Journal of Knot Theory and Its Ramifications

An extension of the Artin Braid Group is considered, with the introduction of new operatores that generate double and triple intersections. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple intersections, and a counter example is given for the case of quadruple intersections. Intersecting knot invariants are also constructed via Markov traces defined on the intersecting braid algebra representations, and the extended Turaev representation is discussed as an example.

We investigate Vassiliev homotopy invariants of string links, and find that in this particular case, most of the questions left unanswered in [3] can be answered affirmatively, In particular, Vassiliev invariants classify string links up to homotopy, and all Vassiliev homotopy string link invariants come from marked surfaces as in [3], using the same construction that in the case of knots gives the HOMFLY and Kauffman polynomials. In addition, the Milnor μ invariants of string links are shown to be Vassiliev invariants, and it is re-proven, by elementary means, that Vassiliev invariants classify braids.

This text presents some recent developments in left distributive algebra induced by set theory and their applications to the combinatorics of braids mainly in terms of order properties and existence of special decompositions.

Let (p/q, n) denote the orbifold with singular set the two bridge knot or link p/q and isotropy group cyclic of orden n. An algebraic curve (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of in PSL . The coordinates x, z, are trace(A²)=x, trace(AB)=z where A and B are the images of canonical generators a, b of . Let (x_n, z_n) be the point of corresponding to the hyperbolic orbifold (p/q, n). We prove the following result: The (orbifold) fundamental group of (p/q, n) is arithmetic if and only if the field Q(x_n, z_n) has exactly one complex place and ϕ(x_n)<ϕ(z_n)<2 for every real embedding .

Consider the angle α for which the cone-manifold (p/q, α) is euclidean. We prove that 2cosα is an algebraic number. Its minimal polynomial (called the h-polynomial) is then a knot invariant. We indicate how to generalize this h-polynomial invariant for any hyperbolic knot.

Finally, we compute h-polynomials and arithmeticity of (p/q, n) with p≦40, and (p/q, n) with p≦99q²≡1 mod p. We finish the paper with some open problems.

How many edges are necessary and sufficient to construct a knot of type K in the cubic lattice? Define the minimal edge number of a knot to be this number of edges. To what extend does the minimal edge number measure the complexity of a knot? What is the behaviour of the minimal edge number under the connected sum of knots, and what is its limiting behaviour? We consider these questions and show that the minimal edge number may be computed using simulated annealing.

This paper studies invariants of 3-manifolds derived from certain finite dimensional Hopf algebras via regular isotopy invariants of unoriented links in the blackboard framing. The invariants are based on right integrals for these Hopf algebras. It is shown that the resulting class of invariants is definitely distinct from the class of Witten-Reshetikhin-Turaev invariants. The invariant associated with the quantum double of a finite group G is treated in this context, and is shown to count the number of homomorphisms of the fundamental group of the 3-manifold to the given finite group G.

We prove that the construction of Vassiliev invariants by expanding the link polynomials P_g,V(q, q⁻¹) at the point q=1 is equivalent to the construction of Vassiliev invariants from Chern-Simons perturbation theory. In both constructions a simple Lie algebra g and an irreducible representation V of g should be specified.

We give an example of a Vassiliev invariant of order six which cannot be obtained by these constructions if we restrict ourselves to simple Lie algebras and do not allow semisimple ones.

The explicit description of primitive elements in the Kontsevich Hopf algebra is given.