Journal of Knot Theory and Its Ramifications

It is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.

Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle $X$ and set $S\subset \mathrm{\text{Hom}}(X,X)$ of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the in-degree quiver polynomial is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide example computations for all oriented surface-links with ch-index up to 10 for choices of quandles and endomorphisms.

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

We show that, for any integers, $g\ge 3$ and $n\ge 2$, there exists a link in $S^3$ such that its complement has a genus $g$ Heegaard splitting with distance $n$.

We introduce the concept of tied links in the solid torus, which generalizes naturally the concept of tied links in $S^3$ previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and we then recover this invariant by using Jones’ method over the bt-algebra of type $B$ and the Markov trace defined on this.