We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants c…
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The paper sets lower bounds for a Kirby-Thompson invariant of 4-manifolds.
problem Determining the Kirby-Thompson invariant of specific 4-manifolds.
method Using trisections, the paper establishes lower bounds and calculates the invariant for specific examples.
result The paper calculates the Kirby-Thompson invariant of the spin of and shows the existence of 4-manifolds with arbitrarily large invariants.
Survey of LMO invariant for 3-manifolds using Kirby structures.
problem Developing a combinatorial approach to the LMO invariant of 3-manifolds.
method Introducing Kirby structures and pre-LMO structures to yield multiplicative invariants.
result Construction and equivalence of two families of invariants, leading to the LMO invariant.
Kirby color defined in Khovanov homology for 4D handlebodies.
problem Invariance of 4D handlebodies under Kirby moves.
method Functoriality and cabling properties of Khovanov homology, handle slide isomorphism.
result Kirby-colored Khovanov homology is invariant under handle slide moves.
This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology …
Simply-connected 4-manifolds are covered by products with a 2-torus.
problem Proving that any closed simply-connected smooth 4-manifold is branched covered by a product of an orientable surface and a 2-torus.
method Natural construction with respect to spin structures, solving Problem 4.113(C) in Kirby's list.
result Closed simply-connected smooth 4-manifolds are 16-fold branched covered by a product of an orientable surface and a 2-torus.
The paper describes handle decompositions and Kirby diagrams for plane algebraic curves.
problem Understanding the topology of the complement of plane algebraic curves.
method Using braid monodromy to refine handle decompositions and Kirby diagrams.
result Explicit handle decompositions and Kirby diagrams for plane algebraic curves are provided.
Note on potential Kirby move type 1 for contact surgery diagrams.
problem Exploring conditions for a contact Kirby move type 1.
method Analyzing necessary conditions for contact surgery diagrams to be candidates for contact Kirby move type 1.
result Existence of a collection of contact positive integral surgery diagrams on Legendrian unknots satisfying the conditions.
This paper proves a conjecture about trisections with a specific length.
problem Proving a conjecture about trisections with a specific length.
method Examining trisections with Kirby-Thompson length 2 and proving the conjecture.
result Proves the conjecture about length 2 trisection being a 4-manifold with length 0.
Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.
problem Creating efficient trisection diagrams for 4-manifolds.
method Algorithm converting Kirby diagrams to trisection diagrams.
result Provides examples of trisection diagrams for 4-manifolds.
New bounds found for complexity of spun knots.
problem Measuring complexity of spun knots.
method Using bridge trisections and distances in the pants complex.
result Bound the Kirby-Thompson invariant of spun knots.
New examples of Kirby-Ramanujam spheres found, leading to contractible 4-manifolds.
problem Finding new examples of Kirby-Ramanujam spheres.
method Tracing the initial steps of Kirby's example and providing additional examples, showing diffeomorphisms and linear independence.
result Found three infinite families of Kirby-Ramanujam spheres that bound contractible 4-manifolds.
Algorithm constructs Kirby diagrams for 4D open books.
problem Constructing Kirby diagrams for 4D open books.
method Algorithm using Heegaard diagrams of pages.
result Diffeomorphic open books constructed with different pages and monodromies.
High-dimensional handlebodies are shown to be products of simpler shapes.
problem Understanding the structure of high-dimensional handlebodies.
method Introducing Kirby diagrams to simplify the structure of -handlebodies.
result High-dimensional handlebodies are products of simpler shapes.
Sharp bounds for Kirby-Thompson invariants of knotted surfaces computed.
problem Computing sharp lower bounds for Kirby-Thompson invariants of knotted surfaces.
method Using dual curve complex distances to compute invariants.
result Exact values of KT-invariants computed for knotted surfaces with bridge number ≤ 6.
This is a survey paper where we expose the Kirby--Siebenmann results on classification of PL structures on topological manifolds and, in particular, the homotopy equivalence TOP/PL=K(Z/2.3) and the Hauptvermutung for manifolds.
