Paper discusses when virtual links are equivalent as twisted links.
problem Determining equivalence of virtual links as twisted links.
method Using stable equivalence classes of links in oriented thickenings of surfaces.
result Necessary and sufficient condition for virtual links to be equivalent as twisted links.
The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.
problem Understanding and distinguishing twisted links.
method Twisted intersection colorings and double coverings.
result There exist infinitely many pairs of twisted links with equivalent double coverings.
New method distinguishes twist equivalence of Seifert surfaces.
problem Distinguishing twist equivalence classes of Seifert surfaces.
method Using bridge spheres and operations on Seifert surfaces.
result Bridge numbers of twist equivalent Seifert surfaces are uniformly bounded.
We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert s…
Paper gives a full-twist inequality for a knot concordance invariant ν^+.
problem Understanding the behavior of the ν^+ invariant under knot cobordisms.
method Developed a full-twist inequality for the ν^+ invariant and extended Wu's cabling formula.
result Extended Wu's cabling formula to all cables and introduced a partial order on ν^+ equivalence classes.
We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of a…
The paper distinguishes non-equivalent branched twist spins using knot determinants.
problem Distinguishing non-equivalent branched twist spins.
method Presented a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinants. Calculated the first elementary ideals and obtained the condition of the knot determinants by substituting -1 for the indeterminate.
result A sufficient condition to distinguish non-equivalent, non-trivial branched twist spins using knot determinants.
Study of Alexander modules for hyperplane arrangements, distinguishing complements.
problem Distinguishing homotopy equivalent but non-homeomorphic hyperplane arrangement complements.
method Analysis of twisted Alexander modules and polynomials.
result Distinguish non-homeomorphic homotopy equivalent arrangement complements.
Homotopy equivalence between formalities with different covariant derivatives.
problem Formality of Dolgushev depends on covariant derivative choice.
method Proved homotopy equivalence of L∞-morphisms twisted by gauge equivalent elements. result Globalized formalities with different covariant derivatives are homotopic.
Proves volume conjecture for twist knots using complex analysis.
problem Volume conjecture for twist knots.
method Equivalence relation, complex analysis, analytic continuation, function of several complex variables.
result Proves volume conjecture for twist knots.
Defines and calculates signature invariants for twisted linking forms.
problem Studying twisted linking forms of knots and three-manifolds.
method Describes how to define and calculate signature invariants associated to a linking form MimesMoF(t)/F[t±1] for F=R,C, where M is a torsion F[t±1]-module. result Classifies such linking forms up to isometry and Witt equivalence and studies their representability by matrices.
The paper shows that certain geometric structures remain unchanged under specific twists.
problem The rational Beauville-Bogomolov-Fujiki lattices of related fibrations are similar.
method Analytic and étale topologies, Hodge structures, and degenerate twistor deformations.
result Isomorphisms of graded vector spaces and Hodge-similar lattices.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
Study shows equivalence between cohomology class existence and polynomial properties for 3D manifolds.
problem Characterizing closed 3D manifolds based on cohomology class existence and polynomial properties.
method Analyzes closed one-forms and twisted Alexander polynomials in relation to cohomology classes.
result Equivalence between cohomology class existence and polynomial properties for most 3D manifolds.
Knots in 3-manifolds are equivalent if isotopic, except in special cases.
problem Understanding when knots in 3-manifolds are equivalent and isotopic.
method Analyzing prime, closed, oriented 3-manifolds and irreducible manifolds, and considering orientation-preserving mapping class groups and homeomorphisms.
result Knots in prime, closed, oriented 3-manifolds are isotopic if and only if the orientation preserving mapping class group is trivial.
We describe a bordered version of totally twisted Khovanov homology. We first twist Roberts's type D structure by adding a "vertical" type D structure which generalizes the vertical map in twisted tangle homology. One of the distinct advantages of our type D structure is that it is homotopy equivalent to a type $…
Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.
problem Understanding and manipulating pseudo-Anosov flows.
method Performing horizontal surgery on pseudo-Anosov flows by cutting along specific annuli and regluing with a Dehn twist.
result Horizontal Goodman surgery on transitive pseudo-Anosov flows yields an almost equivalent flow.
Gluck twists on spheres yield equivalent 4-manifolds under certain conditions.
problem Understanding when Gluck twists on spheres result in homeomorphic or simple homotopy equivalent manifolds.
method Analyzing the Gluck twist operation on 4-manifolds with spheres of different relationships (concordant, homotopic) and examining their resulting manifolds.
result Gluck twists on concordant or homotopic spheres yield equivalent 4-manifolds under specific conditions.
Explains connections between monopoles and modules on elliptic curves.
problem Understanding relationships between different mathematical objects.
method Explains equivalences between monopoles and polystable bundles and modules.
result Monopoles and polystable difference modules on elliptic curves are equivalent.
In this paper, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant algebroids by closed 4-forms introduced in \cite{4form}. We also prove that Dirac struct…
Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the o…
Study of Dehn twists in free groups generates right-angled Artin groups.
problem Understanding dynamics of Dehn twists in free groups.
method Geometry of spheres, tori, and curves in a doubled handlebody; analysis of compatibility conditions.
result Sufficiently large powers of Dehn twists generate right-angled Artin groups.
The paper classifies hyperbolic and satellite T-links formed by twisting.
problem Classifying hyperbolic and satellite T-links formed by twisting.
method Classification through Dehn filling and analysis of parent links.
result Classification of hyperbolic and satellite T-links formed by twisting.
