Characterizes alternating knot exteriors using special surfaces.
problem Understanding topological properties of alternating knots.
method Topological characterisation using special spanning surfaces and a normal surface algorithm.
result Alternating is a topological property of knot exteriors, not just diagrams.
The paper classifies symmetries and determines exterior types of knotted handlebodies.
problem Classifying symmetries and determining exterior types of knotted handlebodies.
method Analysis of hyperbolic knots with non-integral toroidal Dehn surgeries.
result Determines symmetries and exterior types of knotted genus two handlebodies.
The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.
problem Existence of meridional essential surfaces in knot exteriors.
method Analyzing knot exteriors and their surfaces.
result Existence of infinitely many knot exteriors with meridional essential surfaces of any genus and boundary components.
Algorithm finds knot diagrams from exterior triangulations.
problem Finding knot diagrams from their exteriors.
method First practical algorithm for finding diagrams from triangulations of exteriors.
result First diagrams for 23 link exteriors with over 2,500 crossings.
The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.
problem Existence of meridional essential surfaces in hyperbolic knot exteriors.
method Essential tangle decompositions and meridional essential embeddings.
result Infinite collection of hyperbolic knots with meridional essential surfaces of arbitrary genus and boundary components.
Paper introduces an invariant to distinguish handlebody-knot exteriors.
problem Challenges in distinguishing handlebody-knots with homeomorphic exteriors.
method Defined an invariant (annulus diagram) using characteristic submanifold theory and Koda-Ozawa classification for essential annuli.
result The annulus diagram can differentiate handlebody-knot families.
The article explores surfaces in knot exteriors, adding insights to conjectures.
problem Understanding surfaces in knot exteriors.
method Survey and analysis of specific conjectures.
result Added comments on solutions or counterexamples for conjectures.
We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots 614 and 615 are not equivalent. We also show that some genus two handlebody-knot…
The paper generates triangulations of 2-knot complements via spinning 1-knots.
problem Generating triangulations of 2-knot complements.
method Algorithm to generate triangulations of 2-knot complements obtained by spinning 1-knots.
result Triangulations of exteriors of 2-knots from all 1-knots with up to eight crossings.
Study shows (p,q)-cable knots cannot undergo certain types of surgery.
problem Chirally cosmetic surgery on (p,q)-cable knots. method Analyzes JSJ pieces and torus exteriors to prove non-existence of chirally cosmetic surgery.
result Proves (p,q)-cable knots do not admit chirally cosmetic surgery under certain conditions. Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
The study limits the number of 2-holed tori in knot exteriors.
problem Bounding the number of 2-holed tori in knot exteriors.
method Continuing Motegi's program, the paper applies universal bounds to hyperbolic knots.
result There are at most six non-isotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot.
Study essential surfaces in knot exteriors, proving some combinations are realizable.
problem Whether given genus, boundary components, and slope combinations are realizable in knot exteriors.
method Analyzing compact orientable essential surfaces in knot exteriors, proving existence for even boundary components.
result For even boundary components, there exist knots with essential surfaces of specified genus and slope.
The paper finds infinite knots with surfaces of any positive genus.
problem Finding knots with surfaces of arbitrary positive genus.
method Constructing knot exteriors with longitudinal essential surfaces of any positive genus.
result Existence of infinitely many knots with surfaces of any positive genus.
In this paper, we show that, for each non-trivial two bridge knot K and for each g > 2, every genus g Heegaard splitting of the exterior E(K) of K is reducible.
We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal-Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of log |the high…
For certain knots, meridional rank equals bridge number.
problem Prove meridional rank equals bridge number for knots with graph manifold exteriors.
method Proved for a class of knots whose exteriors are graph manifolds.
result Meridional rank equals bridge number for these knots.
Let X be the exterior of connected sum of knots and Xi the exteriors of the individual knots. In \cite{morimoto1} Morimoto conjectured (originally for n=2) that g(X)<σi=1ng(Xi) if and only if there exists a so-called \em primitive meridian \em in the exterior of the connected sum of a proper subset of …
New families of twisted torus knots found with essential surfaces.
problem Whether all twisted torus knots have essential tori.
method Analyzing sequences of twists on torus knots.
result Found two new families of toroidal twisted torus knots.
Framework for reconstructing knot exterior surface complexes.
problem Understanding the mapping class group action on knot exteriors.
method Organizing rigidity questions, using a reduction principle, and classifying automorphisms.
result Common image-kernel bookkeeping and reconstruction framework for knot exteriors.
Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are know…
In this article, we prove that a tunnel number two knot induces a critical Heegaard splitting in its exterior if there are two weak reducing pairs such that each weak reducing pair contains the cocore disk of each tunnel. Moreover, we prove that a connected sum of two 2-bridge knots or more generally that of two $(1,1)…
We show that for each pair of positive integers g and n, there are infinitely many tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each positive integer n, there are tunnel number one knots whose exteriors contain n disj…
Study on cylindrical handlebody-knots with symmetry and rigidity properties.
problem Characterizing symmetry groups of cylindrical handlebody-knots of genus two.
method Classification of essential annuli and analysis of symmetry groups based on Koda-Ozawa theorem.
result Most exteriors of genus two cylindrical handlebody-knots contain no essential disks or tori, and type 3-3 annuli are often unique up to isotopy. Mazur's knot exterior is described by a single regular ideal octahedron, leading to hyperbolic structures related to the Whitehead link.
problem Proving nonhomeomorphism of boundaries of Mazur and Jester manifolds
method Using hyperbolic geometry, Dehn filling, and systolic geodesics
result Proving the boundaries of all Mazur and Jester manifolds are pairwise nonhomeomorphic
A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Natura…
Legendrian knots in tight 3-sphere are uniquely determined by their exteriors.
problem Determining Legendrian knots in tight contact structures.
method Contactomorphism type of exterior, Thurston-Bennequin invariant formula.
result Not all Legendrian links in tight 3-sphere are uniquely determined by their exteriors.
