Journal of Knot Theory and Its Ramifications

We show that over the binary field ${𝔽}_2$, the Bar-Natan perturbation of Khovanov homology splits as the direct sum of its two reduced theories, which we also prove are isomorphic. This extends Shumakovitch’s analogous result for ordinary Khovanov homology, without the perturbation.

In his previous work, the author proved that the canonical decompositions of hyperbolic fibered two-bridge link complements are layered. This implies that they admit taut structures. In this paper, we completely determine, for each hyperbolic fibered two-bridge link, whether the canonical decomposition of its complement is veering with respect to the taut structure.

The Kauffman bracket skein algebra of a compact oriented surface when the variable $A$ in the Kauffman bracket is set equal to $e^{\pi i/N}$, where $N$ is an odd counting number, is a central extension of the ring of $S{L}_2ℂ$-characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

We prove that any 11-colorable knot is presented by an 11-colored diagram where exactly five colors of eleven are assigned to the arcs. The number five is the minimum for all non-trivially 11-colored diagrams of the knot. We also prove a similar result for any 11-colorable ribbon 2-knot.

We prove that any knot or link in any $3$-manifold can be nicely decomposed (split) by a filling Dehn sphere. This has interesting consequences in the study of branched coverings over knots and links. We give an algorithm for computing Johansson diagrams of filling Dehn surfaces out from coverings of $3$-manifolds branched over knots or links.

A balanced spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to S. Kinoshita, Alexander polynomials as isotopy invariants I, Osaka Math. J. 10 (1958) 263–271.), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and $p$-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which $p$ the graph is $p$-colorable, and that a $p$-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group $\mathrm{\Gamma }(p,m,k)$. We finish by proving some properties of the Alexander polynomial.

It is proved by Ozawa that every knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle sphere. In this paper, we show that for every integer $n$ with $n\ge 2$, there is a knot in the 3-sphere which admits an essential free $n$-tangle decomposition and also admits at least $2n-3$ distinct essential tangle spheres. To this end, we give examples of nested essential free $n$-tangles for every integer $n$ with $n\ge 3$.