Journal of Knot Theory and Its Ramifications

We show that a two-bridge ribbon knot $K(m^2,mk\pm 1)$ with $m>k>0$ and $(m,k)=1$ admits a symmetric union presentation with partial knot which is a two-bridge knot $K(m,k)$. Similar descriptions for all the other two-bridge ribbon knots are also given.

The core group of a classical link was introduced independently by Kelly in 1991 and Wada in 1992. It is a link invariant defined by a presentation involving the arcs and crossings of a diagram, related to Wirtinger’s presentation of the fundamental group of a link complement. Two close relatives of the core group are defined by presentations involving regions rather than arcs; one of them is related to Dehn’s presentation of a link group. The definitions are extended to virtual link diagrams and properties of the resulting invariants are discussed.

In 2019, Cho and Nelson introduced an enhancement of the quandle counting invariant, which is quandle coloring quivers. In 2021, Basi and Caprau used it to study $(p,2)$-torus links. Recently, Zhou and Liu used it to study $(p,3)$-torus links. In this paper, we study the quandle coloring space of $(p,4)$- and $(p,5)$-torus links by dihedral quandles and make a conjecture about the quandle coloring quiver of $(p,q)$-torus links, where $q$ is an odd prime number.

In this paper, we introduce and investigate a natural family of metrics on connected components of a rack. The metrics are closely related to certain bi-invariant metrics on the group of inner automorphisms of the rack. We also introduce a bounded cohomology of racks and quandles, relate them to the above metrics and prove a vanishing result for racks and quandles with amenable group of inner automorphisms.