Journal of Knot Theory and Its Ramifications

Let $K$ be a tame knot and consider an $n$ beaded necklace $T^{\circ }$ which is the union of $n$ consecutive disjoint closed round balls (pearls) $B_j$, $j=1,\dots ,n$. An $n$ pearl chain necklace $T$ is the union of $T^{\circ }$ and $K$. We will construct, via the action of a Kleinian group, a sequence of nested pearl chain necklaces $T_k$ whose inverse limit is a wild knot of dynamically defined type $\mathrm{\Lambda }(K,T^0)$. In this paper, we will prove some topological properties of this kind of wild knots; in particular, we generalize the construction of cyclic branched coverings for this case, and we show that there exists a wild knot of dynamically defined type such that ${𝕊}^3$ is an $n$-fold cyclic branched covering of ${𝕊}^3$ along it, for $n\ge 2$.

We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens–Golla $\delta$-invariant to obstruct a knot from being a stabilization of another.

In this paper, we introduce two new polynomial invariants $Q_D(t)$ and $A_D(t)$ for one-component virtual doodles. We will also show that these polynomial invariants are not invariants of flat virtual knots.

A relative trisection of a compact $4$-manifold with connected boundary is said to be a Trivially Extendable (TE)-relative trisection if the boundary monodromy of the relative trisection is the identity map. We call a relative trisection diagram associated to a $TE$-relative trisection of a compact $4$-manifold, a $TE$-relative trisection diagram. Due to a work of Castro and Ozbagci, there is a natural way to associate a trisection diagram to a $TE$-relative trisection diagram. In fact, every closed oriented $4$-manifold can be realized as a $TE$-relative trisection.

In this paper, we study the notion of Murasugi sum of $TE$-relative trisection diagrams. We show that a trisection diagram associated to a Murasugi sum of two $TE$-relative trisection diagrams $\mathcal{𝒟}$ and ${\mathcal{𝒟}}^{\prime }$ is diffeomorphism and handle slide equivalent to the connected sum of the trisection diagrams associated to $\mathcal{𝒟}$ and ${\mathcal{𝒟}}^{\prime }$. This implies that the closed oriented $4$-manifold associated to the Murasugi sum of $TE$-relative trisection diagrams $\mathcal{𝒟}$ and ${\mathcal{𝒟}}^{\prime }$ is the connected sum of the closed $4$-manifolds associated to $\mathcal{𝒟}$ and ${\mathcal{𝒟}}^{\prime }$. Thus, the notion of Murasugi sum can be considered for closed $4$-manifolds in terms of $TE$-relative trisections.

Finally, we discuss an external stabilization of a $TE$-relative trisection of a closed oriented $4$-manifold. We prove that the trisection diagram associated to a stabilization of a $TE$-relative trisection diagram is diffeomorphism and handle slide equivalent to a stabilization of the trisection diagram associated to the $TE$-relative trisection diagram.