Journal of Knot Theory and Its Ramifications

Given two 4-plats K₁ and K₂. Let O be a sum of rational tangles such that N(O+P)=K₁ and N(O+R)=K₂, where P and R are rational tangles and N is the numerator construction on the tangles O+P and O+R, respectively. An algorithm is presented here to solve such a system of equations for O, P and R in the case where O is non rational. If two additional equations of the form N(O+R+R)=K₃ and N(O+R+R+R+) =K₄ are given and one of the K_i is chiral, then there is at most one solution for O and R. If all K_i are achiral then there is at most one solution together with its mirror image.

The aim of this note is to give two applications of Topological Quantum Field Theories to the topology of open manifolds. The invariants we derived may be used to test if an open manifold is simply connected at infinity (as we did for Whitehead's manifold in case of the sl₂(C)-TQFT in level 3) or to construct examples of non-homeomorphic contractible open 3-manifolds.

The writhe of a knot in the simple cubic lattice can be computed as the average linking number of the knot with its pushoffs into four non-antipodal octants. We use a Monte Carlo algorithm to generate a sample of lattice knots of a specified knot type, and estimate the distribution of the writhe as a function of the length of the lattice knots. If the expected value of the writhe is not zero, then the knot is chiral. We prove that the writhe is additive under concatenation of lattice knots and observe that the mean writhe appears to be additive under the connected sum operation. In addition we observe that the mean writhe is a linear function of the crossing number in certain knot families.

The SU(N) quantum invariants for three-manifolds have been established in a combinatorial way by Yokota starting from the skein theory associated with the HOMFLY polynomial invariant of knot theory. Using Yokota's formulation it is here noted that there are distinct three-manifolds that are not distinguished by these invariants and that there are manifolds distinguished by the SU(N) invariants for N ≥ 3 that have the same SU(2) invariants. A by-product of the investigation is a reversing result for the HOMFLY polynomial.

Recent work of Birman and Menasco discusses braid index of satellite links, and introduces a new geometrical invariant of links, which they have called the arc index. Here, we marry the work of Birman and Menasco with that of Rodolph on quasipositivity of knots, to deduce a lower bound for arc index from the Kauffman polynomial.

In this paper, we shall give a general formula for the quantum SO(3)-invariant of Seifert fibered rational homology spheres over a 2-sphere, based on the linear skein theory. We shall derive the Casson-Walker invariant from the quantum SO(3)-invariant. Also, we shall give a formula for the Ohtsuki second invariant.

Let ϕ be an automorphism of a finitely generated free group F, w and element of F, and H the subgroup of F generated by the orbit {ϕⁿ (w), |n ∈ Z}. We describe sufficient conditions ensuring that H is non-finitely generated. Using this we give a simple construction of tori T embedded in S⁴ in such a way that the commutator subgroup of π₁(S⁴ - T) is finitely generated but not finitely presented. Such tori have no minimal Seifert manifolds. An example of an embedded torus with the latter property was given recently by T. Maeda using different methods.

In this paper, we will give an explicit and simple formula for J⁺-invariant of closed 2-vertex curves. As an application, we give an affirmative answer to a problem conjectured by Arnold [A2], related to Legendrian knots in R² × S¹.

A loop S¹ → ℝⁿ is holonomic if it is the (n - 1)-jet extension of a function S¹ → ℝ¹. We prove that for n = 3 any tame link in ℝⁿ is isotopy equivalent to a holonomic one; for n > 3 the space of holonomic links is holotopy equivalent to the space of all differentiable links.

An antichain is associated to each link diagram so that the highest degree of the Kauffman bracket can be determined. As an application, we show that the span of the Kauffman bracket is less than or equal to 4(n - m) for dealternator connected m-alternating diagrams and the upper bound is best possible. This completely solves a conjecture of [1]. Finally, we show that a semi-alternating diagram may be not a minimal diagram which disproves a conjecture of K. Murasugi [10].