2-twist trefoil has 6 crossings, proving non-trivial knotted surface.
problem Computing the crossing number of non-trivial knotted surfaces.
method Bridge trisection and tri-plane diagrams to minimize crossings.
result 2-twist trefoil has crossing number 6.
Minimal charts of type (3,3) are equivalent to a specific subchart.
problem Characterizing minimal charts of type (3,3).
method Analyzing subgraphs and C-move equivalence.
result Minimal charts of type (3,3) are equivalent to a specific subchart.
A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is not isotopic to its orientation reverse. We sketch the proof the classification …
Study lengths of 3-cocycles for specific quandles, finding knot properties.
problem Determining lengths of 3-cocycles for 7-dihedral and octahedral quandles.
method Analyzing specific quandles to find lengths of 3-cocycles.
result 2-twist-spun 52-knot and 4-twist-spun trefoil have triple point number eight. New procedures connect braid charts, triplane diagrams, and braid movies for knotted surfaces.
problem Understanding the braid index and bridge index of knotted surfaces in 4D.
method Introducing rainbow diagrams and new procedures for passing among triplane diagrams, braid movies, and braid charts.
result Inequalities relating braid index and bridge index of 2-knots are obtained.
New method distinguishes knots and knotted surfaces.
problem Distinguishing knots and knotted surfaces.
method Twisted set-theoretic Yang-Baxter solutions and Alexander numbering.
result Distinguished 2-twist spun trefoil from its reverse. The knot quandle of twist-spun trefoils is studied, revealing finite cardinality for small twists.
problem Understanding the knot quandle of twist-spun trefoils.
method Analyzing the quandle structure of 3- to 5-twist-spun trefoils and the relationship with regular tessellations. result The cardinality of the knot quandle of m-twist-spun trefoils is finite if and only if 1≤m≤5. With the aid of a computer, we provide a motion picture of the twist-spun trefoil which exhibits the periodicity well.
New knot quandle structure for twist-spun trefoils discovered.
problem Understanding symmetries of knot-spun structures.
method Defined Schläfli quandles and studied their relationship with knot quandles.
result The knot quandle of the twist-spun trefoil is a central extension of a Schläfli quandle.
Paper shows how to twist knots to make them trivial or non-trivial.
problem Understanding the behavior of knots under twist-spinning operations.
method Analyzes the effect of m1- and m2-twist-spinning on classical knots. result Shows conditions for a knot to be trivial or non-trivial after certain twist-spinning operations.
The paper studies 4-charts with three crossings and their equivalence to a specific knot.
problem Investigating the structure and equivalence of 4-charts with three crossings.
method Examining charts as oriented labeled graphs in a disk, focusing on acyclic components and equivalence through label-orientation-reflection.
result Any linear minimal 4-chart with three crossings is equivalent to a 2-twist spun trefoil knot.
Minimal area of spun trefoil knot is found in 4D cubical space.
problem Finding the minimum area for a spun trefoil knot in 4D cubical space.
method Defined minimal area of cubical 2-knots, used isotopy to find minimum area for spun trefoil.
result Spun trefoil knot requires a specific minimal area in 4D cubical space.
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…
New bounds found for complexity of spun knots.
problem Measuring complexity of spun knots.
method Using bridge trisections and distances in the pants complex.
result Bound the Kirby-Thompson invariant of spun knots.
We define a knot invariant and a 2-knot invariant from any finite categorical group. We calculate an explicit example for the Spun Trefoil.
In 1987 S Cappell and J Shaneson constructed an s-cobordism H from the quaternionic 3-manifold Q to itself, and asked whether H or any of its covers are trivial product cobordism? In this paper we study H, and in particular show that its 8-fold cover is the product cobordism from S^3 to itself. We reduce the triviality…
We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove t…
Proves codimension 2 spun embedding for specific manifolds.
problem Codimension 2 spun embedding for manifolds.
method Proves embedding using open book decompositions and spin structures.
result Proves spun embedding for simply connected spin 5-manifolds and 3-dimensional open books.
Techniques for constructing codimension 2 embeddings and immersions of the 2 and 3-fold branched covers of the 3 and 4-dimensional spheres are presented. These covers are in braided form, and it is in this sense that they are folded. More precisely the composition of the embedding (or immersion) and the canonical proje…
Algorithms compute invariants of 4-manifolds as branched covers.
problem Computing invariants of 4-manifolds as branched covers.
method Diagrammatic algorithms using tri-plane diagrams and permutations.
result Automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction.
Legendrian products can be twist spuns under certain conditions.
problem Understanding when Legendrian products are twist spuns.
method Using Legendrian contact homology and augmentation variety analysis.
result Not all Legendrian products are twist spuns.
Authors find no positive spun triangulations for certain hyperbolic 3-manifolds.
problem Finding positive spun triangulations for hyperbolic 3-manifolds.
method Using Choi's result, they provide examples of closed hyperbolic 3-manifolds and geodesics without positive spun ideal triangulations.
result They provide evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.
Polynomially parameterizes knots and spheres, proving analogous results.
problem Parameterizing knots and spheres using polynomials.
method Analogous to classical knots, parameterized long 2-knots and certain classes of knotted spheres.
result Polynomial parameterizations for knotted spheres constructed.
The paper examines trisection diagrams of spun knots and shows they are standard for certain cases.
problem Whether trisection diagrams induced by the Gluck surgery on specific knots are standard.
method Explicit depiction and analysis of trisection diagrams for spun (2n+1,−2)-torus knots. result Trisection diagrams are standard for spun (2n+1,−2)-torus knots when n=1 and homologically standard for all n. Frame-spun knots are constructed by spinning a knot of lower dimension about a framed submanifold of S^n. We show that all frame-spun knots are slice (null-cobordant).