We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called "projective spectral triple" is Morita equivalent to the well-known commutative spin spectral triple provided that the m…
Solves generalized twisted rabbit problems for higher degree polynomials.
problem When a quadratic polynomial is twisted by a cyclic subgroup, what polynomial is equivalent?
method Uses d2-adic expansion instead of 4-adic for higher degree polynomials. result Provides a solution that depends on the d2-adic expansion of the power of the mapping class element. Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
Minimal charts of type (3,3) are equivalent to a specific subchart.
problem Characterizing minimal charts of type (3,3).
method Analyzing subgraphs and C-move equivalence.
result Minimal charts of type (3,3) are equivalent to a specific subchart.
Two special moves relate any two open book decompositions of a 3-manifold.
problem Relating any two open book decompositions of a 3-manifold.
method Using 4-dimensional Lefschetz fibrations and 3-dimensional contact topology, including Harer's twisting and Giroux-Goodman stable equivalence theorem.
result A complete set of two moves suffices to relate any two open book decompositions of a given 3-manifold.
Homology 3-spheres are shown to be equivalent through specific twists.
problem Proving all homology 3-spheres are J4-equivalent. method Exhibiting an element of J4 acting trivially on the fourth nilpotent quotient of the fundamental group. result All homology 3-spheres are J4-equivalent, providing an explicit example of a non-commutator element in J4. We study open books (or open book decompositions) of a closed oriented 3-manifold which support overtwisted contact structures. We focus on a simple closed curve along which one can perform Stallings twist, called ``twisting loop''. We show that the existence of a twisting loop on the fiber surface of an open book is e…
New proof shows coherent sheaves quasi-equivalent to constructible sheaves on torus.
problem Establishing equivalence between coherent and constructible sheaves on toric varieties.
method Using non-characteristic deformation of sheaves to find twisted polytope sheaves.
result Quasi-equivalence of coherent and constructible sheaves on smooth projective toric varieties.
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
The paper generalizes Kuperberg invariants using twisted Drinfeld doubles.
problem Quantum invariants of knots with additional structure.
method Using twisted Drinfeld doubles and Reshetikhin-Turaev invariants.
result Reidemeister torsion of knot complements as a quantum invariant.
New measure of knot complexity tied to twist number.
problem Understanding knot complexity.
method Defining alternating volume and showing coarsely equivalent to twist number.
result Alternating volume of a knot is coarsely equivalent to its twist number.
The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.
problem Preserving the diffeomorphism type of satellite 2-knots under the Gluck twist.
method Using new descriptions of satellite 2-knots, the paper shows that the Gluck twist does not change the diffeomorphism type of certain satellite 2-knots in three ways.
result The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.
Twisted links are obtained from a base link by starting with a n-braid representation, choosing several (m) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a ce…
We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the…
Twists Gromov and Lefschetz invariants for symplectic and fibered 3-manifolds.
problem Defining and relating twisted invariants for symplectic and fibered manifolds.
method Defining twisted Gromov-Taubes invariants and Lefschetz zeta functions for surface bundles over manifolds, proving their equivalence and interpreting them in terms of Reidemeister torsions.
result Twisted Lefschetz zeta functions and Gromov-Taubes invariants are equivalent for certain bundles, leading to new interpretations of Reidemeister torsions.
Study Alexander polynomials of links in 3-torus.
problem Investigate Alexander polynomials of links in 3-torus.
method Diagrammatic approach, Reidemeister moves, fundamental group, homology group, Alexander polynomials, twisted Alexander polynomials.
result Computed Alexander and twisted Alexander polynomials of links in 3-torus.
We lift the characteristic-2 totally twisted Khovanov homology of Roberts and Jaeger to a theory with integer coefficients. The result is a complex computing reduced odd Khovanov homology for knots. This complex is equivalent to a spanning-tree complex whose differential is explicit modulo a sign ambiguity coming from …
Generalizes Riemann-Hilbert correspondence for curved local systems.
problem Higher Riemann-Hilbert correspondence with scalar curvature.
method Equivalence of dg-categories of curved local systems, graded vector bundles, and representations.
result Equivalence of dg-enhancements of twisted sheaves categories.
For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundament…
New cohomology theory shows compact Lie group actions are Morita invariant.
problem Establishing Morita invariance for cohomology of compact Lie group actions.
method Using bibundles to transfer coefficient systems between Morita equivalent groupoids.
result Twisted Bredon-Illman cohomology is Morita invariant for compact Lie group actions.
This note proves equivalence between Dorfman brackets and lifts, showing universality of the Courant-Dorfman bracket.
problem Characterizing twistings and symmetries of transitive Dorfman brackets.
method Proving equivalence between Dorfman brackets and lifts, intertwining with Courant-Dorfman bracket.
result Universality of the Courant-Dorfman bracket and characterization of Dorfman brackets via lifts.
Paper defines doodles on closed surfaces, unifying classical and virtual theories.
problem Classifying doodles on closed surfaces, especially non-orientable ones.
method Introducing twisted virtual doodles, defining twin groups, and proving Alexander- and Markov-type theorems.
result Unified theory of doodles, showing trivial center and residually finite properties.
These notes discuss various aspect of the ``representation theory'' of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poi…
New handle diagrams show infinite order corks in 4-manifolds.
problem Existence of infinite order corks in 4-manifolds.
method Translated earlier proof into handle diagrams and conditions on torus twists.
result First handle diagrams of infinite order corks.
Study logarithmic flat connections on principal bundles using Lie groupoids.
problem Classify flat connections on principal bundles with logarithmic singularities.
method Use tools from Lie groupoid theory to classify representations and establish van Kampen theorems.
result Obtain a functorial Riemann-Hilbert correspondence for logarithmic connections.