The study describes handle decompositions and Kirby diagrams for line arrangements.
problem Understanding handle decompositions and Kirby diagrams for line arrangements.
method Introduced the divide with cusps and used Lefschetz hyperplane section theorem.
result Described the Kirby diagram for line arrangements.
A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle deco…
Explains Kirby's work on 4-manifold trisections.
problem Understanding trisections of 4-manifolds.
method Near-symplectic constructions.
result Developed a theory of trisections.
Kirby diagrams for exotic R^4's constructed from specific knot complements.
problem Identifying and visualizing exotic 's using Kirby diagrams.
method Provided Kirby diagrams for a family of exotic 's constructed from specific knot complements.
result Generalized Kirby diagrams for a broader family of exotic 's.
From knots to four-manifoldsmath.GT
Survey on knot theory's impact on four-dimensional topology.
problem Understanding invariants of smooth four-manifolds.
method Using Kirby diagrams and Heegaard Floer theory.
result Progress in detecting exotic structures.
We show how to construct a Kirby diagram for a large class of finite volume hyperbolic 4-manifolds constructed by J. Ratcliffe and S. Tschantz.
Kirby diagrams, trisections and gems of PL 4-manifolds: relationships, results and open problemsmath.GT
Review of gem theory's interactions with Kirby diagrams and trisections.
problem Representing and understanding PL 4-manifolds.
method Combining gem theory with Kirby diagrams and trisections.
result New gems representing 4-manifolds requiring 3-handles.
A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed l…
Study of 3-handle attachments in 4D manifolds using Kirby calculus.
problem Inclusion of 3-handles in Kirby diagrams for 4D manifolds.
method Developed moves to extend Kirby calculus, established homological criterion.
result Identified geometric basis of 3-handle attachments, proved uniqueness theorem.
The paper connects Kirby diagrams and 5-colored graphs to represent 4-manifolds.
problem Representing compact 4-manifolds using Kirby diagrams and graphs.
method Algorithmically constructing 5-colored graphs from Kirby diagrams to represent PL 4-manifolds.
result Upper bounds for gem-complexity and regular genus derived from Kirby diagrams.
Round surgery diagrams represent 3-manifolds in .
problem Representing and manipulating 3-manifolds in .
method Introducing round surgery diagrams and defining moves to establish Kirby Calculus.
result Any 3-manifold can be obtained by a round surgery on a framed link in .
New contact Kirby moves complete the set for contact surgery diagrams.
problem Contact surgery diagrams and their relation to contactomorphic contact manifolds.
method Introducing lantern moves and chain moves to complete the set of contact Kirby moves.
result Two contact surgery diagrams represent contactomorphic contact manifolds if and only if they are related by a sequence of specific moves.
Introduces algorithm for Weinstein handlebodies of certain divisors.
problem Constructing Weinstein handlebodies for complements of smoothed toric divisors.
method Explicit coordinates and simple example for algorithm.
result Produces Weinstein Kirby diagrams for complements of smoothed toric divisors.
Generalizes Rochlin's theorem to 4-manifolds with boundary and arbitrary 3-manifold boundaries.
problem Proving divisibility of self-intersection numbers in 4-manifolds with boundary.
method Combining Kirby-Siebenmann invariants, Brown invariants, and pin structures.
result Combinatorial description of Kirby-Siebenmann invariant for 4-manifolds with boundary.
Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as and admits neither 1- nor 3-handles, by using rational blow-downs and K…
A theorem of Kirby states that two framed links in the 3-sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on …
Suppose there are two framed links in a compact, connected 3-manifold (possibly with boundary, or non-orientable) such that the associated 3-manifolds obtained by surgery are homeomorphic (relative to their common boundary, if there is one.) How are the links related? Kirby's theorem gives the answer when the manifold …
New invariant measures knotted surfaces in 4D, revealing unknottedness.
problem Measuring knotted surfaces in 4D.
method Defined an integer invariant for bridge trisections of surfaces in or .
result Invariant implies the surface is unknotted.
