Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.
problem Understanding gaps in genus between strongly invertible knots and their invariant Seifert surfaces.
method Analysis of invariant Seifert surfaces and proof of genus gaps, with variants of Edmonds' theorem.
result Gap between equivariant genus and usual genus can be arbitrarily large for strongly invertible knots.
Table of symmetric diagrams for knots up to 10 crossings.
problem Finding symmetric diagrams for strongly invertible knots.
method Compilation of symmetric diagrams for knots up to 10 crossings.
result Similarity of transversal diagrams to symmetric union diagrams for strongly invertible knots.
Defines knot signature invariant using G-signature theorem.
problem No specific problem stated; focuses on knot theory.
method Uses G-signature theorem to define knot invariant.
result Defines an invariant for strongly invertible knots.
New findings on knot genera using advanced techniques.
problem Understanding the 4-genus of knots, especially strongly invertible and periodic ones.
method Innovative concordance group invariants, Donaldson's theorem, and g-signature.
result Many new examples showing the equivariant 4-genus is larger than the 4-genus.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
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.
New findings on knot operations challenge a long-standing conjecture.
problem Understanding equivariant unknotting numbers of strongly invertible knots.
method Study of symmetric crossing change operations for strongly invertible knots.
result The equivariant unknotting number is not additive under connected sum.
Study on equivariant Q-sliceness for strongly invertible knots.
problem Understanding Q-sliceness for strongly invertible knots.
method Constructive and obstructive approaches using Fox-Milnor condition and equivariant concordance.
result Klein amphichiral knots are equivariant Q-slice in a single Q-homology 4-ball.
Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
Study of strongly invertible Legendrian links in contact 3-space.
problem Characterizing and understanding strongly invertible Legendrian links.
method Equivariant analogs of basic results for strongly invertible and Legendrian links.
result Existence of maximal equivariant Thurston-Bennequin number for strongly invertible links.
The paper calculates the equivariant genus for a specific type of knot.
problem Calculating the equivariant genus of marked strongly invertible knots associated with 2-bridge knots.
method Analyzing invariant Seifert surfaces for marked strongly invertible knots.
result The paper completely determines the equivariant genus for every marked strongly invertible knot with K a 2-bridge knot. Paper finds first infinite family of hyperbolic knots with specific properties.
problem Identifying new hyperbolic knots with specific properties.
method Examined SnapPy census and used knot theory to find new infinite family.
result First infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two.
New bounds on knot unknotting numbers using involutive homology.
problem Bounding the unknotting number of strongly invertible knots.
method Using involutive Bar-Natan homology to establish bounds.
result Identified knots with strict inequality between standard and equivariant unknotting numbers.
We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.
Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.
problem Equivariant algebraic concordance of strongly invertible knots.
method Defining a homomorphism Φ from equivariant concordance group to a new equivariant algebraic concordance group, proving it lifts known homomorphisms and provides new obstructions. result Obtains a new obstruction to equivariant sliceness and novel lower bounds on equivariant slice genus.
Study on knots, genera, and algebraic concordance groups.
problem Understanding the equivariant slice genus of strongly invertible knots.
method Using the Blanchfield form to establish lower bounds and formulate an equivariant algebraic concordance group.
result The equivariant slice genus of an equivariant connected sum of a genus one strongly invertible slice knot is at least n/4.
Proves cosmetic surgery conjecture for strongly invertible knots.
problem Cosmetic surgery conjecture for strongly invertible knots.
method Combines recent results with Khovanov multicurve invariants and Conway tangles.
result Proves equivariant version of the Cosmetic Surgery Conjecture.
The paper defines and calculates an upper bound for the equivariant crossing number of two-bridge knots.
problem Finding the minimum number of crossings in symmetric diagrams for two-bridge knots.
method Defining and calculating c2(K) for two-bridge knots by restricting diagrams to two types. result An algorithm to determine c2(K) for any two-bridge knot and results up to 14 crossings. A construction of a spatial graph from a strongly invertible knot was developed by the second author, and a necessary and sufficient condition for the given spatial graph to be hyperbolic was provided as well. The condition is improved in this paper. This enable us to show that certain classes of knots can yield hyperb…
Study on 2-bridge knots, proving equivariant concordance order is infinite.
problem Equivariant concordance of 2-bridge knots.
method Formula for butterfly polynomial, two proofs of non-equivariant sliceness, new invariant for strongly invertible knots.
result Equivariant concordance order of 2-bridge knots is infinite.
Proves non-solvability of concordance groups using Milnor invariants.
problem Non-solvability of concordance groups of 2-string links and strongly invertible knots.
method Using Milnor invariants to prove non-solvability.
result Proves non-solvability of C(2) and equivariant concordance groups of strongly invertible knots. We construct the first examples of asymmetric L-space knots in S3. More specifically, we exhibit a construction of hyperbolic knots in S3 with both (i) a surgery that may be realized as a surgery on a strongly invertible link such that the result of the surgery is the double branched cover of an alternating link …
Defines real link Floer homology for specific types of links.
problem Developing a new homology theory for certain types of links.
method Combining real Heegaard Floer homology and real sutured Heegaard Floer homology, using real grid diagrams in S3. result Observes structural and property properties of strongly invertible knots.
