We explain that there exists a potential function for the hyperbolic structures of the complement of a hyperbolic knot K which is derived from the Kashaev’s invariant of K and that the A-polynomial of K can be calculated from the function.
Baues introduced a notion of a square ring to describe the structure of a near ring [ΣX , ΣX]. To determine the square ring structure of [ΣX , ΣX] for X =Cα with a metastable element α in terms of the homotopy properties of α, Baues and Iwase introduced a notion of a square form. To determine [ΣX , ΣX] for more general X, we extend the definition of a square form over a ring with torsion. The concrete description of the square ring [ΣX , ΣX] for more general α using our new square forms shall be appeared soon.
In this paper, we first give a diagrammatic analogue of the Young symmetrizer. By using this, the (sl (N,C),ρ)-weight system for an arbitrary finite-dimensional irreducible representation ρ is formulated in a diagrammatic way. The formula is useful for the calculations of the (sl (N,C),ρ)-weight system in the sense that we do not need actual constructions of the representations of sl (N,C) essentially. Hence by using this and the modified Kontsevich integral we can get the quantum (sl (N,C),ρ)-invariant for any finite-dimensional irreducible representation without actual constructions of the representations of sl (N,C). The diagrammatic construction is a generalization of the formula given in “Remarks on the (sl (N,C),ad )-weight system”.
In this paper, we consider geometric structures on a simplicial complex. Rivin considered geometric structures on a 2-simplicial complex which is homeomorphic to an orientable surface when some values are assigned to the edges of the simplicial complex and the assigned values are distributed to the vertices of the 2-faces as their angles. We extend his results to the case where a 2-simplicial complex is not homeomorphic to a surface and also discuss geometric structures on a 3-simplicial complex which is homeomorphic to a 3-manifold.
Knotscape is a useful Linux software for supporting the study of knot theory. However, not a few knot theorists give up to use Knotscape, because they do not use Linux. By preparing some files, it is certified that Knotscape can run not only on Linux but also on other operating systems. In this paper, we introduce the installation of Knotscape on other operation system and some additional help information.
Jeff Weeks’ computer program SnapPea has been used widely in 3-manifold topology. This program computes hyperbolic structures after drillings and Dehn fillings on 3-manifolds, and it provides a variety of associated topological, geometric and arithmetic invariants. In our previous study about Seifert fibered Dehn surgeries on knots, we used SnapPea to investigate a relationship between closed geodesics in hyperbolic knot complements and Seifert fibers after Seifert fibered surgeries on them. We will explain how we used SnapPea in the study and propose some questions inspired by the computer experiments. These experiments were carried out in the joint work with Katura Miyazaki while we were preparing the paper [Miyazaki and Motegi, Comm. Anal. Geom., 7: 551–582].
We classify prime tangles with up to seven crossings consisting of two arcs and some or no loops. Those tangles are mainly distinguished by using the Jones polynomial of the double of a tangle and the Krebes’ invariant.
We introduce an algorithm for generating all generically immersed intervals in the unit disk with a given number of double points N and modulo mirror images. These interval immersions, known as free divides, are associated with knots via a particular map into the 3-sphere, and the gordian unknotting number of a link of a free divide is equal the number of double points of the immersion. So the algorithm provides access to knots for which the gordian number is explicitly known. A computer implementation of our algorithm is described, and we report the results for the total numbers of free divides for N ≤ 8.
In this paper, we shall consider a new kind of drawing software only for surfaces. We introduce pipe structure in pictures of surfaces. And the software Elephant allows us to draw surfaces easily and intuitively. This is a free software and we can get it at http://www.geocities.co.jp/CollegeLife-Labo/9021/.
Circle packings on surfaces with complex projective structures were investigated in the previous paper “Circle packings on surfaces with projective structures”. This paper describes computer experiments performed in the process.
We propose a method for representing the solutions of a certain type of ordinary differential operator L in terms of those of more fundamental differential operators. This method consists of two steps, decomposing L in the ring of differential operators and then describing the projections to the components of this decomposition also in terms of some differential operators. We provide a concrete algorithm for the application of this method and show that this algorithm is successful for a specific example of a 6-th order homogeneous linear ordinary differential operator of Fuchsian type with three singular points.