arXiv research

Essential surfaces in link diagrams on surfaces are crucial for understanding link properties.

The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.

Essential surfaces found in curved 3D shapes.

The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.

Counting essential surfaces in 3-manifolds yields concise formulae and detailed asymptotics.

The second author and Hara introduced the notion of an essential tribranched surface that is a generalisation of the notion of an essential embedded surface in a 3-manifold. We show that any 3-manifold for which the fundamental group has at least rank four admits an essential tribranched surface.

Essential surfaces are detected in higher $\operatorname{SL}_n$-character varieties.

New prime knots found with high-genus surfaces.

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Deh…

We determine all (1,1)-knots which admit an essential meridional surface, namely, we give a construction which produces (1,1)-knots having essential meridional surfaces, and show that if a (1,1)-knot admits an essential meridional surface then it comes from the given construction.

Paper defines a partial order on multibranched surfaces in 3-manifolds.

New invariants measure how far spanning surfaces are from being compressible.

Essential embeddings of metric graphs on hyperbolic surfaces are constructed and studied.

Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.

Detects essential surfaces in knots using character variety intersections.

Study essential surfaces in knot exteriors, proving some combinations are realizable.

Detecting closed essential surfaces in 3-manifolds is challenging.

We construct an algorithm that lists all closed essential surfaces in the complement of a knot that lies on the fiber of a trefoil or figure eight knot. Such knots are Berge knots and hence admit lens space surgeries. Furthermore they may have arbitrarily large hyperbolic volume. Using this algorithm we concoct large v…

The paper finds infinite knots with surfaces of any positive genus.

Khovanov homology detects essential surfaces in knot complements.

Study on shortest arcs on hyperbolic surfaces with boundary.

Counted essential surfaces in a knot's exterior, finding a unique pattern.

Abby Thompson proved that if a link $K$ is in thin position but not in bridge position then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that the if a link is in thin…

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory. We hope the normal s…

Algorithm finds essential surfaces in 3D shapes.

Proves prime theta-curves for knots on minimal genus surfaces.

Characterizes unknotted curves on Seifert surfaces of twist knots.

We prove results showing that the existence of essential maps of surfaces in a manifold M' obtained from a 3-manifold M by Dehn filling implies the existence of essential maps of surfaces in M.

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. Especially we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essenti…

New families of twisted torus knots found with essential surfaces.

An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic view of states of categorified Jones polynomials in thickened surfaces and virtual …

We study a canonical spanning surface obtained from a knot or link diagram depending on a given Kauffman state, and give a sufficient condition for the surface to be essential. By using the essential surface, we can see the triviality and splittability of a knot or link from its diagrams. This has been done on the exte…

Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.

We generalize a theorem of Finkelstein and Moriah and show that if a link $L$ has a $2n$-plat projection satisfying certain conditions, then its complement contains some closed essential surfaces. In most cases these surfaces remain essential after any totally nontrivial surgery on $L$.

The study provides polynomial bounds for essential surfaces in various 3-manifolds.

We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the branched surface). The first three conditions are…

Let $T$ be a graph in a compact, orientable 3--manifold $M$ and let $Γ$ be a subgraph. $T$ can be placed in bridge position with respect to a Heegaard surface $H$. We show that if $H$ is what we call $(T,Γ)$-c-weakly reducible in the complement of $T$ then either a "degenerate" situation occurs or $H$ can be untelescop…

We provide a classification of the essential surfaces of non-negative Euler characteristic in the exteriors of genus two handlebodies embedded in the 3-sphere.

Tollefson described a variant of normal surface theory for 3-manifolds, called Q-theory, where only the quadrilateral coordinates are used. Suppose $M$ is a triangulated, compact, irreducible, boundary-irreducible 3-manifold. In Q-theory, if $M$ contains an essential surface, then the projective solution space has an e…

Geometric finiteness theory for essential surfaces in knot exteriors with geometric bounds.

Let $F$ be a closed essential surface in a hyperbolic 3-manifold $M$ with a toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an essential pleated surface which is not coannular to the boundary torus of $N$ then the depth…

In this paper, we use normal surface theory to study Dehn filling on a knot-manifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knot-manifold that bound normal and almost normal surfaces in a one-vertex triangulation of that knot-manifold. This is combined with existence th…

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative …

Study counts specific surfaces in Montesinos knots with 4 rational tangles.

Culler-Shalen theory extended to arbitrary characteristic.

The paper defines new knot invariants from 2-bridge knots.

The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.