Book introduces hyperbolic geometry for knot theory.
problem Understanding knots through hyperbolic geometry.
method Explains hyperbolic geometry, geometric structures, and techniques.
result Develops three knot invariants from hyperbolic geometry.
Survey of quadrisecants and essential secants in knot geometry.
problem Understanding the geometry of knots through secants.
method Analyzing quadrisecants and essential secants to study knot geometry.
result Applications of secants in knot geometry via total curvature, ropelength, and distortion.
New invariant measures knot geometry, improving volume-volume inequality.
problem Understanding hyperbolic knots through geometry.
method Introducing natural slope based on cusp geometry, using machine learning for detection.
result Improved inequality linking knot signature, volume, and injectivity radius.
This article serves as errata of the book "Energy of knots and conformal geometry", Series on Knots and Everything Vol. 33, World Scientific, Singapore, 304 pages, (2003). (ver. 27/05/2007)
The paper studies hyperbolic geometry of links formed by adding trivial components to knots.
problem Characterizing hyperbolic geometry in links formed by adding trivial components to knots.
method Examining generalized augmented links of knots given by positive braids with at least one full twist.
result Conditions for the hyperbolic geometry of these links are identified.
New knots are found to be non-simple in Legendrian contact geometry.
problem Identifying non-simple Legendrian knots in contact geometry.
method Utilized knot Floer homology and the distinguished surgery triangle.
result Whitehead doubles of the trefoil are Legendrian non-simple.
As an example of the transitions between some of the eight geometries of Thurston, investigated before, we study the geometries supported by the cone-manifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phe…
New metrics measure knots' shapes without changing their orientation.
problem Measuring knots without considering their orientation.
method Defined Möbius invariant metrics on knot space.
result Found conditions for Möbius invariant weighted inner products.
Disproves conjectures about shared surgeries for distinct knots.
problem Knots sharing surgeries for multiple slopes.
method Constructing pairs of distinct knots with shared Dehn surgeries.
result Found pairs of distinct knots with four distinct shared slopes.
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot.…
New method finds exponential growth in knot types from sticks.
problem How many knots can be formed with a fixed number of sticks?
method Polygonal self-intersection to sparse real-algebraic chamber problem, braid construction.
result Factorial-scale upper bound for knot types, optimal growth order.
The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.
problem Checking cosmetic surgeries on knots and 3-manifolds.
method Knot invariants and hyperbolic geometry.
result Verification of no cosmetic surgeries on knots and 3-manifolds.
Expanded Legendrian knot atlas for 10-arc index knots.
problem Lack of Legendrian knot data for knots with high arc index.
method Created an atlas of Legendrian knots up to arc index 10.
result Legendrian knots of arc index 10 have been cataloged.
We review the string/gauge theory duality relating Chern-Simons theory and topological strings on noncompact Calabi-Yau manifolds, as well as its mathematical implications for knot invariants and enumerative geometry.
Study geometric structures of Seifert manifolds with up to three exceptional fibres.
problem Determine Thurston's geometry of Seifert fibered conemanifolds.
method Analyze Seifert manifolds with up to three exceptional fibres and apply to three families.
result Generalize results to all torus knots, associating plots for different geometries.
Study links between surface germs and knot theory in 4D.
problem Understanding the relationship between surface germs and knot theory in R4. method Constructing surface germs XK linked to knots K in S3 and studying their Lipschitz geometry. result Ambient bi-Lipschitz equivalence of surface germs is related to isotopy of knots, and Jones polynomial can recognize non-equivalent germs.
Classifies knots in a special 3D space.
problem Classifying knots in a specific geometric space.
method Complete coarse classification of non-loose torus knots.
result Complete classification of knots in S1imesS2. Paper connects knot theory with cluster algebra via Alexander polynomials.
problem Understanding Alexander polynomials for 2-bridge knots.
method Use of cluster variables and ancestral triangles.
result Alexander polynomials are specializations of cluster variables.
We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.
The study finds bounds on characterizing slopes for all knots.
problem Characterizing slopes for knots in 3D space.
method Combining hyperbolic geometry results with previous work, the study determines explicit bounds for characterizing slopes.
result An optimal bound C(K) for certain satellite knots is found. New bounds on knotting probability of equilateral hexagons found.
problem Determining the knotting probability of equilateral hexagons.
method Symplectic geometry techniques to parametrize and analyze the space of equilateral hexagons.
result New bounds on the knotting probability of equilateral hexagons.
The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple cubic, face centered cubic and body centered cubic lattices are determined. Our r…
We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian simple knot types.
Explains how knots relate to 4D shapes.
problem Understanding 4D shapes through knot theory.
method Combines knot theory with 4D manifold topology.
result Connects 4D shapes to knot theory and other geometries.
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related w…
Formulae for volumes of double twist knots found.
problem Calculating volumes of specific hyperbolic cone-manifolds.
method Explicit formulae for volumes of double twist knots.
result Formulae for volumes of double twist knots found.
