Journal of Knot Theory and Its Ramifications

Diagrams and Reidemeister moves for links in a twisted S¹-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of ℝP³♯ℝP³. In particular, we show that it has torsion. We also present a new computation of the KBSM of S¹ × S² and the lens spaces L(p, 1).

In this paper, following the scheme of [11–13, 16, 17], we study the first non-vanishing self-linkings of knots in both combinatoric and diagrammatic aspects. The method of attack comes from Massey–Milnor linking theory.

In this paper, we investigate a minimal chart with exactly four white vertices.

We generalize a theorem of Burde and de Rham characterizing the zeros of the Alexander polynomial. Given a representation of a knot group π, we define an extension of π, the Crowell group. For any GL_Nℂ representation of π, the zeros of the associated twisted Alexander polynomial correspond to representations of into the group of dilations of ℂ^N.

The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants, we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index.

In 1991, Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of crossing number c(K) which is s(K) ≤ 2c(K). In this paper we give a new upper bound in terms of arc index, and improve Negami's upper bound to . Moreover if K is a nonalternating prime knot, then .

In this paper, we express the Seifert matrix of a periodic link which is presented as the closure of a 4-tangle with some extra restrictions, in terms of the Seifert matrix of the quotient link. As a result, we give formulae for the Alexander polynomial and the determinant of such a periodic link.

We consider several approaches to defining a link homotopy version of the fundamental quandle Q(L) of a link L in S³. We first define the reduced fundamental quandle RQ(L) as a quotient of Q(L). We show that RQ(L) is a link homotopy invariant that carries at least as much information as the meridian-preserving isomorphism class of Milnor's reduced group RG(L). We then show that operator reduction, a plausible alternative approach to defining RQ(L), fails to yield a link homotopy invariant. Finally, we give a geometric characterization of RQ(L), and offer a caveat regarding a seemingly simpler approach.

It is known by few that a trivial knot can be transformed into a lattice knot whose three shadows are all trees (here, three shadows mean the projections of the lattice knot to the directions of ordinary orthogonal axes). Since a knot itself is a loop in the space, this fact may be rather astonishing. It will be interesting to ask whether a non-trivial knot has such a transformation or not. The purpose of this paper is to show that any two-bridge torus knot or link has a transformation into a lattice knot whose three shadows are all trees. The algorithm to construct such a position will be demonstrated.