Study of knots and links in 2-complexes, defining linking numbers and polynomials.
problem Understanding knots and links in 2-dimensional complexes.
method Definition of linking numbers and Kauffman-type bracket polynomials for links in 2-complexes.
result Established relationships between 2-complexes and knots/links in 3-manifolds.
Survey of Turk's head knots and links properties.
problem Characterize Turk's head knots and links.
method Discussion of various results in the mathematical literature.
result Turk's head links are alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime.
Virtual knots and links get multicrossings and petal diagrams.
problem Defining multicrossings for virtual knots and links.
method Extending multicrossings to virtual knots and links, showing petal diagrams exist.
result Petal diagrams exist for all virtual knots.
Study links with annuli using sutured Floer homology.
problem Characterize links with specific cable structures.
method Apply sutured Floer homology techniques.
result Characterizations of links with (n,nm)-cables and (2,2m)-cables. Study shows rank of knot Floer homology detects Hopf links and classifies second smallest links.
problem Detecting and classifying links based on knot Floer homology ranks.
method Analyzes knot Floer homology ranks to identify and classify links.
result Classifies links of the second smallest knot Floer homology.
Characterizes knot groups and symmetric quandles of surface-links.
problem Characterize knot groups and symmetric quandles of surface-links.
method Used plat presentations for surface-links and closed 2-dimensional braids.
result Generalized results to include non-orientable surface-links and showed that dihedral quandles can be realized as symmetric quandles of surface-links.
Survey of intrinsically linked or knotted graphs.
problem Understanding graphs that are inherently knotted or linked.
method Survey of existing results.
result Discussion of known properties of intrinsically linked or knotted graphs.
New invariants defined for framed knots and links.
problem Defining invariants for framed knots and links.
method Introducing birack brackets and categorifying their multiset.
result Quiver-valued invariant defined for framed knots and links.
Abstract summarizes level-rank duality in knot and link invariants.
problem Distinguishing torus knots and links from hyperbolic ones.
method Chern-Simons theory and tables of knot invariants.
result Criterion to distinguish torus knots and links from hyperbolic ones.
New link detection results using knot and link Floer homology.
problem Detecting specific links and knots using Floer homology.
method Inspired by Khovanov homology, uses knot and link Floer homology.
result Detects specific links and knots with high precision.
We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with at most 16 crossings. We also defined semi-meander knots (or knots with ordered Gauss code) and their product.
New tile types for knots and links reduce complexity.
problem Determining the minimum number of tiles needed for knot representations.
method Introduced new tile types and analyzed their impact on knot complexity.
result Corner tile number lies between tile number and 3 times tile number.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain componen…
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is a representation of knot (link) as closure of braid with n threads and length of this braid does not exceed n(4n-5)+2.
We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.
Budney recently constructed an operad that encodes splicing of knots. He further showed that the space of (long) knots is generated over this operad by the space of torus knots and hyperbolic knots, thus generalizing the satellite decomposition of knots from isotopy classes to the level of the space of knots. Infection…
New virtualized Δ-move simplifies virtual knots and links.
problem Simplifying virtual knots and links.
method Introducing a new local deformation called the virtualized Δ-move.
result Virtualized Δ-move is an unknotting operation for virtual knots.
Positive braid knots have simple knot Floer homology.
problem Computing knot Floer homology for positive braids.
method Computed the next-to-top term of knot Floer homology.
result Rank is 1 for any prime positive braid knot.
Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invarian…
We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link …
Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
problem Computing knot symmetric quandle for surface-links.
method Using plat form presentations, the paper computes the knot symmetric quandle for surface-links.
result Infinitely many distinct surface-knots of genus g with plat indices m.
The study develops a word mechanism for knot and link diagrams.
problem Determining incompressible and pairwise incompressible surfaces in knot or link complements.
method Word mechanism applied in knot and link diagrams.
result Finiteness theorem and upper bound on Euler characteristic of surfaces.
Tying knots and linking microscopic loops of polymers, macromolecules, or defect lines in complex materials is a challenging task for material scientists. We demonstrate the knotting of microscopic topological defect lines in chiral nematic liquid crystal colloids into knots and links of arbitrary complexity by using l…
We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or (2,m)-torus knot or link substrate. We show that all knotted or linked products fall into a single family, and prove that the size of this family grows linearly …
We give a simple example showing that a knot or link diagram that lies in the Z2 lattice is not necessarily the projection of a lattice stick knot or link in the Z3 lattice, and we give a necessary and sufficient condition for when a knot or link diagram that lies in the Z2 lat…
Study on the ribbonlength of knots and links, improving upper bounds.
problem Finding the infimal folded ribbonlength of knots and links.
method Creating new folded ribbon knots and applying them to improve upper bounds.
result Improved upper bounds for (2,q)-torus links, twist knots, and pretzel links. In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and…
New method speeds up knot computations in 3D.
problem Computational complexity in knot theory.
method 3D representation of knots for faster computation.
result Savings in computational complexity for knot invariants.
