Akio Kawauchi

ON PSEUDO-RIBBON SURFACE-LINKS

We first introduce the null-homotopically peripheral quadratic function of a surface-link to obtain a lot of pseudo-ribbon, non-ribbon surface-links, generalizing a known property of the turned spun torus-knot of a non-trivial knot. Next, we study the torsion linking of a surface-link to show that the torsion linking of every pseudo-ribbon surface-link is the zero form, generalizing a known property of a ribbon surface-link. Further, we introduce and algebraically estimate the triple point cancelling number of a surface-link.

TORSION LINKING FORMS ON SURFACE-KNOTS AND EXACT 4-MANIFOLDS

We focus an interest on the torsion linking of a surface-knot. It is a knot invariant independent of the surface-knot group and its peripheral subgroup. It is identified with the torsion linking of any associated closed 4-manifold with infinite cyclic first homology. In the case of such a 4-manifold with an exact leaf, the linking of the leaf is identified with an orthogonal sum of it and a hyperbolic linking .

Mutative hyperbolic homology 3-spheres with the same Floer homology

We construct a large class of finitely many hyperbolic homology 3-spheres making the following invariants equal, simultaneously, the integral homology, the quantum SU(2) invariants, the hyperbolic volume, the hyperbolic isometry group, the η-invariant, the Chern-Simons invariant, and the Floer homology.

TOPOLOGICAL IMITATION, MUTATION AND THE QUANTUM SU(2) INVARIANTS

It is proved that any two mutative closed oriented 3-manifolds have the same quantum SU(2) invariant. By a constructive argument of topological imitation, we construct finitely many mutative hyperbolic imitations of any given closed oriented 3-manifold with certain arbitrariness of isometry groups whose quantum SU(2) invariants are close to the original one.

Three dimensional homology handles and circles

This paper will extend the known propertes of the Alexander polynomials of classical knot complements to the properties of the Alexander polynomials of arbitrary (possibly non-orientable) compact 3-manifolds with infinite cyclic first homology groups. In particular, the Alexander polynomial will always have a reciprocal property. The existence of the corresponding manifolds and the other related results will be shown.

UNKNOTTING ORIENTABLE SURFACES IN THE 4-SPHERE

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an ...

On Transforming a Spatial Graph into a Plane Graph

This talk is an improved revision of the talk (see [1]) given at the workskp Knots and soft-matter physics, Kyoto, August, 2008 on a complexity of a spatial graph with an emphasis on a transformation of spatial graph into a plane graph. In a research of proteins, molecules, or polymers, it is important to understand geometrically and topologically spatial graphs possibly with degree one vertices including knotted arcs. For every spatial graph without degree one vertices, we introduce the complexity and related topological invariants, called the warping degree, the γ-warping degree, and the (γ,Γ)-warping degree by revising the similar invariants in [1]. The concept of complexity explains a path from any given spatial graph to a plane graph. Similarly, the unknotting number and related concepts γ -unknotting number, Γ -unknotting number, (γ ,Γ )-unknotting number (generalizing the usual unknotting number of a knot) are explained. These invariants are used to define semi-topological invariants for a spatial graph with degree one vertices, meaningful even for a knotted arc.

Enumerating prime links by a canonical order

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conways table of prime links of 10 crossings.

An intrinsic Arf invariant on a link and its surface-link analogue

Abstract We define a modulo one rational number invariant of order up to 2 as the Arf invariant of the Z 2 -hyperbolic quadratic function of every infinite cyclic covering of every (possibly non-orientable) compact 3-manifold, and analogously a modulo one rational number invariant of order up to 4 as an invariant of the Q / Z -quadratic function of every infinite cyclic covering of every compact oriented 4-manifold. Typically the invariants become an invariant of an arbitrary oriented link in S 3 and an invariant of an arbitrary oriented surface-link in S 4 .

Knot theory of spatial graphs

The topological study of spatial graphs is considered to be a natural extension of knot theory, although it has not been paid much attention until quite recently. In this chapter, we regard two notions on “equivalence” of graphs. The first one is a notion naturally extending positive-equivalence of links and is called equivalence. The second one is a notion which is useful when we study the exterior of a spatial graph and is called neighborhood-equivalence. Since the importance of the first concept is motivated by recent developments in molecular chemistry, we devote the first section to some comments on the topology of molecules. In 15.2 we discuss some results on the first notion, and in 15.3 some results on the second notion, including an explanation of recent developments on the tunnel number.