Journal of Knot Theory and Its Ramifications

The space of Gauss diagram formulas that are knot invariants is introduced by Goussarov–Polyak–Viro in 2000; it is extended to nanophrases by Gibson–Ito in 2011. However, known invariants in concrete presentations of Gauss diagram formulas are very limited, even in the one-component case. This paper gives a recipe to obtain explicit forms of Gauss diagram formulas that are invariants of virtual links with base points or tangles. As an application, we introduce a new construction of Gauss diagram formulas of $3$-bouquets and how to give link invariants that do not change with base point moves, including a reconstruction of the Milnor’s triple linking number.

Plucking polynomial of a plane rooted tree with a delay function $\alpha$ was introduced in 2014 by Przytycki. As shown in this paper, plucking polynomial factors when $\alpha$ satisfies additional conditions. We use this result and ${\mathrm{\Theta }}_A$-state expansion introduced in our previous work to derive new properties of coefficients $C(A)$ of Catalan states $C$ resulting from an $(m\times n)$-lattice crossing $L(m,n)$. In particular, we show that $C(A)$ factors when $C$ has arcs with some special properties. In many instances, this yields a more efficient way for computing $C(A)$. As an application, we give closed-form formulas for coefficients of Catalan states of $L(m,3)$.

A surface-link is a closed surface embedded in the 4-space, possibly disconnected or non-orientable. Every surface-link can be presented by the plat closure of a braided surface, which we call a plat form presentation. The knot symmetric quandle of a surface-link $F$ is a pair of a quandle and a good involution determined from $F$. In this paper, we compute the knot symmetric quandle for surface-links using a plat form presentation. As an application, we show that for any integers $g\ge 0$ and $m\ge 2$, there exist infinitely many distinct surface-knots of genus $g$ whose plat indices are $m$.

The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of collections of open curves in 3-space using the theory of knotoids and linkoids, which are equivalence classes of diagrams with open arcs. In this paper, new invariants of linkoids are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. This gives rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.

Gay and Meier asked whether or not a trisection diagram constructed by the Gluck twist on a spun or a twist spun 2-knot obtained from some method is standard. In this paper, we depict the trisection diagrams explicitly when the 2-knot is the spun $(2n+1,2)$-torus knot, where $n\ge 1$, and show that the trisection diagram is standard when $n=1$. Moreover, we introduce a notion of homologically standard for trisection diagrams and show that the trisection diagram is homologically standard for all $n$.