Journal of Knot Theory and Its Ramifications

In this paper, we prove that two cubic knots K₁, K₂ are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use this fact to describe a cubic knot in a discrete way, as a cyclic permutation of contiguous vertices of the ℤ³-lattice (with some restrictions); moreover, we describe a regular diagram of a cubic knot in terms of such cyclic permutations.

We compare the invariants for classical knots and links defined using the Juyumaya trace on the Yokonuma–Hecke algebras with the HOMFLYPT polynomial. We show that these invariants do not coincide with the HOMFLYPT except in a few trivial cases.

This article is a revised version of the author's earlier paper on a TOP-splitting of a closed connected oriented 4-manifold with infinite cyclic fundamental group. We show that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if it is virtually TOP-split. As a consequence, we see that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if the intersection form is indefinite. This also implies that every closed connected oriented smooth spin 4-manifold with infinite cyclic fundamental group is TOP-split.

We consider the question of when is the closed manifold obtained by elementary surgery on an n-knot Seifert fibered over a 2-orbifold. The possible bases are strongly constrained by the fact that knot groups have weight 1. We present a new family of 2-knots with solvable groups, overlooked in earlier work. The knots in this new family are neither invertible nor amphicheiral, and the weight orbits for the knot groups are parametrized by ℤ. There are no known examples in higher dimensions.

We prove that given any three polygonal unknots in ℝ³, then we may form the Borromean rings out of them through rigid motions of ℝ³ applied to the individual components together with possible scaling of the components. We also prove that if at least two of the unknots are planar, then we do not need scaling. This is true even for a set of three polygonal unknots that are arbitrarily close to three circles, which themselves cannot be used to form the Borromean rings.

The knot Floer complex together with the associated concordance invariant ε can be used to define a filtration on the smooth concordance group. We show that the indexing set of this filtration contains ℕ × ℤ as an ordered subset.

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).