EXPLICIT FORMULAE FOR CHERN-SIMONS INVARIANTS OF THE HYPERBOLIC J(2n, 2m) KNOT ORBIFOLDS

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1 EXPLICIT FORMULAE FOR CHERN-SIMONS INVARIANTS OF THE HYPERBOLIC J2n, ) KNOT ORBIFOLDS JI-YOUNG HAM, JOONGUL LEE Abstract. We calculate the Chern-Simons invariants of the hyperbolic J2n, ) not orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of J2n, ) not. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee s methods to a bi-infinite family. We dealt with even slopes just as easily as odd ones. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic J2n, ) not orbifolds. For the fundamental group of J2n, ) not, we tae and tailor Hoste and Shanahan s. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert s canonical 2-bridge diagram or not. 1. Introduction Chern-Simons invariants of hyperbolic not orbifolds are computed explicitly for a few infinite families in 9, 10, 11 using the Schläfli formula. In 8, Chern-Simons invariants of hyperbolic alternating not orbifolds are computed explicitly using ideal triangulations by extending Neumann s Methods. This paper is written to compare the Chern-Simons invariants computed in 8 with those of this paper. We wish to provide an efficient algorithm to compute the Chern-Simons invarinats of hyperbolic orbifolds of nots and lins with SnapPy in a near future. In this paper, we present the explicit formulae for Chern-Simons invariants of the hyperbolic J2n, ) not orbifolds and we present them numerically for some of J2n, ) not orbifolds. We expect our methods directly extend to any two-bridge not of the Conway s notation C 1, 2,..., 2r ). Note that J2n, ) not is equal to C, 2n) not. For a two-bridge hyperbolic lin, there exists an angle α 0 2π 3, π) for each lin K such that the cone-manifold Kα) is hyperbolic for α 0, α 0 ), Euclidean for α = α 0, and spherical for α α 0, π 25, 13, 19, 26. We will use the Chern- Simons invariant of the lens space L 4nm, 2n 1) ) calculated in 15. Let us denote J2n, ) by T2n. The following theorem gives the Chern-Simons invariant formulae for T2n. The following theorem applies to the hyperbolic T2n nots. Let S v) be the Chebychev polynomials defined by S 0 v) = 1, S 1 v) = v and S v) = vs 1 v) S 2 v) for all integers. Theorem 1.1. Let X2n α), 0 α < α 0 be the hyperbolic cone-manifold with underlying space S 3 and with singular set T2n of cone-angle α. Let be a positive 2010 Mathematics Subject Classification. 57M25, 57M27. Key words and phrases. Chern-Simons invariant, J2n, ) not, orbifold, Riley-Mednyh polynomial, orbifold covering. 1

2 2 Ji-young Ham, Joongul Lee Figure 1. J2n, ) not integer such that -fold cyclic covering of X 2n 2π Simons invariant of X2n 2π ) mod 1 by the following formula: 2π cs X2n α0 4π 2 2π 4π 2 4π 2 π )) ) is hyperbolic. Then the Chernif is odd) is given if is even or mod cs L 4nm, 2n 1) )) 2 2 log M 2 )) S n v) S n 1 v)) S n 1 v) S n 2 v)) Im S n v) S n 1 v)) M 2 dα S n 1 v) S n 2 v)) Im log M 2 )) S n v 1 ) S n 1 v 1 )) S n 1 v 1 ) S n 2 v 1 )) α 0 S n v 1 ) S n 1 v 1 )) M 2 dα S n 1 v 1 ) S n 2 v 1 )) Im log M 2 )) S n v 2 ) S n 1 v 2 )) S n 1 v 2 ) S n 2 v 2 )) α 0 S n v 2 ) S n 1 v 2 )) M 2 dα, S n 1 v 2 ) S n 2 v 2 )) π where for M = e iα 2, x = v M 2 M 2 ImS n v) S n 1 v))s n 1 v) S n 2 v))) 0 32, Lemma 3.9), x 1 = v 1 M 2 M 2, and x 2 = v 2 M 2 M 2 are zeroes of Riley-Mednyh polynomial φ 2n x, M) which is given in Theorem 2.3 and x 1 and x 2 approach common x as α decreases to α 0 and they come from the components of x and x. 2. J2n, ) nots A general reference for this section is 16. A not is J2n, ) not if it has a regular two-dimensional projection of the form in Figure 1. J2n, ) not has 2n right-handed vertical crossings and left-handed horizontal crossings. Recall that we denote it by T2n. One can easily chec that the slope of T2n is /4nm ) which is equivalent to the not with slope 2n 1) ) /4nm ) 28. We will use the following fundamental group of J2n, ) not in 16. The following proposition can also be obtained by reading off the fundamental group from the Schubert normal form of T2n with slope /4nm ) 28, 27 which answers Hoste-Shanahan s question completely for J2n, ) nots. A little caution is needed. Since the slope is even, instead of ɛ j = 1) 2jm 4nm+1 for the power of t in w, we need to use ɛ j if the same orientation of two meridians, s and t, is preferred. LetX2n be S 3 \J2n, ). Proposition 2.1. π 1 X 2n ) = s, t sw m t 1 w m = 1,

