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In this paper we study further when tangles embed into the unknot, the unlink or a split link. In particular, we study obstructions to these properties through geometric characterizations, tangle sums and colorings. As an application we determine when each prime 2-string tangle with up to seven crossings embeds into the unknot, the unlink or a split link., Comment: 27 pages, 18 figures; to be published in Topology and its Applications

We prove that the tunnel number of a satellite chain link with a number of components higher than or equal to twice the bridge number of the companion is as small as possible among links with the same number of components. We prove this result to be sharp for satellite chain links over a 2-bridge knot., Comment: 16 pages, 14 figures; To appear in Algebraic and Geometric Topology

In this work, we introduce a new graph convexity, that we call Cycle Convexity, motivated by related notions in Knot Theory. For a graph $G=(V,E)$, define the interval function in the Cycle Convexity as $I_{cc}(S) = S\cup \{v\in V(G)\mid \text{there is a cycle }C\text{ in }G\text{ such that } V(C)\setminus S=\{v\}\}$, for every $S\subseteq V(G)$. We say that $S\subseteq V(G)$ is convex if $I_{cc}(S)=S$. The convex hull of $S\subseteq V(G)$, denoted by $Hull(S)$, is the inclusion-wise minimal convex set $S'$ such that $S\subseteq S'$. A set $S\subseteq V(G)$ is called a hull set if $Hull(S)=V(G)$. The hull number of $G$ in the cycle convexity, denoted by $hn_{cc}(G)$, is the cardinality of a smallest hull set of $G$. We first present the motivation for introducing such convexity and the study of its related hull number. Then, we prove that: the hull number of a 4-regular planar graph is at most half of its vertices; computing the hull number of a planar graph is an $NP$-complete problem; computing the hull humber of chordal graphs, $P_4$-sparse graphs and grids can be done in polynomial time.

We show a combinatorial argument in the diagram of large class of links, including satellite and hyperbolic links, where for each of which the tunnel number is the minimum possible, the number of its components minus one., Comment: 12 pages, 7 figures

We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.

For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $\varphi_1$ and $\varphi_2$, and two non-equivalent covers $p_i: \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $\varphi_1$ to $\tilde S$ with respect to $p_1$ coincides with that of $\varphi_2$ with respect to $p_2$. The mapping tori of the $\varphi_i$ and their lift provide examples of pairs of hyperbolic $3$-manifolds so that the first is covered by the second in two different ways., Comment: 9 pages, 2 figures

In this paper we define alternating Kauffman states of links and we characterize when the induced state surface is a fiber. In addition, we give a different proof of a similar theorem of Futer, Kalfagianni and Purcell on homogeneous states., Comment: 11 pages, 10 figures; Accepted for publication in Algebraic and Geometric Topology

In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted strings., Comment: 16 pages, 20 figures; In this version we correct three arguments on the proofs of the tangles of Theorems 1.1 and 1.2 being essential. The results of the original paper remain unaltered