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial.…
We give a new criterion for a given knot to be a Montesinos knot by using the Rasmussen invariant and the signature. We apply the criterion to study Seifert fibered surgery on a strongly invertible knot, and show that a (p,q,q)-pretzel knot with integers p,q≥2 admits no Seifert fibered surgery.
A new knot invariant measures crossings in three orthogonal directions.
problem Defining a new knot invariant for certain knot diagrams.
method Defining the simultaneous crossing number for knots with doubly transvergent diagrams.
result The limit of the ratio of the new invariant to the usual crossing number is at most 8.
Specialized knot theory theorems for strongly involutive links.
problem Classical Alexander and Markov theorems for links.
method Equivariant closure map for strongly involutive links.
result Surjective equivariant closure map up to equivalence of strongly involutive links.
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertib…
A bound on knot unknotting using equivariant signature.
problem Equivariant unknotting of knots.
method Analysis of strongly invertible knots and application of equivariant unknotting moves.
result The equivariant signature provides a lower bound for the equivariant unknotting number.
New method uses Floer homology to create exotic disk pairs.
problem Creating exotic disk pairs in 4-manifolds.
method Singular instanton Floer homology with Chern--Simons filtration.
result Constructs exotic pairs of slice disks and knots.
A new polynomial invariant for strongly involutive links.
problem Characterizing strongly involutive links using polynomial invariants.
method Introducing a two-variable polynomial invariant \(P^e\) with equivariant skein relations.
result Specialisation of \(P^e\) recovers the graded Euler characteristic of a spectral sequence.
We refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply-graded theory, arising from a pair of filtrations. We focus primarily on strongly invertible knots and show, for instance, that this refinement is able to detect mutation.
New invariants prove exotic slice disks for knots.
problem Proving exotic slice disks for knots.
method Involutive Khovanov homology and derived invariants.
result Reprove exotic slice disks for knots Jn. New symplectic annular Khovanov homology connects knot theory to Floer homology.
problem Understanding the relationship between knot theory and Floer homology.
method Introducing a new version of symplectic annular Khovanov homology and establishing spectral sequences.
result Established spectral sequences linking different knot homologies.
Flip symmetry on knot diagrams affects Khovanov homology.
problem Understanding the flip map on Khovanov homology.
method Analyzing the behavior of the flip map on unlinks and using it to determine the involution.
result The flip map is the identity map over \(\mathbb{F}_2\), confirming a conjecture.
Classifies tight contact structures with special symmetries.
problem Classifying tight contact structures with specific symmetries.
method Proves classification results for tight contact structures in 3-space, ball, and sphere with a new integral torsion.
result New integral torsion dictates a splitting between equivalence classes.
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
We show that quasi-alternating links arise naturally when considering surgery on a strongly invertible L-space knot (that is, a knot that yields an L-space for some Dehn surgery). In particular, we show that for many known classes of L-space knots, every sufficiently large surgery may be realized as the two-fold branch…
Study equivariant 4-genus of knots in symmetric 4-manifolds.
problem Understanding equivariant 4-genus of knots in symmetric 4-manifolds.
method Developed techniques for constructing slice disks via equivariant tubing construction.
result Equivariant 4-genus can differ from standard and equivariant 4-genus of 4-manifolds.
We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic …
For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…
The study calculates and analyzes alternating surgeries for various knots.
problem Identifying and understanding alternating surgeries on knots.
method Algorithmic computation and structural analysis of alternating surgery slopes.
result The set of alternating surgery slopes is algorithmically computable and exhibits interesting phenomena.
Operator on tangles derived from knot 2-cabling.
problem Understanding tangle operators induced by knot 2-cabling.
method Induced operator on 4-ended tangles via 2-cabling of a knot, passing to Khovanov theory.
result Full description of operator's restriction to cap-trivial tangles, inspired by concordance invariants.
New invariants refine link homology, showing large genus differences.
problem Understanding genus differences in equivariant cobordisms.
method Refined Bar-Natan homology for involutive links, constructing new numerical invariants.
result Difference between equivariant and isotopy-equivariant slice genera can be arbitrarily large.
A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…
Proves spectral sequence for real Heegaard Floer homology.
problem Real Heegaard Floer homology and its localization.
method Proves existence of a localization spectral sequence.
result Existence of spectral sequence for real Heegaard Floer hat variant.
The paper defines new knot genera and finds bounds for stabilization distances.
problem Finding bounds for stabilization distances of symmetric surfaces.
method Defining new knot genera and using them to find bounds.
result Constructs unknotted symmetric 2-spheres without symmetric 3-ball bounds.