Masaaki Wada

Twisted Alexander polynomial for finitely presentable groups

of the indeterminates tl,. . . , t, with coefficients in R called the twisted Alexander polynomial of Iassociated to p. The twisted Alexander polynomial is well-defined up to a factor of Et;’ . . t:, where E E R” is a unit of R and e,, . . . , e, are integers. The twisted Alexander polynomial is a generalization of the Alexander polynomial (cf. [3]) in the following sense. Let Ibe a finitely presentable group whose abelianization ~1: r -+ (t) is of rank 1. Then the Alexander polynomial of r is written as

Conjugacy invariants of möbius transformations

We define conjugacy invariants of Clifford matrices, which have been introduced by Ahlfors to generalize the Mobius transformation groups PSL (2,R) and PSL (2, C) to higher dimensions. Our conjugacy invariants are a generalization of the trace square functions. We also show explicitly how these invariants determine the conjugacy class of a Mobius transformation.

Computerized three-dimensional analysis of the heart and great vessels in normal and holoprosencephalic human embryos.

The developing heart and great vessels undergo drastic morphogenetic changes during the embryonic period. To analyze the normal and abnormal development of these organs, it is essential to visualize their structures in three and four dimensions, including the changes occurring with time. We have reconstructed the luminal structure of the hearts and great vessels of staged human embryos from serial histological sections to demonstrate their sequential morphological changes in three dimensions. The detailed structures of the embryonic heart and major arteries in normal and holoprosencephalic (HPE) human embryos could be reconstructed and visualized, and anatomical structures were analyzed using 3D images. By 3D analysis, cardiac anomalies such as double‐outlet right ventricle and malrotation of the heart tube were identified in HPE embryos, which were not easily diagnosed by histological observation. Reconstruction and analysis of 3D images are useful for the study of anatomical structures of developing embryos and for identifying their abnormalities. Anat Rec, 2007.

CONSTRUCTING LINKS BY PLUMBING FLAT ANNULI

We show that every oriented link is the boundary of a surface obtained from an embedded disk by plumbing a flat annulus a finite number of times.

Drawing bers embeddings of the teichmüller space of once-punctured tori

We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmüller space of once-punctured tori. The coordinate plane is chosen in such a way that the accessory parameter is hidden in the relative position of the origin. Our algorithm consists of two steps. For each point in the coordinate plane, we first compute the corresponding monodromy representation by numerical integration along certain loops. Then we decide whether the representation is discrete by applying Jørgensens theory on the quasi-Fuchsian space of once-punctured tori.

OPTi's Algorithm for Discreteness Determination

We summarize how OPTi draws the parameter space. Each point in the picture of a parameter space corresponds to a group, and the program colors the point according to whether the group is discrete or indiscrete. Applying Jørgensens inequality to certain sets of generators, OPTi first tries to decide indiscreteness of the group. If the process fails for generators up to a certain depth, the program then tries to construct the Ford region. When it succeeds in constructing the Ford region, Poincarés polyhedron theorem guarantees the discreteness of the group.

An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let M be a hyperbolic 3-manifold obtained by identifying the faces of n convex ideal polyhedra P1, . . . , Pn. If the faces of P1, . . . , Pn−1 are glued to Pn, then M can be decomposed into ideal tetrahedra by subdividing the Pi’s.

The Schwarzian and Möbius Transformations in Higher Dimensions

The relationship between the Schwarzian and Mobius transformations is well-known in dimensions 1 and 2. However, in dimension n ≥ 3, even the “proper” definition of the Schwarzian is not clear. In this paper, we introduce a “natural” generalization of the Schwarzian using the Clifford algebra and show that it vanishes exactly for Mobius transformations. The situation is simplest for non-singular transformations of the Euclidean space although the framework can be applied, with as light modification, to maps as general as immersions between any Riemannian manifolds.

CODING LINK DIAGRAMS

A method of coding diagrams of knots, links and tangles is introduced. Also, how to draw a diagram for a given code is explained.

Parabolic Representations of the Groups of Mutant Knots

It is proved that the groups of the Kinoshita-Terasaka knot and the Conways knot have the same number of parabolic representations in SL2(F) for every finite field F. Although the proof is based on the fact that the two knots are mutants of each other, it is also shown that the groups of mutant knots do not in general have the same number of parabolic representations.