Soo Hwan Kim

Weak and strong convergence of the Ishikawa iteration process with errors for two asymptotically nonexpansive mappings

In this paper, we prove the weak and strong convergence of the Ishikawa iterative scheme with errors to a common fixed point for two asymptotically nonexpansive mappings in a uniformly convex Banach space under a condition weaker than compactness. Our theorems improve and generalize recent known results in the literature.

TORUS KNOTS AND 3-MANIFOLDS

We consider a family of words in a free group of rank n which determine 3-manifolds ℳn(p,q). We prove that the fundamental groups of ℳn(p,q) are cyclically presented, and that ℳn(p,q) is the n-fold cyclic covering of the 3-sphere branched over the torus knots T(p,q) if p is odd and q≡±2(mod p). We also obtain an explicit Dunwoody parameters for the torus knots T(p,q) for odd p and q≡±2(mod p).

ON THE 2-BRIDGE KNOTS OF DUNWOODY (1,1)-KNOTS

Abstract. Every (1,1)-knot is represented by a 4-tuple of integers (a,b,c,r),where a>0,b≥ 0,c≥ 0,d= 2a+b+c,r∈ Z d ,and it is well knownthat 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 knotsand determine all 4-tuples representing any given 2-bridge knot. 1. IntroductionIn this note all manifolds will be assumed to be closed, connected and ori-entable and all (1,1)-knots are non-oriented if there is no special reference.In [6] Dunwoody introduced a family of 3-manifolds depending on six integerparameters which induce a class of knots. It was shown that all knots inducedby Dunwoody manifolds are (1,1)-knots in [21]. Moreover all (1,1)-knots areinduced by Dunwoody manifolds in [4]. In [12] and [21] a type of 4-tuplesrepresenting all 2-bridge knots was described. We here determine a type of4-tuples representing all 2-bridge knots and their dual and mirror images froma different point of view. We also recall that a type of 4-tuples representing thetorus knot T(p,q) was determined in [1] and [15] when either q ≡ ±1 mod por q ≡ ±2 mod p.Let (V

On the Polynomial of the Dunwoody (1, 1)-knots

There is a special connection between the Alexander polynomial of (1, 1)-knot and the certain polynomial associated to the Dunwoody 3-manifold ([3], [10] and [13]). We study the polynomial(called the Dunwoody polynomial) for the (1, 1)-knot obtained by the certain cyclically presented group of the Dunwoody 3-manifold. We prove that the Dunwoody polynomial of (1, 1)-knot in is to be the Alexander polynomial under the certain condition. Then we find an invariant for the certain class of torus knots and all 2-bridge knots by means of the Dunwoody polynomial.

On Hyperbolic 3-Manifolds Obtained by Dehn Surgery on Links

We study the algebraic and geometric structures for closed orientable 3-manifolds obtained by Dehn surgery along the family of hyperbolic links with certain surgery coefficients and moreover, the geometric presentations of the fundamental group of these manifolds. We prove that our surgery manifolds are 2-fold cyclic covering of 3-sphere branched over certain link by applying the Montesinos theorem in Montesinos-Amilibia (1975). In particular, our result includes the topological classification of the closed 3-manifolds obtained by Dehn surgery on the Whitehead link, according to Mednykh and Vesnin (1998), and the hyperbolic link 𝐿𝑑

An irreducible Heegaard diagram of the real projective 3-space p3

We give a genus 3 Heegaard diagram H of the real projective space p3, which has no waves and pairs of complementary handles. So Negamis result that every genus 2 Heegaard diagram of p3 is reducible cannot be extended to Heegaard diagrams of p3 with genus 3.

An iterative method for total quasi- Φ -asymptotically nonexpansive semi-groups and generalized mixed equilibrium problems in Banach spaces

In this paper, we introduce an iterative scheme by the modified Halpern- Mann-type for total quasi- ? -asymptotically nonexpansive semi-groups and prove the strong convergence theorem using our new iterative process which converges strongly to a common element of the set of common fixed points for total quasi- ? -asymptotically nonexpansive semi-groups and the set of solutions of generalized mixed equilibrium problem in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property using the properties of the generalized f-projection operator. Our results extend and improve the corresponding recent results in the literature.

On the Stability of -Derivations and Lie -Algebra Homomorphisms on Lie -Algebras: A Fixed Points Method

We investigate new generalized Hyers-Ulam stability results for -derivations and Lie -algebra homomorphisms on Lie -algebras associated with the additive functional equation:

The Dual and Mirror Images of the Dunwoody 3-Manifolds

Recently, in 2013, we proved that certain presentations present the Dunwoody -manifold groups. Since the Dunwoody -manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody -manifolds. It is well known that every -knots in a lens space can be represented by the set of the 4-tuples (Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody -manifolds, we proved the fact that the certain subset of representing all -bridge knots of -knots is determined completely by using the dual and mirror -decompositions (S. H. Kim and Y. Kim (2011)). In this paper, we show how to obtain the dual and mirror images of all elements in as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).

ON HYPERBOLIC 3-MANIFOLDS WITH SYMMETRIC HEEGAARD SPLITTINGS

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. There are two well known results about the realization of closed 3-manifolds. One is that any closed orientable 3-manifold can be obtained by Dehn surgeries on the components of an oriented link in the 3-sphere. The other one says that any closed 3-manifold can be represented as a branched covering of some link in the 3-sphere. So if we consider a link in the 3-sphere, we can construct many classes of closed orientable 3-manifolds by considering its branched coverings or Dehn surgeries along it. The description of closed 3-manifolds as polyhedral 3-balls, whose finitely many boundary faces are glued together in pairs, is a further standard way to construct 3-manifolds (see (3), (4), (10), (11), and (12)). If the polyhedral 3-ball admits a geometric structure and the face identifica- tion is performed by means of geometric isometries, then the same geometric structure is inherited by the quotient manifold (see (10), (12), and (15)). Many authors have studied the connections between the face identification procedure and the representation of closed 3-manifolds as branched coverings of the 3- sphere. In (10) Helling, Kim and Mennicke considered a family of polyhedral 3-balls Pn depending on a positive integer n, and for any coprime positive in- tegers n and k, they defined a pairwise gluing of faces in the boundary of Pn yielding a closed orientable 3-manifold Mn,k. In the sequel, they proved that Mn,k is an n-fold strongly cyclic covering of the 3-sphere branched over the Whitehead link and classified, up to isometry, those coverings. More general