This paper confirms a bound for folded ribbonlength of 2-bridge knots.
problem Bounding the folded ribbonlength of 2-bridge knots.
method Investigated the folded ribbonlength of 2-bridge knots and proved a linear upper bound.
result The folded ribbonlength of a 2-bridge knot K is bounded above by 2c(K)+2. New polynomial connects knot genus to 3-manifold geometry.
problem Detecting knot genus from 3-manifold geometry.
method Ideal triangulation and super-Ptolemy assignments.
result New polynomial conjecturally agrees with torsion polynomial.
We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.
The L2-Alexander invariant detects knots more precisely than genus and simplicial volume.
problem Detecting individual knots using genus, simplicial volume, and L2-Alexander invariant. method Hybrid techniques from hyperbolic geometry and topology.
result The L2-Alexander invariant detects more knots than the pair (genus, simplicial volume). 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.
The study analyzes neural network predictions of knot invariants and finds that braid representations work best.
problem Understanding and predicting knot invariants using neural networks.
method Investigated different knot representations and invariants, proposed a cosine similarity score.
result Braid representations are best for predicting knot invariants, and some invariants are easier to learn than others.
This paper studies CR geometry of transversal curves in the 3-sphere.
problem Investigating CR geometry of transversal curves in the 3-sphere.
method Using local CR invariants of the 3-sphere, four global invariants are considered: phase anomaly, CR spin, Maslov index, and CR self-linking number.
result Closed critical curves of the simplest CR invariant variational problem for generic transversal curves are studied.
Study bounds on cusp volumes of alternating knots on surfaces.
problem Bounding cusp volumes of knots on surfaces.
method Analyzing hyperbolic knots with alternating projections on embedded surfaces.
result Two-sided bounds on cusp area in terms of twist number and surface genus.
The paper finds hyperbolic small knots in many 3-manifolds.
problem Finding small knots in 3-manifolds.
method Explicit examples of hyperbolic small knots in spherical 3-manifolds.
result Explicit examples of hyperbolic small knots in most spherical 3-manifolds.
Infinite knots have non-integer trace values.
problem Finding knots with non-integer trace values.
method Proved existence of infinitely many non-homeomorphic hyperbolic knot complements with specific trace properties.
result Infinitely many non-homeomorphic hyperbolic knot complements with non-integer trace values.
A polynomial knot is a smooth embedding κ:ℜ→ℜn whose components are polynomials. The case n=3 is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological knot. This paper contains basic results for these knots as well as many examples.
New infinite family of knots found with unique properties.
problem Finding asymmetric L-space knots with specific properties.
method Generalizing known asymmetric L-space knot t12533 to form an infinite family.
result First infinite family of asymmetric hyperbolic L-space knots of braid index 4.
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based only on a diagram. The results are obtained by finding geometric information on ge…
New examples of gordian unlinks show different rope geometries.
problem Tackling the existence of non-trivial unknots with prescribed length and thickness.
method Provided the first examples of gordian unlinks with thickness in [1,2).
result Thinner normal tubes lead to different rope geometries.
Withdrawn and replaced by two related manuscripts: (1) "Stabilization in the braid groups I:MTWS", published in Geometry and Topology Volume 10 (2006), 413-540, arXiv:math.GT/0310279, and (2) "Stabilization in the braid groups II: Transversal simplicity of knots", Geometry and Topology Volume 10 (2006), to appear, arXi…
Constructs a spectrum for knot Floer homology without holomorphic geometry.
problem Computing knot Floer homology without using holomorphic geometry.
method Combinatorial definition and inductive construction of models for moduli spaces.
result Conjectures that the filtered homotopy type of the spectrum is an invariant of the knot.
The paper uses algebraic geometry tools to study knot invariants and Dehn surgeries.
problem Understanding algebraic and number theoretic properties of canonical components of character varieties.
method Utilizes quaternion Azumaya algebras and Brauer groups of curves over number fields.
result Constructs new knot invariants using algebraic geometry.
Study open book coverings and their applications in contact geometry.
problem Understanding the structure of open book foliations and their impact on contact geometry.
method Using open book foliations to explore coverings and virtually overtwisted contact manifolds.
result Open book coverings produce transverse knots with depth greater than 1.
New method detects and compares folding pathways of knotted proteins.
problem Understanding the function of knots in protein folding.
method Topological analysis of protein knotoid distributions and entanglement.
result Reveals unique folding pathway for shallow knotted Carbonic Anhydrases.
We solve the Jones conjecture, which states that the exponent sum in a minimal braid representation of a knot in S^3 is a knot invariant, by proving a generalized version of the original one. We apply contact geometry to study this problem in knot theory.
It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. In this paper we pro…
Introduces knot theory via surface perspectives.
problem Understanding knots through surface geometry.
method Explains isotopies, Reidemeister moves, and Seifert surfaces.
result Introduces a group structure on knots.