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

Abstract

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).

Parameterizations of 1-Bridge Torus Knots

“…In [6], the first homology of the exterior of a 1-bridge torus knot was given in terms of an abstract description of a 1-bridge torus knot via a mapping class of twice punctured torus and also a necessary and sufficient condition for the existence of k-fold branched cover of a lens space along a 1-bridge torus knot was given. However it is hard to apply their result to a 1-bridge torus knot given by a diagram, for example.…”

“…Let (r, s, t, ρ) ǫ be integers satisfying the assumption in Theorem 2. 6. Then we obtain a simple arc α on T with ends x, y.…”

(1, 1)-knots via the mapping class group of the twice punctured torus

“…The statement derives from a straightforward application of [5,Theorem 7], For example, the fundamental group of the n-fold cyclic branched covering …”

“…These knots are very important in the light of some results and conjectures involving Dehn surgery on knots (see in particular [9] and [25]). Moreover, the strict connection between cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups have been pointed out in [5], [12] and [21].…”

“…As a notable application, in Sections 4 and 5 we obtain standard representations for the two most important classes of (1, 1)-knots in S 3 : the torus knots and the two-bridge knots. Moreover, applying certain results obtained in [5], we give explicit cyclic presentations for the fundamental groups of all cyclic branched coverings of torus knots of type (k, ck +2), with c, k > 0 and k odd.…”