Journal of Knot Theory and Its Ramifications

An infinite family of invariants of multicomponent links in 3-manifolds is introduced and used to prove the non-splitting and non-equivalence of textile structures.

We construct an infinite series of invariants of Fiedler type (i.e. composed of oriented arrow diagrams arranged by elements of H₁(M³)) for multicomponent links in M³ = M² × R¹, M² orientable with π₁(M²) ≠ {1}.

In this paper, it is shown that there are no nonconstant Goussarov–Polyak–Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem of [1] which states none of the Birman coefficients of the Jones–Kauffman polynomial are of GPV finite type.

In this paper, we give an algorithm to calculate the minimal self-intersection number of paths in a compact surface with boundary representing a given element of the free group F(x₁, x₂, …, x_n). In particular, this algorithm says whether or not a word in x₁, x₂, …, x_n is representable by a simple path. Our algorithm is simpler than similar algorithms given previously. In the case of a disk with n holes the problem is equivalent to the problem of deciding which relators can appear in an Artin n-presentation.

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom(K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σ_x of Hom(K, Σ) onto itself: σ_xρ(a) = ρ(xax^-1) ∀ a ∈ K, ρ ∈ Hom(K, Σ). As proven in [5], the dynamical system (Hom(K, Σ), σ_x) is a shift of finite type. In the case when Σ is abelian, Hom(K, Σ) is finite.

In this paper we calculate the periods of orbits of (Hom(K, ℤ/p), σ_x), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.

Lin's definition of twisted Alexander knot polynomial is extended for finitely presented groups. Cha's fibering obstruction theorem is generalized. The group of the 2-twist-spun trefoil knot is seen to have a faithful representation that yields a trivial twisted Alexander polynomial.

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

We prove that an Alexander quandle of prime order is generated by any pair of distinct elements. Furthermore, we prove for such a quandle that any ordered pair of distinct elements can be sent to any other such pair by an automorphism of the quandle.

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.