Journal of Knot Theory and Its Ramifications

C.F. Gauss gave a necessary condition for a word to be the intersection word of a closed normal planar curve and he gave an example which showed that his condition was not sufficient. M. Dehn provided a solution to the planarity problem [3] and subsequently, different solutions have been given by a number of authors (see [9]). However, all of these solutions are algorithmic in nature. As B. Grünbaum remarked in [7], “they are of the same aesthetically unpleasing character as MacLane’s [1937] criterion for planarity of graphs. A characterization of Gauss codes in the spirit of the Kuratowski criterion for planarity of graphs is still missing”. In this paper we use the work of J. Scott Carter [2] to give a necessary and sufficient condition for planarity of signed Gauss words which is analogous to Gauss’s original condition.

An interesting class of knots have complement with a remarkably simple topological description. This class includes all the arborescent knots with only even weights hence, in particular, the two bridge knots and many knots of ten or fewer crossings. For these knots, there are choices of minimal genus Seifert surfaces S such that all taut, depth one foliations of the knot complement, having S as sole compact leaf, can be classified up to isotopy. These foliations correspond exactly to the lattice points over the open faces of the unit ball in a Thurston-like norm on the relative homology of the complement of S.

We linearize the Artin representation of the braid group given by (right) automorphisms of a free group providing a linear faithful representation of the braid group. This result is generalized to obtain linear representations for the coloured braid groupoid and pure braid group too. Applications to some areas of two-dimensional physics are mentioned.

The polygons on the cubic-lattice have played an important role in simulating various circular molecules, especially the ones with relatively big volumes. There have been a lot of theoretical studies and computer simulations devoted to this subject. The questions are mostly around the knottedness of such a polygon, such as what kind of knots can appear in a polygon of given length, how often it can occur, etc. A very often asked and long standing question is about the minimal length of a knotted polygon. It is well-known that there are knotted polygons on the lattice with 24 steps yet it is unproved up to this date that 24 is the minimal number of steps needed. In this paper, we prove that in order to obtain a knotted polygon on the cubic lattice, at least 24 steps are needed and we can only have trefoils with 24 steps.

We show that every oriented link is the boundary of a surface obtained from an embedded disk by plumbing a flat annulus a finite number of times.

We construct a polynomial link invariant as Markov trace on certain one parameter algebras and we prove that it is equal to the invariant corresponding to the exeptional Lie algebra of type G₂. We use braid representatives to calculate the invariant for several knots and links.

In [8] R.F. Williams showed that all knots in the Lorenz template are prime. His proof included the cases where any number of positive twists were added to one of the template’s branches. However [8] does give an example of a composite knot in a template with a single negative twist. Below we will show that in all the negative cases composite knots do exist, and give a mechanism for producing many examples. This problem was cited in a list of problems in dynamics in [1, problem 4.2].