Standard trisection diagrams found for a specific type of knot.
problem Finding standard trisection diagrams for Gluck twists.
method Using a specific method to obtain trisection diagrams for spun (p+1,p)-torus knots. result Standard trisection diagrams were found for the spun (p+1,p)-torus knots. We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}.
The concept of a normal surface in a triangulated, compact 3-manifold was generalised by Thurston to a spun-normal surface in a non-compact 3-manifold with ideal triangulation. This paper defines a boundary curve map which takes a spun-normal surface to an element of the direct sum of the first homology groups of the v…
Much work has been done on the existence and uniqueness of broken Lefschetz fibrations such as those by Auroux et al., Gay and Kirby, Lekili, Akbulut and Karakurt, Baykur, and Williams, but there has been a lack of explicit examples. A theorem of Gay and Kirby suggests the existence of a broken Lefschetz fibration of S…
Study Morse-Novikov numbers for frame spun knots and surface-links.
problem Compute Morse-Novikov numbers for frame spun knots and surface-links.
method Apply circle-valued Morse theory to frame spun knots and surface-links.
result Obtain a formula relating the Morse-Novikov numbers of frame spun knots and their complements.
The paper constructs multisections for m-spun 3-manifolds in higher dimensions.
problem Finding multisections for higher-dimensional manifolds.
method Extending the concept of spun 3-manifolds to higher dimensions, constructing multisections and diagrams.
result Infinitely many examples of non-diffeomorphic multisected manifolds in all dimensions.
The paper constructs non-diffeomorphic trisections of 4-manifolds.
problem Constructing distinct trisections of the same genus on a fixed 4-manifold.
method Using spun Seifert fiber spaces and Meier's spun trisections, the paper constructs 2k−1 non-diffeomorphic (3k,k)-trisections. result The Nielsen classes of the generators of the fundamental group are isotopy invariants of the trisection.
We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T2-knots and turned spun T2-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an u…
Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation be…
Khovanov homology detects trefoil knots.
problem Identifying knots using homology.
method Spectral sequence to knot Floer homology.
result Reduced 2-coloured Khovanov homology detects trefoil.
In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map fr…
Khovanov homology fails to differentiate certain slice disks.
problem Differentiating roll-spun slice disks from trivial ones.
method Using Khovanov homology and Morse theory.
result Khovanov homology cannot distinguish roll-spun slice disks from trivial ones.
Study shows patterns in Stein fillability of trefoil surgeries.
problem Determining Stein fillability of contact structures.
method Analyzing surgeries on the trefoil knot.
result Patterns in Stein fillability identified.
The paper proves properties of branched covers of specific knots and tori.
problem Investigating the smoothness and diffeomorphism of specific 4-manifolds.
method Analyzing double branched covers of twist-roll spun knots and turned twisted tori, applying techniques to show diffeomorphism.
result Proves that certain 4-manifolds are diffeomorphic to standard manifolds.
We construct a 2+1 dimensional classical gauge theory on manifolds with spin structure whose action is a refinement of the Atiyah-Patodi- Singer eta-invariant for twisted Dirac operators. We investigate the properties of the Lagrangian field theory for closed, spun 3-manifolds and compact, spun 3-manifolds with boundar…
By 2-twist-spinning the knotted graph that represents the knotted handlebody 52, we obtain a knotted foam in 4-dimensional space with a non-trivial quandle cocycle invariant.
Characterizes groups of branched twist-spun knots.
problem Understanding the groups of branched twist-spun knots.
method Characterization through 3-manifold groups and conjectural algebraic approach.
result Each group is the group of at most finitely many branched twist spins.
Proved colored HOMFLY-PT polynomials for specific knots.
problem Calculating colored HOMFLY-PT polynomials for specific knots.
method Rigorous mathematical proof for trefoil, figure-eight, and twist knots.
result Colored HOMFLY-PT polynomials expressed as sums for different knots.
Study of codimension-1 embeddings in 3-manifolds using twist maps and push maps.
problem Embedding 3-manifolds in specific topological spaces.
method Sphere twist maps and push maps to construct codimension-1 spun embeddings.
result Every closed orientable 3-manifold admits a codimension-1 spun embedding in specific spaces.
Banchoff's sphere splits trefoil knots in 3-manifolds.
problem Understanding how Dehn spheres split trefoil knots in 3-manifolds.
method Constructing and analyzing filling Dehn spheres and their diagrams.
result Banchoff's sphere splits the trefoil knot in S3. The classical trefoil is famous for having a three-colouring which distinguishes it from the unknot. The three-colouring is also notorious for not distinguishing the right handed from the left handed trefoil. However with a bit of tweaking the three colours can also be used for this task. What lies behind the method is…
Khovanov homology can identify trefoil knots.
problem Detecting trefoil knots using Khovanov homology.
method Floer homology, contact geometry, open books, instanton Floer setting, bypass exact triangle, spectral sequence.
result Khovanov homology detects trefoils.
Embeds all contact 3-manifolds into specific 5-manifolds.
problem Embedding all contact 3-manifolds into fixed contact 5-manifolds.
method Spun embeddings and Lefschetz fibrations.
result Embeds all contact 3-manifolds into a Stein fillable contact structure on the twisted S3-bundle over S2 and a unique overtwisted contact structure on S3imesS2.