Journal of Knot Theory and Its Ramifications

We study a subclass of alternating links for which the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.

In the present paper, we will study the creation of projective planes by Dehn surgery on knots in the 3-sphere. It is shown that a projective plane cannot be created by Dehn surgery on a crosscap number two knot. As a corollary, we will prove that crosscap number two knots satisfy the projective space conjecture, which asserts that the projective 3-space cannot be obtained by Dehn surgery on a non-trivial knot in the 3-sphere.

Let M be a simple 3-manifold with a toral boundary component. It is known that if two Dehn fillings on M along the boundary produce a reducible manifold and a toroidal mainfold, then the distance between the filling slopes is at most three. This paper gives a remarkably short proof of this result.

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(α) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a central role.

Let O_d denote the ring of integers in . An orientable finite volume cusped hyperbolic 3-manifold M is called arithmetic if the faithful discrete representation of π₁(M) into PSL(2,C) is conjugate to a group commensurable with some Bianchi group PSL(2,O_d). If M is a closed orientable 3-manifold, we say a link L ⊂ M is arithmetic if M\L is arithmetic.

In the paper, we show that there exist closed orientable 3-manifolds which do not contain anarithmetic knot. Our methods give much more precise informations for non-hyperbolic 3-manifolds. For certain Lens Spaces, we can give fairly complete statements.

We consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in S³, which are elements of the braid groups B_g,n. We prove a geometric version of the Markov theorem for braid equivalence in the handlebody, which is based on the L-moves. Using this we then prove two algebraic versions of the Markov theorem. The first one uses the L-moves. The second one uses the Markov moves and conjugation in the groups B_g,n. We show that not all conjugations correspond to isotopies.

We introduce a computer tool called Knot2000(K2K) which was developed for the purpose of support for the research of knot theory. K2K is a package on Mathematica in which consists of 19 functions and it has already been opened to the public with other external programs and data files. In this paper, we will describe focusing on the usages of each functions and some examples of effective ways to use K2K, and show its availability.

We have established in [S3] an algorithm with a new data structure that decomposes gluing homeomorphisms of 3-manifolds given by planar Heegaard diagrams into a product of canonical Dehn's twists. To support this study, we developed a computer software called Decomposition of Planar Homeomorphisms (Genus 3) that automatically decomposes the self homeomorphis of a closed connected surface given by any planar Heegaard diagram of genus 3 into a product of canonical Dehn's twists. In this paper, we demonstrate the content and the implementation that this software holds and also show its availability.

This paper begins with a review of the extension of A'Campo's divides to the more general construction of oriented divides, using the same disk model of S³. It is shown that all links in S³ can be represented by oriented divides (in contrast with divide links) and simple methods are presented for moving between links and their oriented divide presentations.

Using the properties of the disk model, oriented divides are extended by three extra equivalence rules, and it is shown that this extended theory of oriented divides is equivalent to knot theory in S³. Fueled by this result, the concept of oriented divide braid is introduced, together with a group construction which parallels the classical braid group representation of geometric braids. This leads to results analagous to Alexander's well-known theorem concerning braid presentations, and also the Markov theorem.