Counted essential surfaces in a knot's exterior, finding a unique pattern.
problem Counting essential surfaces in a knot's exterior.
method Counted essential surfaces by genus, using Euler totient function. Showed normal surfaces are connected by counting their components. Used Agol, Hass, and Thurston's tools to convert component counting into orbit counting.
result Found a unique pattern in the number of essential surfaces by genus.
Geometric finiteness theory for essential surfaces in knot exteriors with geometric bounds.
problem Understanding the topology of essential surfaces in knot exteriors with geometric constraints.
method Developed a relative geometric finiteness theory using bounded geometry and thickness conditions.
result Every bounded-geometry slice contains only finitely many pair-isotopy classes, and topology is recoverable from finite geometric data.
Classifies essential annuli in a genus two handlebody exterior.
problem Classifying essential annuli in a genus two handlebody exterior.
method Building on JSJ-graph classification and essential annuli classification.
result Characterizes the numbers of different types of essential annuli in an infinite family.
The paper connects surface embeddings to handlebody-knots and explores their properties.
problem Understanding the relationship between surface embeddings and handlebody-knots.
method Using Fox's theorem, the paper associates handlebody-knots to surface embeddings and constructs new surfaces.
result For every genus two prime bi-knotted surface, one handlebody-knot is irreducible and the other is reducible.
Given integers g_i > 1 (i=1,...,n) we prove that there exist infinitely may knots K_i in S^3 so that g(E(K_i)) = g_i and the Heegaard genus of the exterior of the connected sum of K_1,...,K_n is the sum the Heegaard genera of K_1,...,K_n, that is: g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). (Here, E() denotes the ex…
Study incompressible surfaces to find cable knots' representativity and waist.
problem Computing the representativity and waist of cable knots.
method Study incompressible surfaces in cable knots' exteriors.
result Compute representativity and waist for most cable knots.
The study examines how knots relate based on a specific map property.
problem Understanding the complexity of knots through a map property.
method Examines knots through a degree 1 map property between their exteriors.
result Supports the idea that more complex knots have a higher map property.
Study bounds leading coefficients of L2-Alexander torsions for 3-manifolds.
problem Bounding leading coefficients of L2-Alexander torsions. method Upper and lower bounds using hyperbolic volumes and relative L2-torsions. result Equal bounds for numerous families of knot exteriors, including 2-bridge knots.
Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.
problem Classifying essential annuli in genus two handlebody-knots.
method Introducing τ- and ρ-tangles and good rectangles, classifying these structures.
result Categorization of atoroidal 3-decomposable genus two handlebody-knots based on essential annuli.
Brunnian theta curves in 3D spheres have hyperbolic exteriors.
problem Characterizing Brunnian theta curves in 3D spheres.
method Sutured manifold theory, classification of annuli, and Thurston's hyperbolic geometry.
result Brunnian theta curves of low bridge number have hyperbolic exteriors with totally geodesic boundary.
Supose that Y is a lens space with ∣H1(Y;Z)∣ prime, and Y does not contain a genus one fibered knot. We show that Y contains a knot whose exterior is a once-punctured torus bundle if and only if Y is the result of p/q-surgery on the trefoil. This partially answers a question posed by Ken Baker in…
New knots bound multiple non-isotopic ribbon disks.
problem Finding knots that bound multiple non-isotopic ribbon disks.
method Classification of fibered, homotopy-ribbon disks for generalized square knots.
result Infinitely many knots bound infinitely many pairwise non-isotopic ribbon disks.
We describe a procedure for creating infinite families of hyperbolic knots having unique minimal genus Seifert surface. A large subset of these knots have the further property that the surface cannot be the sole compact leaf of a depth one foliation of the knot exterior.
We exhibit an algorithm to determine the bridge number of a hyperbolic knot in the 3-sphere. The proof uses adaptations of almost normal surface theory for compact surfaces with boundary in ideally triangulated knot exteriors.
Study shows torsion in knot module for specific Montesinos knots.
problem Torsion in Kauffman bracket skein module of Montesinos knots.
method Analyzes Kauffman bracket skein module over Z[q±21] for specific knots. result Provides a negative answer to Problem 1.92 in Kirby's list.
The paper introduces new invariants for genus one knots and surfaces.
problem Understanding invariants of genus one knots and surfaces.
method Investigating properties of the Alexander form of 3-manifolds to extract invariants of Seifert surfaces.
result Extracted invariants of genus one Seifert surfaces from the Alexander form of their exteriors.
The goal of this paper is to offer a comprehensive exposition of the current knowledge about Heegaard splittings of exteriors of knots in the 3-sphere. The exposition is done with a historical perspective as to how ideas developed and by whom. Several new notions are introduced and some facts about them are proved. In …
We give (1) an upper bound on the denominators of numerical boundary slopes and (2) an upper bound on the differences between two numerical boundary slopes, for Montesinos knot exteriors.
A knot K in a closed connected orientable 3-manifold M is called a 1-genus 1-bridge knot if (M,K) has a splitting into two pairs of a solid torus V_i (i=1,2) and a boundary parallel arc in it. The splitting induces a genus two Heegaard splitting of the exterior of K naturally, i.e., K has an unknotting tunnel. However …
This paper classifies knots with simple curves in genus 2 handlebodies.
problem Classifying knots with specific curves in handlebodies.
method Analyzing R-R diagrams and 2-handle additions.
result Classification of knots with proper power curves in genus 2 handlebodies.
In this article, we give an explicit formula to compute the non-abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.