Study calculates instanton Floer homology for surgeries on L-space knots.
problem Computing the framed instanton Floer homology of surgeries on L-space knots.
method Used Baldwin-Sivek contact invariant and proved homogeneity of the invariant.
result Framed instanton Floer homology agrees with Heegaard Floer homology for L-space knots.
New proof shows L-space knots are fibered and have specific properties.
problem Proving L-space knots are fibered and have strong quasipositive properties.
method Conceptually simpler proof using Heegaard Floer homology, translated to instanton Floer homology.
result Generalization of L-space knot properties to other Floer homologies.
New proof shows certain 3D shapes can't be instanton L-spaces.
problem Proving certain 3D shapes cannot be instanton L-spaces.
method Proving by gluing complements of knots and analyzing their fundamental groups.
result Proven that the resulting 3-manifold cannot be an instanton L-space.
Computes upper bounds for instanton knot homology.
problem Computing the exact dimension of instanton knot homology.
method Developed an algorithm using knot diagrams and Heegaard diagrams.
result Algorithm provides sharp bounds for knots up to seven crossings.
New invariants from framed instanton homology for knot concordance.
problem Concordance of knots and their properties.
method Framed instanton homology to define invariants.
result Computations and bounds on knot concordance invariants.
Instantons prove knots are fibered and strongly quasipositive.
problem Proving properties of knots using instantons and L-space surgeries.
method New cobordism maps in framed instanton Floer homology.
result Proves fibered and strongly quasipositive knots have specific homomorphisms.
Enhances knot surgery formulae for instanton Floer homology.
problem Calculating instanton Floer homology for various 3-manifolds.
method Integral surgery formulae and rational surgery formulae for null-homologous knots.
result Characterizes instanton Floer homology of specific 3-manifolds and knots.
The paper examines 2-torsion in instanton Floer homology for knots and 3-manifolds.
problem Existence of 2-torsion in instanton Floer homology for knots and 3-manifolds.
method Analyzes framed and singular instanton Floer homology, uses lemmas from Heegaard Floer theory.
result Shows existence of 2-torsion in instanton Floer homology for various knots and 3-manifolds.
New insights into knot surgeries via instanton 2-torsion.
problem Understanding rational surgeries on knots and their implications.
method Extending earlier results on integral surgeries to rational surgeries using framed instanton homology.
result If framed instanton homology is 2-torsion-free, the knot is an instanton L-space knot and the surgery parameter is greater than 2g(K)-1.
Proves exact triangle linking knot instanton Floer homology to surgeries.
problem Relating knot instanton Floer homology to surgeries.
method Proves an exact triangle using integer coefficients.
result Poincaré Homology Sphere is not an instanton L-space with integer coefficients.
This work extends knot homology theory to links, proving exact triangles and categorifying link signatures.
problem Extending knot homology theory to links and proving exact triangles.
method Equivariant singular instanton Floer theory, circle-equivariant Morse-Floer theory, cobordism constructions.
result Established unoriented skein exact triangles and categorified link signatures.
Enhanced Euler characteristic improves knot homology detection.
problem Detecting knots in instanton homology.
method Decomposing sutured instanton homology with torsion.
result Enhanced Euler characteristic equals SFH Euler characteristic.
Sutured instanton homology uses Heegaard diagrams to bound homology dimensions.
problem Bounding the dimension of sutured instanton homology.
method Using admissible Heegaard diagrams, proving a bound on the number of generators of the associated sutured Heegaard Floer complex.
result Strong L-spaces are instanton L-spaces.
We study knots in S3 with infinitely many SU(2)-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into SU(2) has cyclic image. We show that for every such nontrivial knot K, its set of SU(2)-cyclic slopes is bounded and has a unique limit point, whic…
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.
(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.
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.
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.
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 3D shapes can't fit in certain 4D spaces.
problem Proving 3-manifolds can't embed in symplectic 4-manifolds.
method Constructing L-spaces and using Heegaard Floer correction terms, instanton moduli spaces.
result Infinitely many 3-manifolds cannot embed in symplectic 4-manifolds.
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.
We call a knot in the 3-sphere SU(2)-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in SU(2) are binary dihedral. This is a generalisation of being a 2-bridge knot. Pretzel knots with bridge number ≥3 are not SU(2)-simple. We provide an …
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. 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.
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 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.
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.
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.
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 …
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.
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 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.
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 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.
We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns which produce satellite L-space knots.
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. 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.
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 …
Given an L-space knot we show that its Upsilon function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Upsilon function is, in the case of L-space knots, contained in the d-invariants of large surgeries. Generalizations…
In this paper we look at the knot complement problem for L-space Z-homology spheres. We show that an L-space Z-homology sphere Y cannot be obtained as a non-trivial surgery along a knot K⊂Y. As a consequence, we prove that knots in an L-space Z-homology sphere are determined …
For any alternating knot, it is known that the double branched cover of the 3-sphere branched over the knot is an L-space. We show that the three-fold cyclic branched cover is also an L-space for any genus one alternating knot.
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…
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…
Formula for τ-invariant of satellite knots derived from L-space satellite operators.
problem Calculating the τ-invariant of satellite knots using L-space satellite operators.
method Algorithm to compute knot Floer complexes and formula for τ-invariant.
result Formula recovers existing formulas and proves new properties of τ-invariant.
We show there exist infinitely many knots of every fixed genus g≥2 which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot T(2,2g+1) of the same genus and they are fibred and strongly quasipositive.
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. If a knot is a nontrivial connected sum of positive torus knots, then it is not concordant to an L-space knot.
Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be cla…