingston, C., Twisted Alexander invariants, Reidemeister torsion, and Casson‐Gordon invariants.. Topology,. 38. (1999),. no.. 3, 635‐661.. [8] Kitano, T., Twisted Alexander polynomial and Reidemeister torsion. Pacific. (1996),. no.. J.. Math., 174. 2, 431‐442.. [9] Kitano, T., Twenty years of twisted Alexander polynomials, Sugaku, 65 (2013), 360−384. (in Japanese). [10] Kitano, T., Goda, H., and Morifuji, T., Twisted Alexander invariants, Sugaku Memoirs vol.5, The Mathematical Society of Japan, 2006 (in Japanese).. [11] Kojima, S., The geometry of 3‐dimension, Asakurasyoten, 2002 (in Japanese). [12] Lin, X.S., Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin. 17. (2001),. no.. 3, 361‐380.. [13] Menal‐Ferrer, P. and Porti, J., Twisted cohomology for hyperboilc three manifolds. Osaka J. Math. 49 ,. (2012), 741‐769..
(12) 120. [14] Menal‐Ferrer, P. and Porti, J., Higher‐dimensional Reidemeister torsion invariants for. cusped hyperbolic. 3‐manifolds. J.. Topol.. ,. 7. (2014),. no.. 1, 69‐119.. [15] Morifuji,T., Representations of knot groups into \mathrm{S}\mathrm{L}(2,\mathrm{C}) and twisted Alexander polyno‐ mials, Handbook of Group Actions (Vol. I), Advanced Lectures in Mathematics 31 (2015) 527‐576.. [16] Müller, W., Analytic torsion and Soc., 6 (1993),. no.. \mathrm{R} ‐torsion for unimodular. representations.. J. Amer. Math.. 3, 721‐753.. [17] Müller, W., The asymptotics of the Ray‐Singer analytic torsion of hyperbolic 3‐manifolds, Metric and differential geometry,. 317‐352, Progr. Math., 297, Birkhäuser/Springer, Basel,. 2012.. [18] Ohtsuki, T.,. Knot. invariants, Kyoritsushuppan, 2015 (in Japanese).. [19] Wada, M., Twisted Alexander polynomial for finitely presentable groups. Topology, 33. (1994),. 2, 241‐256.. no.. [20] Yamaguchi, Y., On the non‐acyclic Reidemeister torsion for knots, Dissertation. University. [21] Yamaguchi, Y., A relationship between the non‐acyclic Reidemeister torsion and the. acyclic. Department. Reidemeister torsion. Ann. Inst. Fourier. (Grenoble),. 58. (2008),. no.. a zero. of. 1, 337‐362.. of Mathematics. Tokyo University. of Agriculture and. 2‐24‐16 Naka‐cho,. Tokyo. at the. of Tokyo, 2007.. Technology. Koganei. 184‐8588. Japan \mathrm{E} ‐mail address:. godaecc.. tuat.. ac.. jp. \ovalbox{\t \smal REJECT}_{\near ow5_{\backslash}^{1}\ovalbox{\t \smal REJECT}\mp}^{\leftrightar ow}\mathrm{T}$\lambda$\ovalbox{\t \smal REJECT}\cdot\mathrm{x}^{$\beta$\Leftrightar ow\Re\Leftrightar ow\mathcal{I}\ovalbox{\t \smal REJECT}\mathrm{f}\mathrm{f}\mathrm{l}_{J}^{\ovalbox{\t \smal REJECT}_{\mathrm{L} \ovalbox{\t \smal REJECT} \mathrm{A}_{\square}\mathrm{H}\exist \backslash\not\in^{\backslash}.
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