3 Chern-Simons invariants of genus one two-bridge nots 3 where w = t 1 s) n ts 1 ) n The Chebychev polynomial. Let S v) be the Chebychev polynomials defined by S 0 v) = 1, S 1 v) = v and S v) = vs 1 v) S 2 v) for all integers. The following explicit formula for S v) can be obtained by solving the above recurrence relation 33. S n v) = ) n i 1) i v n 2i i 0 i n 2 for n 0, S n v) = S n 2 v) for n 2, and S 1 v) = 0. The following proposition 2.2 can be proved using the Cayley-Hamilton theorem 30. a b Proposition , Proposition 2.4 Suppose V = SL c d 2 C). Then V S v) ds = 1 v) bs 1 v) cs 1 v) S v) as 1 v) where v = trv ) = a + d The Riley-Mednyh polynomial. Let and let ρs) = M 1 0 M 1, ρt) = M 0 2 v M 1, c = 0 2 v ) 1. 2 v 0 Then from the above Proposition 2.2, we get the following Theorem 2.3. Let ρs) = S, ρt) = T and ρw) = W. Then trt 1 S) = v = trt S 1 ). Let v = x + M 2 + M 2. Theorem 2.3 can be found in 8, 32. We include the proof for readers convenience. Theorem , 32 ρ is a representation of π 1 X2n ) if and only if x is a root of the following Riley-Mednyh polynomial, φ 2n x, M) = S m z) xs n 1 v) S n v) + 1 v)s n 1 v)) S m 1 z). Proof. Since T 1 S = T S 1 = Hence 1 M 1 MM 2 + x 2) + M 2 ) M 2 + x 1 + M 2 1 M M 1 M 2 + x 2) + M 2 ) M 2 + x 1 + M 2 T 1 S) n Sn v) v 1)S = n 1 v) M 1 S n 1 v), Mv 2)S n 1 v) S n v) S n 1 v) T S 1 ) n Sn v) v 1)S = n 1 v) MS n 1 v) M 1 v 2)S n 1 v) S n v) S n 1 v) W = T 1 S) n T S 1 ) n W = 11 W 12 where 2 v)w 12 W 22, and

