Addendum to "Spherical structures on torus knots and links"
aa r X i v : . [ m a t h . G T ] J un Addendum to“Spherical structures on torus knots and links” ∗ Alexander Kolpakov
Abstract
The present paper considers an infinite family of cone-manifoldsendowed with spherical metric. The singular strata is the torus knotor link t( p, q ) depending on gcd( p, q ) = 1 or gcd( p, q ) >
1. In thelatter case one obtains a link with gcd( p, q ) components. Cone anglesalong all components of the singular strata are supposed to be equal.Domain of existence for a spherical metric is found and a volumeformula is presented.
Three-dimensional cone-manifold is a metric space obtained from a collectionof disjoint simplices in the space of constant sectional curvature k by isometricidentification of their faces in such a combinatorial fashion that the resultingtopological space is a manifold (also called the underlying space for a givencone-manifold).Such the metric space inherits the metric of sectional curvature k on theunion of its 2- and 3-dimensional cells. In case k = +1 the correspondingcone-manifold is called spherical (or admits a spherical structure). By anal-ogy, one defines euclidean ( k = 0) and hyperbolic ( k = −
1) cone-manifolds.The metric structure around each 1-cell is determined by a cone anglethat is the sum of dihedral angles of corresponding simplices sharing the 1-cell under identification. The singular locus of a cone-manifold is the closureof all its 1-cells with cone angle different from 2 π . For the further account ∗ Addendum to “Spherical structures on torus knots and links”, arXiv:1008.0312.
1e suppose that every component of the singular locus is an embedded circlewith constant cone angle along it. For the further account, see [1].The present paper is an addendum to [2] and comprises more generalcase of torus knot and link cone-manifolds. The cone angles are supposed tobe equal for all components of the singular strata. Denote a cone-manifoldof torus knot type singularity and cone angle(s) α by T p,q ( α ). This cone-manifold is rigid: it is Seifert fibred due to [3] and the base is a turnover ofcone angles α , πp and πq . Domains of existence for a spherical metric aregiven in terms of cone angle(s) and volume formula is presented. ( p, q ) knots Denote by t( p, q ) a torus knot or link depending on the case of gcd( p, q ) = 1or gcd( p, q ) >
1. In the latter case one obtains a link with gcd( p, q ) compo-nents, see [4].As far as t( p, q ) and t( q, p ) torus links are isotopic one may assume that p ≤ q without loss of generality. Denote T p,q ( α ) a cone–manifold with singu-lar set the torus knot t( p, q ) and cone angle α along its component(s).The following theorem holds for T p,q ( α ) cone-manifolds: Theorem 1
The cone-manifold T p,q ( α ) , ≤ p ≤ q admits a spherical struc-ture if π (cid:18) − p − q (cid:19) < α < π (cid:18) − p + 1 q (cid:19) . The volume of T p,q ( α ) equals Vol T p,q ( α ) = p · q (cid:18) α − π (cid:18) − p − q (cid:19)(cid:19) . Proof.
We prove this theorem using the result on two–bridge torus linksobtained earlier in [2] and the covering theory. Every two–bridge torus linkis a (2 , p ) torus link as shown at the Fig. 1.Divide the proof into three subsequent steps: Using the Reidemeister moves rearrange the diagram of T , p ,given at the Fig. 1 link in order to place one of its components around theother. The diagram obtained is depicted at the Fig. 2. Denote by T , p ( α, β )a cone-manifold with cone angles α and β along the components of T , p .2igure 1: Link T , p Figure 2: Link T , p rearranged The diagram of T p,q is depicted at the Fig. 3. It is clearly seenthat T p,q ( α ) forms a q –folded cyclic covering of T , p ( α, πq ) branched alongits central component, as depicted at the Fig. 2. The formula of [2, Theorem 2] provides thatVol T , p ( α, β ) = 12 p (cid:18) α + β · p − π ( p − (cid:19) . As far as T p,q ( α ) is a q -folded cyclic branched covering of T , p ( α, πq ) onehas thatVol T p,q ( α ) = q · Vol T , p (cid:18) α, πq (cid:19) = p · q (cid:18) α − π (cid:18) − p − q (cid:19)(cid:19) . T p,q Recall, that the number of components equals gcd( p, q ). By the Schl¨afliformula (see [1]), the singular strata component’s length for T p,q ( α ) equals ℓ α = 1gcd( p, q ) · dd α Vol T p,q ( α ) = lcm( p, q ) (cid:18) α − π (cid:18) − p − q (cid:19)(cid:19) . The proof is completed. (cid:3)
References [1]
D. Cooper, C. Hodgson, and S. Kerckhoff , “Three-dimensional orbifolds and cone-manifolds”, with a postface by Sa-dayoshi Kojima. Tokyo: Mathematical Society of Japan, 2000.(MSJ Memoirs; 5)[2]
A. Kolpakov, A. Mednykh , “Spherical structures on torus knotsand links” // Siberian Math. J. 2009. V. 50, N. 5, pp. 856-866.arXiv:1008.0312[3]
J. Porti , “Spherical cone structures on 2-bridge knots and links”// Kobe J. of Math. 2004. V. 21. N. 1. P. 61–70. Available on-line.[4]