Journal of Knot Theory and Its Ramifications

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.

In this paper, we give a bound for the number of boundary slopes of orientable immersed proper π₁-injective surfaces of given genus g in an orientable Haken 3-manifold M with a torus boundary, where the bound is independent of M, and a function of g and m, the number of the Jaco–Shalen–Johannson decomposition tori of M.

We construct a multi-variable polynomial invariant for virtual knots and links by using the concepts of a decorated virtual magnetic graph diagram and a weight map. We show that the invariant is a variety of virtual link polynomial with multiple variables introduced in [16] by the author and it gives a sharpened evaluation of the virtual crossing number of a virtual knot or link.

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class of m-string braid groups B_m(𝕊²\{x₁,…,x_n}) of the n-punctured sphere includes the usual Artin braid groups B_m (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (for n = 3).

We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B_m is completely determined for all m ∈ ℕ (respectively, for all m ≠ 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients.

When n ≥ 2, we prove that the lower central series (respectively, derived series) of B_m(𝕊²\{x₁,…,x_n}) is constant from the commutator subgroup onwards for all m ≥ 3 (respectively, m ≥ 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B₂(𝕊²\{x₁,x₂}) admits various interpretations, as the Baumslag–Solitar group BS(2,2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B₂(𝕊²\{x₁,x₂}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product ℤ₂ * ℤ. Further, its lower central series quotients Γ_i/Γ_{i + 1} are direct sums of copies of ℤ₂, the number of summands being determined explicitly. In the case m ≥ 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Γ_i/Γ_{i + 1} are 2-elementary finitely-generated groups.

We consider the quantum Teichmüller space of the punctured surface introduced by Chekhov–Fock–Kashaev, and formalize it as a noncommutative deformation of the space of algebraic functions on the Teichmüller space of the surface. In order to apply it in 3-dimensional topology, we put more attention to the details involving small surfaces.