4 4 Ji-young Ham, Joongul Lee W 11 = S 2 nv) + 2 2v)S n v)s n 1 v) M 2 2v M 2 v + v 2 )S 2 n 1v) W 12 = M 1 M)S n v)s n 1 v) + Mv M M 1 )S 2 n 1v) W 22 = S 2 nv) 2S n v)s n 1 v) M 2 M 2 v)s 2 n 1v). Let z = TrW ). Then, since S 2 nv) vs n v)s n 1 v) + S 2 n 1v) = 1 by 31, Lemma 2.1 or by induction), z = W 11 + W 22 = 2S 2 nv) vs n v)s n 1 v) + S 2 n 1v)) + 2M 2 + 2M 2 2v M 2 v M 2 v + v 2 )S 2 n 1v) = 2 + v 2)xS 2 n 1v). By Proposition 2.2, we have W ) m Sm z) W = 22 S m 1 z) W 12 S m 1 z). 2 v)w 12 S m 1 z) S m z) W 11 S m 1 z) Therefore trsw m c)/ 2 v = M M 1 )W 12 S m 1 z) + S m z) W 11 S m 1 z) ) gives φ 2n x, M) Longitude Let l = w m w ) m, where w is the word obtained by reversing w. Let L = ρl) 11. Then l is the longitude which is null-homologus in X2n. Recall ρw) = W. Let W = ρw ). It is easy to see that W can be written as W W = 22 W12 2 v) W 12 W11 where W ij is obtained by W ij by replacing M with M 1. Similar computation was introduced in 16. Hence, W 11 = S 2 nv) + 2 2v)S n v)s n 1 v) M 2 2v M 2 v + v 2 )S 2 n 1v) W 12 = M M 1 )S n v)s n 1 v) + M 1 v M 1 M)S 2 n 1v) W 22 = S 2 nv) 2S n v)s n 1 v) M 2 M 2 v)s 2 n 1v). Definition. The complex length of the longitude l is the complex number γ α modulo 4πZ satisfying trρl)) = 2 cosh γ α 2. Note that l α = Reγ α ) is the real length of the longitude of the cone-manifold X2n α). The following lemma was introduced in 16 with slightly different coordinates. Lemma W 21 L + W 21 = 0. Theorem 3.2. L = M 2 S n v) S n 1 v)) S n 1 v) S n 2 v)) S n v) S n 1 v)) M 2 S n 1 v) S n 2 v)).

5 Chern-Simons invariants of genus one two-bridge nots 5 Proof. By directly computing W 21 L + W 21 = 0 in Lemma 3.1 and substituting S n v) + S n 2 v) for vs n 1 v), the theorem follows. 4. Schläfli formula for the generalized Chern-Simons function The general references for this section are 14, 15, 34, 21, 9, 10 and 11. In 14, Hilden, Lozano, and Montesinos-Amilibia defined the generalized Chern- Simons function on the oriented cone-manifold structures which matches up with the Chern-Simons invariant when the cone-manifold is the Riemannian manifold. On the generalized Chern-Simons function on the family of J2n, ) conemanifold structures we have the following Schläfli formula. Theorem 4.1. Theorem 1.2 of 15) For a family of geometric cone-manifold structures, X2n α), and differentiable functions αt) and βt) of t we have di X2n α) ) = 1 4π 2 βdα. 5. Proof of the theorem 1.1 For n 1 and M = e i α 2, φ 2n x, M) have n component ) zeros. The component )) which passes through x 1, x 2 ) = 2 2 cos, 2 2 cos at α = π+1) 4nm+1 π 1) 4nm+1 π is the geometric component by 15, Theorem 2.1, 2 x 1 > 0 and 2 x 2 > 0. For each T2n, there exists an angle α 0 2π 3, π) such that T2n is hyperbolic for α 0, α 0 ), Euclidean for α = α 0, and spherical for α α 0, π 25, 13, 19, 26. Denote by DX2n α)) be the set of zeros of the discriminant of φ 2n x, e i α 2 ) over x. Then α 0 will be one of DX2n α)). On the geometric component we can calculate the Chern-Simons invariant of an orbifold X2n 2π ) mod 1 1 if is even or mod 2 if is odd), where is a positive integer such that -fold cyclic covering of X2n 2π ) is hyperbolic: )) )) 2π 2π cs X2n I X2n mod 1 ) I X2n π) ) π 4π 2 β dα mod 1 ) 1 cs L4nm, 2n 1) )) 2 α0 Im 2 log 4π 2 4π 2 2π π α 0 Im π 4π 2 Im α 0 mod 1 2π M 2 S n v) S n 1 v)) S n 1 v) S n 2 v)) S n v) S n 1 v)) M 2 S n 1 v) S n 2 v)) log M 2 S n v 1 ) S n 1 v 1 )) S n 1 v 1 ) S n 2 v 1 )) S n v 1 ) S n 1 v 1 )) M 2 S n 1 v 1 ) S n 2 v 1 )) log M 2 S n v 2 ) S n 1 v 2 )) S n 1 v 2 ) S n 2 v 2 )) S n v 2 ) S n 1 v 2 )) M 2 S n 1 v 2 ) S n 2 v 2 )) if is even or mod 1 2 ) if is odd )) dα )) dα )) dα where the second equivalence comes from Theorem 4.1 and the third equivalence comes from the fact that I X2n π) ) 1 2 cs L4nm, 2n 1) )) mod 2) 1, Theorem 3.2, and geometric interpretations of hyperbolic and spherical holonomy representations.

