Journal of Knot Theory and Its Ramifications

We address the concept of stick number for knots and links under various restrictions concerning the length of the sticks, the angles between sticks, and placements of the vertices. In particular, we focus on the effect of composition on the various stick numbers. Ultimately, we determine the traditional stick number for an infinite class of knots, which are the (n,n-1)-torus knots together with all of the possible compositions of such knots. The exact stick number was previously known for only seven knots.

A topological characterization of all possible links composed of the periodic orbits of a Non Singular Morse-Smale flow on S³ has been made by M. Wada.

The presence of symmetry forces the appearance of given types of links. In this paper we introduce a geometrical tool to represent these type of links when a symmetry around two axes is considered on NMS systems: mosaics.

On the other hand, we use mosaics to study what kind of bifurcation can occur in this type of system maintaining the symmetry.

We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of Rep(U_q(sl₂)) (q a principal 4r^th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2].

We give several generalizations of the light bulb trick. These show that in a variety of settings there is a unique isotopy class of arcs connecting two points.

We derive from the perturbation expansion of the Wilson loop expectation value in the Chern-Simons theory an explicit combinatorial expression for a third-order finite link invariant, thereby generalising the knot invariant considered in a previous article.

The minimal number of straight line segments required to form a given knot or link in ℝ³ is determined for a family of torus knots and links.

The best known examples of Vassiliev invariants are the coefficients of a Jones-type polynomial expanded after exponential substitution. We show that for a given knot, the first N Vassiliev invariants in this family determine the rest for some integer N.

In this note we show that the rank of the trip matrix of a positive knot diagram is exactly twice the genus of the associated positive knot. From this, we give a quick proof of the following result of Murasugi: The term of lowest degree in the Jones polynomial of a positive knot is 1 · t^g, where g is the genus of the knot.