Study concordance of alternating torus knots to L-space knots.
problem When are linear combinations of alternating torus knots concordant to L-space knots?
method Proved Allen's conjecture for alternating torus knots and established a necessary condition.
result Linear combinations of alternating torus knots are concordant to L-space knots if and only if they are a single torus knot.
New hyperbolic knots not concordant to algebraic ones found.
problem Identifying knots not concordant to algebraic knots.
method Constructing hyperbolic L-space knots.
result Found hyperbolic knots that are not concordant to algebraic knots.
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.
Classifies 3-braids as L-space knots.
problem Identifying L-space knots among 3-braids.
method Classification based on L-space properties.
result Closed 3-braids classified as L-space knots.
New knots found that can't be turned into L-space after surgery.
problem Finding knots that can't be transformed into L-space after surgery.
method Examined algebraically concordant torus knots and fibred knots.
result Infinitely many knots of genus g≥2 cannot be turned into L-space. A knot in the 3-sphere is called an L--space knot if it admits a nontrivial Dehn surgery yielding an L--space. Like torus knots and Berge knots, many L--space knots admit also a Seifert fibered surgery. We give a concrete example of a hyperbolic, L-space knot which has no exceptional surgeries, in particular, no Seifer…
Study shows some knots can't be concordant to L-space knots.
problem Identifying knots that cannot be concordant to L-space knots.
method Used methods by Friedl, Livingston, and Zentner.
result Found knots that are not algebraically concordant to L-space knots.
The study confirms conjectures about slopes of knots using knot Floer homology.
problem Verifying conjectures about non-integer characterizing slopes of knots.
method Using knot Floer homology, the study verifies conjectures for specific classes of knots.
result Almost all slopes are characterizing for many knots, and infinitely many for L-space knots. Paper classifies CFK∞ type of almost L-space knots.
problem Understanding almost L-space knots and their properties.
method Utilizes Heegaard Floer homology and knot Floer homology.
result Classifies CFK∞ type of almost L-space knots. Paper constructs infinitely many non-braid positive hyperbolic L-space knots.
problem Proving the non-braid positivity of hyperbolic L-space knots.
method Constructs infinitely many hyperbolic L-space knots that are not braid positive.
result Distinct examples from Baker and Kegel's constructions.
Characterizes Legendrian knots in lens spaces.
problem Classifying Legendrian knots in lens spaces.
method Splitting lens spaces and using convex Heegaard decomposition.
result All Legendrian torus knots in universally tight lens spaces are classified.
Two knots with unique surgery properties.
problem Characterizing strongly invertible L-space knots.
method Examined surgeries and knot properties.
result Found knots whose surgeries are never Khovanov thin.
It is known that connected sums of positive torus knots are not concordant to L-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial L-space knots other than the torus knots themselves…
Study shows space writhe closely correlates with knot signature in polymers.
problem Understanding the relationship between space writhe and knot signatures in knotted polymers.
method Performed Langevin dynamics simulations of knotted polymers to measure space writhe.
result Space writhe is strongly correlated with knot signature in complex knots.
A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
The paper classifies a special family of knots in lens spaces using knot Floer homology.
problem Classifying constrained knots in lens spaces.
method Parameterization by five integers, characterization via spinc structures, and knot Floer homology calculations. result Complete classification of constrained knots based on knot Floer homology.
New knots share same Upsilon invariant despite different Alexander polynomials.
problem Identifying concordant knots via Upsilon invariant.
method Examined hyperbolic L-space knots and their Upsilon invariants.
result Infinitely many pairs of hyperbolic L-space knots with distinct Alexander polynomials share the same Upsilon invariant.
The study examines quasi-alternating surgeries on knots and their properties.
problem Understanding quasi-alternating surgeries and their relation to L-space knots.
method Exploration of quasi-alternating surgeries, classification of knots, and analysis of slopes.
result All SnapPy census L-space knots except two admit quasi-alternating surgeries.
We find braid positive presentations for most L-space knots, except one, and explore related knot properties.
problem Characterizing L-space knots and their braid positivity.
method Examined SnapPy census knots, used HOMFLY polynomial analysis, and investigated Alexander polynomials.
result One L-space knot is not braid positive, and a 1-parameter family of hyperbolic L-space knots might not be.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight …
New infinite family of hyperbolic L-space knots with specific semigroups.
problem Characterizing semigroups of L-space knots.
method Defined formal semigroups from Alexander polynomials and analyzed hyperbolic knots.
result Found an infinite family of hyperbolic L-space knots with semigroups generated by five elements.
Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots …
By a fixed continuous map from a 3-space to itself, a knot in the 3-space may be mapped to another knot in the 3-space. We analyze possible knot types of them. Then we map a knot repeatedly by a fixed continuous map and analyze possible infinite sequences of knot types.
If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.
The study connects twist positivity to L-space knots and concordance.
problem Understanding the relationship between twist positivity and knot concordance.
method Analyzing Alexander polynomials and braid indices of knots.
result For twist positive L-space knots, the braid index equals the bridge index.
Expanding on prime knots with 6 or less mosaic tiles, this paper analyzes those with 7 tiles.
problem Determining the tile number and space-efficiency for prime knots with mosaic number 7.
method Extending the methods of Heap and Knowles (2017) to include prime knots with mosaic number 7.
result Identifying the possible tile numbers and space-efficient layouts for all prime knots with mosaic number 7.
Study on pretzel knots showing cyclic branched covers are L-spaces.
problem Understanding cyclic branched covers of pretzel knots and their properties.
method Analyzing pretzel knots Kk and their n-fold cyclic branched covers for all n≥1. result The n-fold cyclic branched covers of pretzel knots Kk are L-spaces for all n≥1. Study shows surgeries on certain knots yield left-orderable 3-manifolds.
problem Integral Dehn surgeries on genus one fibered knots in lens spaces.
method Classification of genus one fibered knots by Baker and analysis of surgeries.
result All integral Dehn surgeries on these knots yield left-orderable 3-manifolds.
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.
L-space knots lack essential Conway spheres, proven with Floer theory.
problem Essential Conway spheres in L-space knots.
method Floer theoretic invariant for tangles.
result L-space knots have no essential Conway spheres.
A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient…
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.
New examples of non-simple knots in Lens spaces show rich botany.
problem Understanding the variety of Legendrian and transversal knots in Lens spaces.
method Presented new families of non-simple knots in tight Lens spaces.
result More non-isotopic Legendrians topologically isotopic to the n-twist knot in Lens spaces than in S3. The paper classifies knots in real projective 3-space and introduces new geometric tools.
problem Classifying knots in real projective 3-space and understanding their properties.
method Structural theorem, space bending surgery, genus definition, non-cancellation theorem.
result The genus detects knottedness and classifies knots in real projective 3-space.
We consider the following question: when is the manifold obtained by gluing together two knot complements an L-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an L-space. We extend this result to allow for arbitrary integer framings. We find that splicing two in…
Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either ±1 or 0 and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's p…
New description of (1,1) L-space knots leads to non-left-orderable surgeries.
problem Characterizing and understanding (1,1) L-space knots. method Analyzing coherent reduced (1,1)-diagrams to describe (1,1) L-space knots and proving non-left-orderable fundamental groups. result Any L-space obtained by Dehn surgery on a (1,1)-knot in S3 has a non-left-orderable fundamental group. It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parame…
New invariant for surface-knots in 4D space.
problem Invariants for surface-knots in 4D space.
method Diagrammatic approach in 3D space.
result Invariant constructed for surface-knots.
Defines a measure of knot concordance using cobordism distance.
problem Measuring how close knots are to being linearly dependent.
method Cobordism distance on cyclic subgroups of knot concordance group.
result Projective space of knot concordance group with integer-valued metric.
This paper uses sheaf theory to model virtual knots geometrically.
problem Defining and understanding virtual knots in a geometric framework.
method Sheaf theory applied to virtual knot diagrams to model them geometrically.
result A geometric model for virtual knots formalizes the intuitive notion of knots in a variable ambient space.
Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.
problem Contact cosmetic surgeries for Legendrian knots in L-spaces.
method Adapting techniques from S3 to L-spaces, incorporating Heegaard Floer theory constraints.
result Contact cosmetic surgery conjecture holds for non-trivial Legendrian knots, except for Lagrangian slice knots.
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…
Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a continuation of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens spac…
New definition of knot invariants for higher-dimensional knots.
problem Defining invariants for long knots in high-dimensional spaces.
method Flexible definition as counts of diagrams, extending to asymptotic homology.
result Generalized invariants for long knots in more general spaces.
The paper identifies knots in specific lens spaces based on their complements.
problem Determining knots in homology lens spaces based on their complements.
method Analyzing knots in homology lens spaces with specific conditions.
result Knots are determined by their complements under certain conditions.
The article confirms two quasi-alternating surgeries for 9 asymmetric L-space knots.
problem Understanding quasi-alternating surgeries on asymmetric L-space knots.
method Using the Montesinos trick to confirm known surgeries.
result Confirmation of two quasi-alternating surgeries for each of 9 asymmetric L-space knots.