6 6 Ji-young Ham, Joongul Lee The following theorem gives the Chern-Simons invariant of the Lens space L4nm+ 1, 2n 1) ). Theorem 5.1. Theorem 1.3 of 15) cs L 4nm, 2n 1) )) m n 4nm mod 1). 6. Chern-Simons invariants of the hyperbolic J2n, ) not orbifolds and of its cyclic coverings The table 1 gives the approximate Chern-Simons invariant of T2n for n between 1 and 4, m between 1 and 4 with n m. Since T2 2, T4 4, T6 6, T8 8 are amphicheiral nots, their Chern-Simons invariants are zero as expected. We used Simpson s rule for the approximation with in Simpson s rule) intervals from 0 to α 0 and in Simpson s rule) intervals from α 0 to π. The table 2 gives the approximate Chern-Simons invariant of the hyperbolic orbifold, cs X2n 2π )) for n between 1 and 4, m between 1 and 4 with n m, and for between 3 and 10, and of its cyclic covering, cs M X2n ) ) except amphicheiral nots. We used Simpson s rule for the approximation with in Simpson s rule) intervals from 2π/ to α 0 and in Simpson s rule) intervals from α 0 to π. We used Mathematica for the calculations. We record here that our data in Table 1 and those obtained from SnapPy match up up to existing decimal points and our data in Table 2 and those presented in 8 match up up to existing decimal points when Chern-Simons invariants presented in 8 are divided by 2π 2 and then read by modulo 1/r for r even and are divided by π 2, read by modulo 1/r and then divided by 2 for r odd. Table 1: Chern-Simons invariant of X2n 4 with n m except amphicheiral nots. 2n α 0 cs ) X2n for n between 1 and 4, m between 1 and Acnowledgements. The authors would lie to than Alexander Mednyh, Hyu Kim, Nathan Dunfield. References 1 Shiing-shen Chern and James Simons. Some cohomology classes in principal fiber bundles and their application to Riemannian geometry. Proc. Nat. Acad. Sci. U.S.A., 68: , 1971.

7 Chern-Simons invariants of genus one two-bridge nots 7 Table 2: Chern-Simons invariant of the hyperbolic orbifold, cs X2n 2π )) for n between 1 and 4, m between 1 and 4 with n m, and for between 3 and 10, and of its cyclic covering, cs M X2n ) ) except amphicheiral nots. cs X4 2 2π )) cs M X4 2 ) ) cs X8 2 2π )) cs M X8 2 ) ) cs X8 4 2π )) cs M X8 4 ) ) cs X6 2 2π )) cs M X6 2 ) ) cs X6 4 2π )) cs M X6 4 ) ) cs X8 6 2π )) cs M X8 6 ) ) Jinseo Cho, Hyu Kim, and Seonwha Kim. Optimistic limits of ashaev invariants and complex volumes of hyperbolic lins. J. Knot Theory Ramifications, 239), Jinseo Cho and Jun Muraami. The complex volumes of twist nots via colored Jones polynomials. J. Knot Theory Ramifications, 1911): , Jinseo Cho, Jun Muraami, and Yoshiyui Yoota. The complex volumes of twist nots. Proc. Amer. Math. Soc., 13710): , Daryl Cooper, Craig D. Hodgson, and Steven P. Kerchoff. Three-dimensional orbifolds and cone-manifolds, volume 5 of MSJ Memoirs. Mathematical Society of Japan, Toyo, With a postface by Sadayoshi Kojima. 6 Marc Culler, Nathan Dunfield, Jeff Wees, and Many others. SnapPy. edu/t3m/snappy/. 7 Oliver Goodman. Snap. 8 Ji-Young Ham, Hyu Kim, Joongul Lee, and Seobeom Yoon. On the volume and the chernsimons invariant for the hyperbolic alternating not orbifolds

