Related papers: Non-parallel essential surfaces in knot complement…
A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism…
We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at…
We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected)…
A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$…
This paper provides two obstructions to small knot complements in $S^3$ admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture…
We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have…
Let $D$ be a cellular alternating link diagram on a closed orientable surface $\Sigma$. We prove that if $D$ has no removable nugatory crossings then each checkerboard surface from $D$ is $\pi_1$-essential and contains no essential closed…
Let K be a knot in S^3 of genus g and let n>0. We show that if rk HFK(K,g) < 2^{n+1} (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient a_g of its Alexander polynomial…
Kupavskii, Volostnov, and Yarovikov have recently shown that any set of $n$ points in general position in the plane has at least as many (partial) triangulations as the convex $n$-gon. We generalize this in two directions: we show that…
Kuratowski's theorem says that the minimal (under subgraph containment) graphs that are not planar are the subdivisions of $K_5$ and of $K_{3,3}$. Here we study the minimal (under subdigraph containment) strongly-connected digraphs that are…
Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K)…
An ear in a triangulation $T$ of a convex $n$-gon $P$ is a triangle of $T$ that shares two sides with $P$ itself. Certain enumerational and structural problems become easier when one considers only triangulations with few ears. We…
We construct 1-parameter families of non-periodic embedded minimal surfaces of infinite genus in $T \times \mathbb{R}$, where $T$ denotes a flat 2-tori. Each of our families converges to a foliation of $T \times \mathbb{R}$ by $T$. These…
A persistent lamination for a knot K is an essential lamination in the complement of K, which remains essential after every non-trivial Dehn surgery along K. Having a persistent lamination implies, for example, that every manifold obtained…
This article is an English translation of Japanese article "Musubime to Kyokumen", Math. Soc. Japan, Sugaku Vol. 67, No. 4 (2015) 403--423. It surveys a specific area in Knot Theory concerning surfaces in knot exteriors. In version 2, we…
The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots…
We systematically investigate properties of various triangle centers (such as orthocenter or incenter) located on the four faces of a tetrahedron. For each of six types of tetrahedra, we examine over 100 centers located on the four faces of…
Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of…
In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher,…
In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S^3 for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a…