Journal of Knot Theory and Its Ramifications

A Lissajous knot is defined to be one isotopic to a knot which admits a parametrization (for 0≤t≤2π)

Motivation for considering Lissajous knots came originally from the study of DNA molecular configurations. We will show that a Lissajous knot necessarily has Kervaire invariant zero so that the trefoil, figure-8 and the (2,5) torus knot are not Lissajous. The knot 5₂ can be realized with n_x = 2, n_y = 3, n_z = 7.

We prove that the number of independent Vassiliev knot invariants of order n is less than (n − 1)! — thus strengthening the a priori bound (2n − 1)!!

We define invariants < L)₋₁ and F_L(A) for links L in Σ × R where Σ is a surface by using the idea of Kauffman bracket. We compute < L)₋₁ in case of Σ = P², a projective plane. In case of Σ = T², a torus, we show examples of computations of F_L(A) for a certain class of links L in T² × R related to genus 1 Heegaard diagrams of lens spaces and a relation among them. This relation implies that F_L(A) is a topological invariant for lens spaces in a certain sense.

This paper provides necessary and sufficient conditions for a representation of any 3-braid to be minimal (with regard to the number of crossings) and includes an algorithm to obtain such a representation, the number of distinct minimal representations of a given 3-braid, as well as a unique canonical form for each braid in B₃.

Also presented are necessary and sufficient conditions for any 3-string braid word to be a minimal representation of its conjugacy class. A canonical form for each conjugacy class in B₃ is given.

We show that the degenerations of tunnel numbers of knots under connected sum can be arbitrarily large.

We give an algorithm to simplify the calculation of the inavriants of links in a solid torus, which are defined by means of the representations of and the height function. The explicit formula of the initial data for this procedure is presented. Lastely, we observe that the invariant in the simplest level gives an example belonging to a class of 1-trivial dichromatic link invariants different from that defined by Hoste and Kidwell.

If a set of local moves can transform every knot into a trivial knot, it is called a generalized unknotting operation. The author collects generalized unknotting operations and classify them up to local equivalence.

The weight test of Gersten and Pride is modified, to test whether a given abstract Wirtinger presentation occurs as the geometric Wirtinger presentation of a knot. A more refined version of this test then shows that the groups presented by certain abstract Wirtinger presentations are hyperbolic and so do not occur as knot groups. We give an infinite class of examples of such groups with symmetric Alexander polynomial.

A ribbon n-knot Kⁿ is constructed by attaching m bands to m + 1n-spheres in the Euclidean (n + 2)-space. There are many way of attaching them; as a result, Kⁿ has many presentations which are called ribbon presentations. In this note, we will induce a notion to classify ribbon presentations for ribbon n-knots of m-fusions (m ≥ 1, n ≥ 2), and show that such classes form a totally ordered set in the case of m = 2 and a partially ordered set in the case of m ≥ 1.