On a Class of Cyclically Presented Groups

Kim, Ann Chi; Kim, Yang-Kok; Vesnin, Andrei; Baik, Young Gheel; Johnson, Johnson David L.

doi:10.1515/9783110807493-016

Cited by 12 publications

(19 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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]). …”

Section: Introductionmentioning

confidence: 99%

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

Cattabriga¹,

Mulazzani²

2003

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold stronglycyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus n. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type (k, hk ± 1).

show abstract

Section: Introductionmentioning

confidence: 99%

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

Cattabriga¹,

Mulazzani²

2003

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…For particular values of the surgery coefficients (including the classes of manifolds cited above), the periodic Takahashi manifolds turn out to be nfold cyclic coverings of S 3 , branched over two-bridge knots of genus one 1 , whose parameters are obtained using Kirby-Rolfsen calculus [18] (compare the analogous result of [11], obtained by a different approach). Observe that in [19] a characterization of all periodic Takahashi manifolds as n-fold cyclic coverings of S 3 , branched over the closure of certain rational 3-string braids, is presented, but the result is incorrect, as we show in Remark 1.…”

Section: Introductionmentioning

confidence: 99%

On periodic Takahashi manifolds

Mulazzani¹

2001

Tsukuba J. Math.

View full text Add to dashboard Cite

In this paper we show that periodic Takahashi 3-manifolds are cyclic coverings of the connected sum of two lens spaces (possibly cyclic coverings of S 3 ), branched over knots. When the base space is a 3-sphere, we prove that the associated branching set is a two-bridge knot of genus one, and we determine its type. Moreover, a geometric cyclic presentation for the fundamental groups of these manifolds is obtained in several interesting cases, including the ones corresponding to the branched cyclic coverings of S 3 .Mathematics Subject Classification 2000: Primary 57M12, 57R65; Secondary 20F05, 57M05, 57M25.

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%