Journal of Knot Theory and Its Ramifications

An n-string tangle (B³,T) is a 3-ball B³ which contains n properly embedded arcs T={t_i}, it is called rational if there is a homeomorphism of pairs from (B³,T) to (D,P)×I where D is the unit disk, P is any set of n points in the interior of D and I is the unit interval. In this article we extend the classification of the 3-braid group, , obtained by using Kauffman bracket polynomial, to other families of rational 3-tangles.

The fundamental group of a 2-bridge knot has a particularly nice presentation, having only two generators and a single relation. For certain families of 2-bridge knots, such as the torus knots, or the twist knots, the relation takes on an especially simple form. Exploiting this form, we derive a formula for the A-polynomial of twist knots. Our methods extend to at least one other infinite family of (non-torus) 2-bridge knots. Using these formulae we determine the associated Newton polygons. We further prove that the A-polynomials of twist knots are irreducible.

In this paper, we define edge sum of some trivializable graphs. The main theorem of this paper is that an edge sum of some trivializable graphs at handle edges is trivializable and using this theorem we actually construct a larger class of trivializable graphs.

We show that a certain natural class of tangles 'generate the collection of all tangles with respect to composition'. This result is motivated by, and describes the reasoning behind, the 'uniqueness assertion' in Jones' theorem on the equivalence between extremal subfactors of finite index and what we call 'subfactor planar algebras' here. This result is also used to identify the manner in which the planar algebras corresponding to M⊂M₁ and N^op⊂M^op are obtained from that of N⊂M.

Our results also show that 'duality' in the category of extremal subfactors of finite index extends naturally to the category of 'general' planar algebras (not necessarily finite-dimensional or spherical or connected or C*, in the terminology of Jones).

We study bifurcations of Voronoi diagrams on the plane. The generic bifurcations of Voronoi diagrams for moving points are classified into four types.

A braid has an associated family of Voronoi diagrams. If we admit only three types among four in the associated generic family of Voronoi diagrams, the braid type reduces to a braid with k half twists for some integer k.

Let K be a classical knot in ℝ³. We can deform the diagram of K to that of a trivial knot by changing the overcrossings and undercrossings at some double points of the diagram of K. We consider the same problem for higher dimensinal knots. Let n≥2 and π:ℝⁿ⁺²=ℝⁿ⁺¹×ℝ→ℝⁿ⁺¹ denote the natural projection map. A pseudo-ribbon n-knot is an n-knot f:Sⁿ→ℝⁿ⁺² such that the self-intersection set of π◦f:Sⁿ→ℝⁿ⁺¹ consists of only double points and is homeomorphic to a disjoint union of (n-1)-spheres. We prove that for n≠3,4, the projection (π◦f)(Sⁿ)⊂ℝⁿ⁺¹ of any pseudo-ribbon n-knot f is the projection of a trivial n-knot.

We find families of prime diagrams of knots with arbitrary extreme coefficients in their Jones polynomials. Some graph theory is presents in connection with this problem, generalizing ideas by Yongju Bae and Morton [4] and giving a positive answer to a question in their paper.

We continue to study the algebraic property of the linear representation of the mapping class group of a closed oriented surface of genus 2 constructed by V. F. R. Jones. We consider the perturbation of the representation at the involved parameter t=1. This perturbation naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients. Our arguments follow the same line of our previous paper which dealt with the perturbation at t=-1. However, the obtained results may still suggest a new aspect of the representation.

A classical theorem of R. H. Bing states that a closed connected 3-manifold M is homeomorphic to the 3-sphere if and only if every knot in M is contained in a 3-ball. We give a simple proof of this characterization based on the surgery presentation of 3-manifolds.