8 8 Ji-young Ham, Joongul Lee 9 Ji-Young Ham and Joongul Lee. Explicit formulae for Chern-Simons invariants of the twist not orbifolds and edge polynomials of twist nots. Mat. Sb., 2079): , Ji-Young Ham and Joongul Lee. Explicit formulae for Chern Simons invariants of the hyperbolic orbifolds of the not with Conway s notation C2n, 3). Lett. Math. Phys., 1073): , Ji-Young Ham, Joongul Lee, Alexander Mednyh, and Alesey Rassazov. On the volume and the chern-simons invariant for the 2-bridge not orbifolds. arxiv , Ji-Young Ham, Alexander Mednyh, and Vladimir Petrov. Trigonometric identities and volumes of the hyperbolic twist not cone-manifolds. J. Knot Theory Ramifications, 2312): , 16, Hugh Hilden, María Teresa Lozano, and José María Montesinos-Amilibia. On a remarable polyhedron geometrizing the figure eight not cone manifolds. J. Math. Sci. Univ. Toyo, 23): , Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia. On volumes and Chern-Simons invariants of geometric 3-manifolds. J. Math. Sci. Univ. Toyo, 33): , Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia. Volumes and Chern-Simons invariants of cyclic coverings over rational nots. In Topology and Teichmüller spaces Katinulta, 1995), pages World Sci. Publ., River Edge, NJ, Jim Hoste and Patric D. Shanahan. A formula for the A-polynomial of twist nots. J. Knot Theory Ramifications, 132): , Paul Kir and Eric Klassen. Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of T 2. Comm. Math. Phys., 1533): , Paul A. Kir and Eric P. Klassen. Chern-Simons invariants of 3-manifolds and representation spaces of not groups. Math. Ann., 2872): , Sadayoshi Kojima. Deformations of hyperbolic 3-cone-manifolds. J. Differential Geom., 493): , Robert Meyerhoff. Hyperbolic 3-manifolds with equal volumes but different Chern-Simons invariants. In Low-dimensional topology and Kleinian groups Coventry/Durham, 1984), volume 112 of London Math. Soc. Lecture Note Ser., pages Cambridge Univ. Press, Cambridge, Robert Meyerhoff and Daniel Ruberman. Mutation and the η-invariant. J. Differential Geom., 311): , G. D. Mostow. Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Inst. Hautes Études Sci. Publ. Math., 34):53 104, Walter D. Neumann. Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-manifolds. In Topology 90 Columbus, OH, 1990), volume 1 of Ohio State Univ. Math. Res. Inst. Publ., pages de Gruyter, Berlin, Walter D. Neumann. Extended Bloch group and the Cheeger-Chern-Simons class. Geom. Topol., 8: electronic), Joan Porti. Spherical cone structures on 2-bridge nots and lins. Kobe J. Math., 211-2):61 70, Joan Porti and Hartmut Weiss. Deforming Euclidean cone 3-manifolds. Geom. Topol., 11: , Robert Riley. Parabolic representations of not groups. I. Proc. London Math. Soc. 3), 24: , Horst Schubert. Knoten mit zwei Brücen. Math. Z., 65: , William Thurston. The geometry and topology of 3-manifolds. boos/gt3m, 1977/78. Lecture Notes, Princeton University. 30 Anh T. Tran. Reidemeister torsion and Dehn surgery on twist nots. arxiv: , To appear in Toyo Journal of Mathematics. 31 Anh T. Tran. Twisted Alexander polynomials of genus one two-bridge nots. arxiv: , Anh T. Tran. Volumes of double twist not cone-manifolds. arxiv: , Anh T. Tran. The A-polynomial 2-tuple of twisted whitehead lins. arxiv: , Tomoyoshi Yoshida. The η-invariant of hyperbolic 3-manifolds. Invent. Math., 813): , 1985.

9 Chern-Simons invariants of genus one two-bridge nots 9 35 Christian K. Zicert. The volume and Chern-Simons invariant of a representation. Due Math. J., 1503): , address: jiyoungham1@gmail.com. Department of Mathematics Education, Hongi University, 94 Wausan-ro, Mapo-gu, Seoul, 04066, Korea address: jglee@hongi.ac.r