On the cyclic coverings of the knot 5₂

Abstract

We construct a family of hyperbolic 3-manifolds whose fundamental groups admit a cyclic presentation. We prove that all these manifolds are cyclic branched coverings of S 3 over the knot 5 2 and we compute their homology groups. Moreover, we show that the cyclic presentations correspond to spines of the manifolds.

Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

“…In particular, for h = 1 and k = 2 we have that t(k, hk + 1) = t(2, 3) is the trefoil knot and the fundamental group of its n-fold cyclic branched covering is G n (x 3 x 1 x − 1 2 ), which is clearly isomorphic to the Sieradski group S(n) described in Section 1.…”

“…induced by a Heegaard diagram of a closed orientable 3-manifold) is of considerable interest in geometric topology and has already been examined by many authors (see [10,23,25,26,27,28,31]). Moreover, the connections between cyclic coverings of S 3 branched over knots and cyclic presentations of their fundamental groups, induced by suitable Heegaard diagrams, have recently been discussed in several papers (see [1,5,6,7,9,13,14,16,18,19,21,32]). …”

Seifert manifolds and (1, 1)-knots

“…The cyclic branched coverings of knots in S 3 with cyclically presented fundamental group have been thoroughly investigated in the recent years by many authors (see [1][2][3][4][5][6][7][8][9][10]). Their results have been included in an organic and more general context in [11], where it is proved that the fundamental group of each n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation encoded by a genus n Heegaard diagram.…”

On Branched Coverings of Lens Spaces

“…The goal of this paper is to study the topological and geometrical properties of certain classes of group presentations depending on a finite number of integers. This kind of work can be found in several papers (see, for example, [1][2][3][4][5][6][8][9][10][12][13][14][15][16][17]19,21,27,29]), where there are many connections between the theory of cyclically presented groups (see § 3 for the definition) and the topology of cyclic branched coverings of knots. Given a knot K in the 3-sphere S 3 , we say that a closed connected 3-manifold M is an n-fold cyclic branched covering of K if M is the n-fold cyclic covering of S 3 branched over K (see, for example, [26]).…”

“…Sieradski and Fibonacci manifolds admit nice combinatorial representations as quotients of polyhedral 3-balls by pairwise identifications of oppositely oriented boundary faces. Some natural generalizations of these polyhedral schemata were given, for example, in [1,5,12,15]. In the present paper we will consider more general tessellations of the boundary of a 3-ball, including those obtained in the quoted papers as particular cases.…”