• 대한수학회보
• /
• 제40권1호
• /
• pp.101-108
• /
• 2003

In this paper we study the connections between cyclic presentations of groups and cyclic branched coverings of (1, 1)- knots. In particular, we prove that every π-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus π.

• 대한수학회지
• /
• 제35권4호
• /
• pp.961-980
• /
• 1998

The cyclically presented groups which arise as fundamental groups of cyclic branched coverings of the knot $5_2$ are studied. The fundamental polyhedra for these groups are described. Moreover the cyclic covering manifolds are obtained in terms of Dehn surgery and as two-fold branched coverings of the 3-sphere.

• 대한수학회보
• /
• 제54권5호
• /
• pp.1851-1857
• /
• 2017

We study relations between two descriptions of closed orientable 3-manifolds: as branched coverings and as Heegaard splittings. An explicit relation is presented for a class of 3-manifolds which are branched cyclic coverings of connected sums of lens spaces, where the branching set is an axis of a hyperelliptic involution of a Heegaard surface.

• East Asian mathematical journal
• /
• 제38권1호
• /
• pp.107-127
• /
• 2022

In this paper, we discuss the relationship between doubly-pointed Heegaard diagrams of (1, 1)-knots in lens spaces and Dunwoody 3-manifolds, and then give explicit representations of n-fold cyclic branched coverings of all (1, 1)-knots in S³ up to 10 crossings in Rolfsen's knot table as Dunwoody 3-manifolds.

• 대한수학회지
• /
• 제46권6호
• /
• pp.1119-1137
• /
• 2009

We construct a family of hyperbolic 3-manifolds by pairwise identifications of faces in the boundary of certain polyhedral 3-balls and prove that all these manifolds are cyclic branched coverings of the 3-sphere over certain family of links with two components. These extend some results from [5] and [10] concerning with the branched coverings of the whitehead link.

• 대한수학회논문집
• /
• 제20권4호
• /
• pp.803-812
• /
• 2005

We construct an infinite family of closed 3-manifolds M(2m+ 1, n, k) which are identification spaces of certain polyhedra P(2m+ 1, n, k), for integers $m\;\ge\;1,\;n\;\ge\;3,\;and\;k\;\ge\;2$. We prove that they are (n / d)- fold cyclic coverings of the 3-sphere branched over certain links $L_{(m,d)}$, where d = gcd(n, k), by handle decomposition of orbifolds. This generalizes the results in [3] and [2] as a particular case m = 2.

• 대한수학회지
• /
• 제47권3호
• /
• pp.585-598
• /
• 2010

We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

• 대한수학회논문집
• /
• 제19권3호
• /
• pp.531-544
• /
• 2004

We study a family of 2-bridge knots with 2-tangles in the 3-sphere admitting a genus two 1-bridge splitting. We also observe a geometric relation between (g - 1, 1)-splitting and (g,0)- splitting for g = 2,3. Moreover we construct a family of closed orientable 3-manifolds which are n-fold cyclic coverings of the 3-sphere branched over those 2-bridge knots.

• 대한수학회지
• /
• 제50권2호
• /
• pp.425-444
• /
• 2013

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

• 대한수학회보
• /
• 제48권1호
• /
• pp.197-211
• /
• 2011

Every (1, 1)-knot is represented by a 4-tuple of integers (a, b, c, r), where a > 0, b $\geq$ 0, c $\geq$ 0, d = 2a+b+c, $r\;{\in}\;\mathbb{Z}_d$, and it is well known that all 2-bridge knots and torus knots are (1, 1)-knots. In this paper, we describe some conditions for 4-tuples which determine 2-bridge knots and determine all 4-tuples representing any given 2-bridge knot.