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.
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.
Classifies reversible and strongly reversible elements in quaternionic groups.
problem Classifying reversible and strongly reversible elements in quaternionic groups.
method Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
result Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
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 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.
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.
Classifies involutions on alternating prime non-split links.
problem Classifying involutions on alternating prime non-split links.
method Using the category of flypes between reduced alternating diagrams.
result Quotient of an alternating periodic link is alternating; freely 2-periodic alternating links have an even number of components.
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.
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 paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
The paper classifies 10 antipodal pairings of self-dual maps.
problem Understanding the antipodal pairings of strongly involutive polyhedra.
method Classification of self-dual pairings and construction of polyhedra.
result Determination of 10 antipodal pairings among 24 self-dual pairings.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
Study involutions on 3D small covers, proving quotient spaces are linked 2-spheres.
problem Understanding quotient spaces of 3D small covers by involutions.
method Analyzing quotient spaces and using topological properties.
result Quotient spaces of 3D small covers by involutions are linked 2-spheres.
Study of equivariant movie moves for involutive links.
problem Equivariant cobordisms between involutive links.
method Equivariant Morse theory and singularity theory.
result 39 equivariant movie moves for isotopic cobordisms.
Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.
This paper classifies reversible and strongly reversible elements in affine groups.
problem Classifying reversible and strongly reversible elements in affine groups.
method Identifying affine transformations and using conjugacy by involutions.
result Classification of reversible and strongly reversible elements in affine groups.
New real invariants for 3-manifolds and links.
problem Defining real structures in Floer homology.
method Introducing real spin-c structures and applying monopole Floer homology.
result Invariants for links via double branched covers.
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.
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…
New invariants detect corks obstructing homology ball extensions.
problem Detecting corks obstructing homology ball extensions.
method Adapting Heegaard Floer homology formalism for local equivalence, using monotonicity and surgery.
result Established new families of corks and proved various examples are strongly non-extendable.
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. We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our …
Pairs of elements in quaternionic hyperbolic space have zero measure of being strongly doubly reversible.
problem Characterizing pairs of elements in quaternionic hyperbolic space that are strongly doubly reversible.
method Analyzing conjugacy conditions and using Haar measure.
result The set of strongly doubly reversible pairs has Haar measure zero in $\PSp(n,1) imes \PSp(n,1)$.
New spin on Khovanov-Rozansky homology categorifies spin link polynomial.
problem Categorify spin link polynomial using colored Khovanov-Rozansky homology.
method Equip Λ^n-colored sl_{2n} Khovanov-Rozansky homology with an involution.
result Categorifies spin-colored so_{2n+1} quantum link polynomial for n=1,2,3.
Links can be transformed into many others using a specific operation.
problem Understanding the relationship between strongly quasipositive links and their concordance.
method Used a satellite operation with a slice knot to transform links.
result Strongly quasipositive links can be transformed into infinitely many other links.
The paper classifies involutions on S^4, proving linearities under certain conditions.
problem Classifying involutions on S^4 with specific fixed-point sets.
method Combining surgery theory, Schoenflies theorem, and equivariant topology.
result Linear involutions on S^4 with 1-dimensional fixed-point sets are proven.
Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.
problem Characterizing the homeomorphism group of telescoping 2-manifolds.
method Introduced telescoping 2-manifolds, studied homeomorphism groups, and used commutator subgroup properties.
result Homeomorphism group of telescoping 2-manifolds is strongly distorted.
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.
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.
Study non-fibered links' relation to tight contact structures.
problem Understanding non-fibered links and their tight contact structures.
method Analyze non-fibered links with induced partial open books and contact structures.
result Strongly quasipositive non-fibered links induce tight contact structures, but the converse is not always true.
Proves alternating quasipositive links are positive and strongly quasipositive.
problem Characterizing alternating quasipositive links.
method Analyzing alternating diagrams with specific crossing conditions.
result Alternating quasipositive links are positive and strongly quasipositive.
The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if L is an oriented quasi-alternating link with a quasi-alternatin…
The study establishes a link between fibered links and their concordance invariants.
problem Determining the conditions for fibered strongly quasi-positive links.
method Proved that an n-component fibered link L is strongly quasi-positive if and only if τ(L)=g3(L)+n−1. result Explicitly determined fibered prime links with at most 9 crossings and their properties.
The paper explores fibered and quasi-positive links, introducing new families and invariants.
problem Understanding the structure of L-space links and their properties. method Using the H-function as a concordance link invariant.
result Introduced a subfamily of fibered strongly quasi-positive L-space links and an infinite family of non-quasi-positive L-space links. Characterizes a subset of links using quasipositive and homogeneous properties.
problem Understanding the properties of T-positive links.
method Characterization through strongly quasipositive and T-homogeneous braids.
result T-positive links are precisely the strongly quasipositive links that are closures of T-homogeneous braids.
We generalize an algorithm of Rudolph to establish that every link is topologically concordant to a strongly quasipositive link.
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 show that a link in an open book can be realized as a strongly quasipositive braid if and only if it bounds a Legendrian ribbon with respect to the associated contact structure. This generalizes a result due to Baader and Ishikawa for links in the three-sphere. We highlight some related techniques for determining wh…
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.
Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case it has positive Conway polynomial (that is, all its coefficients…
Study d-invariants of L-space double branched covers of arborescent links.
problem Determine d-invariants of double branched covers of arborescent links. method Use immersed curves description of bordered Floer homology and involution of curves.
result Spin d-invariants of Σ2(L) are determined by the signatures of L. The paper classifies reversible and strongly reversible elements in Hermitian isometry groups.
problem Classifying reversible and strongly reversible elements in Hermitian isometry groups.
method Classification through group theory and algebraic manipulation.
result New classification of strongly reversible elements in Sp(n).
We classify isotopy classes of irreducible Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2 matrix with 0 in the lower right hand corner (as always is true when the absolute value of the trace is 3), then any irreducible splitting is strongly irreducible and of genu…
We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if δn=σ1σ2…σn−1 is the dual …
The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.
Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover π:M \to N we obtain the following bound on the Heegaard genus of N: g(N) \leq 4g…
Study shows how knot Floer homology and bordered Floer theory are linked.
problem Understanding the relationship between knot Floer homology and bordered Floer theory.
method Proves local equivalence and establishes algebraic formulas.
result Involutive knot Floer homology and bordered Floer theory determine each other under certain conditions.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.