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.
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.
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. 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.
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.
Abstract: Holographic principle applied to knots and links.
problem Understanding polynomial invariants and skein relations for knots and links.
method Formulated holographic principle for knots and links; used recurrence relations for polynomial invariants of torus knots and links.
result Derivation of Jones skein relation and its generalization.
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.
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.
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.
New knots arise from infections by n-component string links that can't be achieved by (n-1)-component infections.
problem Understanding the differences in concordance classes of knots through infections.
method Using satellite constructions and infections to study concordance classes of knots.
result Existence of knots that cannot be obtained by infecting along (n-1)-component string links.
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.
Explains how knots and links can be shown using braids.
problem None explicitly stated; focuses on representation methods.
method Diagrammatic representations of knots and links via braids.
result Explains how braids can represent knots and links.
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.
A 2-link with trivial components has a nonribbon band-sum.
problem Characterizing nonribbon 2-links.
method Examining specific 2-links and their band-sums.
result Found a nonribbon 2-link with trivial components and a nonribbon band-sum.
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.
Study of cosmetic bandings on knots and links, including a counterexample.
problem Cosmetic bandings on knots and links and their impact on knot types.
method Examples and counterexamples of cosmetic bandings, hyperbolic knot analysis.
result A counterexample to a conjecture about chirally cosmetic surgeries.
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…
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.
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.
Study homology of torus links and colored knots.
problem Computing homology of specific link types.
method Khovanov-Rozansky homology for positive torus links and colored knots.
result Computed homology for a family of links including torus links and colored 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.
Directed graphs in tournaments can link or knot in various ways.
problem Understanding intrinsic linking and knotting in directed graphs, specifically tournaments.
method Analyzing the minimum number of vertices required for intrinsic linking or knotting properties in tournaments.
result Bounds on the minimum number of vertices required for various intrinsic linking or knotting properties in tournaments.
The paper extends quantum link invariants to rotational virtual knots and links.
problem Understanding quantum invariants for rotational virtual knots and links.
method Established background virtual knot theory, defined rotational virtual knots, and introduced extensions of quantum link invariants.
result A non-trivial link L is not detected by any quantum invariant formulated in this paper.
Stable-ribbon surface-links have unique handle-irreducible summands.
problem Characterize handle-irreducible summands of stable-ribbon surface-links.
method Combining old results on stably trivial surface-links and surface-knots with infinite cyclic fundamental groups.
result Every handle-irreducible summand of a stably trivial surface-link is a trivial 2-link.
Invariants for singular knots and links using non-commutative cocycles.
problem Creating invariants for singular knots and links.
method Defining universal group and functions for non-commutative cocycles, and computing examples.
result Invariants for singular knots and links defined using non-commutative cocycles.
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
The paper simplifies knot and link diagrams with triple-crossings.
problem Generating and classifying minimal triple-crossing knot and link diagrams.
method Systematic method to generate minimal triple-crossing projections, introducing new diagrammatic moves.
result Classification of knots and links with triple-crossing number up to five, derivation of minimal generating set of moves.
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 …
New lattice stick knot condition identified.
problem Determining if 2D lattice knots project to 3D lattice sticks.
method Provided a necessary and sufficient condition.
result Identified a condition for 2D lattice knots to project to 3D lattice sticks.
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.
Paper discusses a new link invariant from virtual link theory.
problem Developing a new invariant for twisted links.
method Using JKSS invariant of virtual links and double coverings.
result Derived a new invariant for twisted links.
Electromagnetic fields can be knotted and stable, linked to quasipositive links.
problem Understanding the topology of electromagnetic fields and their stability.
method Constructing complex algebraic plane curves to create quasipositive links containing the original link.
result Electromagnetic fields can be knotted and stable, corresponding to Legendrian knots.
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.
Virtual knot invariants are linked to link invariants in 3D space.
problem Relating virtual knot invariants to link invariants in 3D space.
method Virtual covers and fiber stabilization techniques.
result Alexander polynomials of almost classical knots are specializations of multi-variable Alexander polynomials of certain links.
Study on finite n-quandles of knots and links, computing graphs and groups.
problem Understanding finite n-quandles of specific knot types.
method Computed Cayley graphs and automorphism groups.
result Detailed data for two-bridge and torus knots and links.
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 …
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. Delta Diagrams show any knot or link can be represented with 3-5 sided regions.
problem Representing knots and links with specific geometric constraints.
method Defined Delta Diagrams and proved any knot/link can be represented.
result Proved any knot/link can be represented by Delta Diagrams.
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…
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.
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.
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.
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.