Journal of Knot Theory and Its Ramifications

Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An example is given of a slice open virtual string that is not ribbon. Definitions are extended to open n-strings.

Let G be a finitely generated group, and let λ ∈ G. If there exists a knot k such that πk = π₁(S³\k) can be mapped onto G sending the longitude to λ, then there exists infinitely many distinct prime knots with the property. Consequently, if πk is the group of any knot (possibly composite), then there exists an infinite number of prime knots k₁, k₂, … and epimorphisms ⋯ → πk₂ → πk₁ → πk each perserving peripheral structures. Properties of a related partial order on knots are discussed.

In 1993, Birman conjectured that the desingularization map from the singular braid monoid to the integral group ring of the braid group determined by and is injective. The conjecture, which has recently been proven true by Paris (2004), may be generalized to all Artin groups. In this article we prove that the generalized conjecture holds for one of the infinite families of Artin groups of spherical type, namely I₂(p).

We give a recurrence formula for calculating the Alexander polynomials of 2-bridge links by using a special type of Conway diagram. As an application we give sufficient and necessary conditions for the periodic covering link over a 2-bridge link to be a fibered link. We also give a recurrence formula to calculate the reduced Alexander polynomials of periodic covering links over 2-bridge links.

It is well-known that the coefficient of z^m of the Conway polynomial is a Vassiiev invariant of order m. In this paper, we show that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K.

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings.

We study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We also show that mutation does not preserve homology cobordism classes of 3-manifolds. A significant consequence is that there exist 3-manifolds which have the same quantum SU(2)-invariants but are not homology cobordant. These results are obtained by investigating the effect of mutation on the Milnor -invariants, or equivalently the Massey products.

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.