Journal of Knot Theory and Its Ramifications

We consider the number of almost unknotted embeddings of graphs in Z³. We show that the number of such embeddings is the same, to exponential order, as the number of unknotted embeddings. We also consider some higher dimensional analogues, ie almost unknotted embeddings of surfaces which are p-dimensional analogues of Θ-graphs in Z^p+2. We describe a lattice version of the spinning construction which establishes the embeddability of such surfaces in Z^p+2 and show that the number of embeddings is the same, to exponential order, as the number of unknotted embeddings. The proofs of our upper bounds feature a novel application of the classical Loomis–Whitney inequality.

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

We complete the proof of Oka's conjecture on the Alexander polynomial of an irreducible plane sextic. We also calculate the fundamental groups of irreducible sextics with a singular point adjacent to J₁₀.

Suppose k is a connected sum of two knots, one of which admits a 2-string essential free tangle decomposition, then the exterior of k contains an incompressible surface of genus n for each positive integer n.

The cyclotomic Birman–Wenzl–Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We show that the cyclotomic BMW algebras are free modules over any admissible, integral ground ring, and that they are isomorphic to cyclotomic versions of the Kauffman tangle algebras.

Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalized knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well-known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n ≥ 2. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n ≥ 2, carries more information about K than the isomorphism type of the fundamental group. The appendix contains some results on representations of the trefoil group in PSL(2, p) that are needed for the proof.

This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.