Narrow Results

Given a long knot diagram D and a finite quandle Q, we consider the set of all quandle colorings of D with a fixed color q of its initial arc. Using this set we define the family of quandle automorphisms which is a knot invariant. For every element x ∈ Q one can consider the formal sum , taken over all . Such formal sums can be applied to a tangle embedding problem and recognizing non-classical virtual knots.

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper, we construct a more subtle invariant: a family of biquandle endomorphisms obtained from the set of colorings and longitudinal information.

Let $K$ be a hyperbolic knot in the $3$-sphere. If a $p/q$-Dehn surgery on $K$ produces manifold with an embedded Klein bottle or essential $2$-torus, then we prove that $\left|p\right|\le 8g_K-3$, where $g_K$ is the genus of $K$. We obtain different upper bounds according to the production of a Klein bottle, a non-separating $2$-torus, or an essential and separating $2$-torus. The well known examples which are the figure eight knot and the pretzel knot $K(-2,3,7)$ reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed $3$-manifolds.

We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander module by singling out these peripheral elements provides a significantly stronger link invariant. In particular, the enhanced module determines all linking numbers in a link; in contrast, the module alone does not even detect how many linking numbers are 0.

In this paper, we explain how the medial quandle of a classical or virtual link can be built from the peripheral structure of the reduced Alexander module.

In this paper, we answer a question raised in “Peripheral elements in reduced Alexander modules” [J. Knot Theory Ramifications 31 (2022) 2250058]. We also correct a minor error in that paper.

An open problem in link-homotopy of links in S³ is classification using peripheral invariants, analogous to that of Waldhausen for links up to ambient isotopy. An approach to such a classification was outlined by Levine, but shown not to be feasible by the author. Here, we develop an approach to finding classification counterexamples. The approach requires non-injectivity of a group homomorphism that is completely determined by minimal-weight commutator numbers (equivalent to the first non-vanishing invariants of Milnor). For non-injectivity, the minimal-weight commutator numbers must all be non-zero, and satisfy a certain system of polynomials, which we compute for 4- and 5-component links.