Study minimum ribbonlength of immersed flat knots and links.
problem Finding the minimum ribbonlength for immersed planar knots and links.
method Embedding into disk diagram space to find length minimizers.
result Computed minimal ribbonlength for some knot and link diagrams.
The paper studies knots in projective space using virtual link theory.
problem Understanding knots in three-dimensional projective space.
method Associate virtual links to projective links and apply virtual knot theory techniques.
result Equivalent projective links correspond to equivalent virtual links modulo a flype move.
Computed linking number of modular knots and Lorenz links.
problem Computing the linking number of modular knots in a specific space.
method Using correspondence between modular links and Lorenz links, and intersection number involving Conway topographs.
result Computed linking number of modular knots and compared to a previous formula.
We formulate the holographic principle for knots and links. For the "space" of all knots and links, torus knots T(2m+1,2) and torus links L(2m,2) play the role of the "boundary" of this space. Using the holographic principle, we find the skein relation of knots and links with the help of the recurrence relation for pol…
Study Khovanov homology of positive links and L-space knots, finding vanishing conditions.
problem Understanding Khovanov homology of positive links and L-space knots.
method Analyzing Khovanov homology groups in specific gradings and extending results to (p,q)-cables and Heegaard Floer L-space knots.
result Khovanov homology of positive links and L-space knots vanishes under certain conditions.
Study extends knot polynomials to links, identifying them with known invariants.
problem Extending knot polynomials to links.
method Applying Reshetikhin-Turaev functor to braided Hopf algebras with automorphisms.
result Identifies some knot polynomials with known link invariants.
Study of cable links of uniformly thick knots, revealing new isotopy phenomena.
problem Understanding Legendrian isotopy in cable links of uniformly thick knots.
method Introduced new technique of Legendrian surgeries to classify Legendrian knots in negative cables of twist knots.
result Found new phenomena of stabilized Legendrian links that are smoothly isotopic but not Legendrian isotopic.
The paper explores orderability in link quandles and provides various results.
problem Orderability of link quandles and its interconnections with link groups.
method Develops a general theory of orderability, proves specific cases, and explores interconnections.
result Link quandles of most non-trivial torus links are not right-orderable.
The paper extends BPS invariants for framed knots and links.
problem Investigating BPS invariants for framed knots and links.
method Using the dual A-polynomial and framing change formula, the paper extends the relationship between algebraic curves and BPS invariants to framed knots and links.
result Explicit formulas for extremal A-polynomials and BPS invariants of framed knots, and numerical calculations for framed Whitehead links and Borromean rings.
We develop a model characterizing all possible knots and links arising from recombination starting with a twist knot substrate, extending previous work of Buck and Flapan. We show that all knot or link products fall into three well-understood families of knots and links, and prove that given a positive integer n, the…
We consider vector fields on knot/link complements in S3 which are transverse to the fibres of a fibration of the complement over a circle. We prove that a large class of fibred knots/links, including all non-torus fibred 2-bridge knots, has the following property: any vector field transverse to the fibres of the fi…
Knot lattice homology invariant of smooth knot type in rational homology spheres.
problem Invariance of knot lattice homology in rational homology spheres.
method Proving knot lattice homology invariant through doubly-filtered homotopy type.
result Knot lattice homology invariant of smooth knot type in rational homology spheres.
New invariant distinguishes singular knots and links.
problem Classifying singular knots and links.
method Using oriented singquandles and weight functions at crossings.
result Distinguishes singular granny knot from singular square knot.
Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted l…
Modular knots follow Chebotarev law from surgeries on hyperbolic fibered links.
problem Understanding knots that behave like prime numbers.
method Analyzing planetary links and surgeries on hyperbolic fibered links in S3. result Modular knots around torus knots Ka,b follow the Chebotarev law. Study on quandle coloring quivers for (p, 2)-torus knots and links.
problem Understanding quandle colorings of (p, 2)-torus knots and links.
method Introduced quandle coloring quivers and studied them for dihedral quandles.
result Characterized quandle coloring quivers for (p, 2)-torus knots and links.
Investigates BNSR invariants of link and knot groups, proving specific properties.
problem Characterizing finiteness properties of normal subgroups in link and knot groups.
method Analyzes BNSR invariants of link and knot groups, proving specific properties.
result Proves specific conditions for finiteness properties of link and knot groups.
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.