Software simplifies triangulations of 4-manifolds, revealing exotic structures.
problem Understanding smooth 4-manifolds from discrete and algorithmic perspectives.
method New software tools, including an algorithm for triangulations from Kirby diagrams and a heuristic for simplification.
result Presented new triangulations of exotic pairs, corks, and plugs, including the smallest known K3 surface.
Algorithm decides homeomorphism of 4-manifolds.
problem Deciding homeomorphism of closed, simply connected 4-manifolds.
method Represent 4-manifolds by Kirby diagrams, compute Kirby-Siebenmann invariant, and use algorithms for Kirby diagrams and intersection forms.
result Algorithmic decision for homeomorphism of 4-manifolds.
Akbulut and Kirby conjectured that two knots with the same -surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.
Paper constructs S-equivalent genus one knots distinguishable by Jones polynomial.
problem Problem 1.6 in Kirby's problem list (K3) about S-equivalence of genus one knots.
method Constructs a sequence of genus one knots that are S-equivalent but distinguishable by Jones polynomial.
result Genus one knots can be S-equivalent yet distinguishable by Jones polynomial.
Study obstructs symplectic structures on Mazur manifolds.
problem Obstructing symplectic structures on Mazur manifolds.
method Using Heegaard Floer homology cobordism maps.
result Obstructs the existence of symplectic structures on Mazur manifolds.
Standard trisection diagrams found for Mazur type 4-manifolds.
problem Finding standard trisection diagrams for Mazur type 4-manifolds.
method Doubling a relative trisection diagram and using an algorithm from Kirby diagrams to trisection diagrams.
result Certain trisection diagrams of Mazur type 4-manifolds are standard.
Connected sums defined for codimension two locally flat submanifolds in higher dimensions.
problem Defining connected sums for codimension two locally flat submanifolds in various dimensions.
method Using results from higher dimensional topological manifolds and four-manifolds, defining connected sums for codimension two locally flat submanifolds.
result A well-defined connected sum exists up to orientation preserving homeomorphism.
We give a construction of Kirby weight systems associated to sl(2) and valued into the finite field Z/pZ. We show that it is possible to apply this sequence of weight systems on the universal invariant of framed link. We also show that the corresponding sequence admits a Fermat limit, which defines an asymptotic ration…
Upper bounds for surface-links in the Yoshikawa table are estimated.
problem Estimating Kirby-Thompson invariants of surface-links.
method Using tri-plane diagrams and L-, L*-invariants.
result Upper bounds for surface-links in the Yoshikawa table are obtained.
Equivalence of a Bridged Link Calculus and Kirby's Calculus on Non-Simply Connected 3-Manifoldsmath.GT
We recall an extension of Kirby's Calculus on non-simply connected 3-manifolds given in [FR], and the surgery calculus of bridged links from [Ke], which involves only local moves. We give a short combinatorial proof that the two calculi are equivalent, and thus describe the same classes of 3-manifolds. This makes the p…
The abstract discusses constructing 3d N=2 gauge theories using surgeries and M5-branes.
problem Constructing geometrically 3d N=2 gauge theories.
method Using surgeries and M5-branes wrapping on plumbing three-manifolds.
result Various dualities of 3d theories can be interpreted as Kirby moves and equivalent surgeries.
Study topological 4-manifolds with specific fundamental groups.
problem Classify topological 4-manifolds with 4-dimensional fundamental group.
method Use algebraic topology, Poincaré duality, and Kirby-Siebenmann invariant.
result Two manifolds are homeomorphic if they are s-cobordant and have same Kirby-Siebenmann invariant.
One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as . In this paper we prove that the number of diffeomorphism classes grows at least as …
New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.
problem Defining a symplectic invariant for Weinstein domains.
method Contact cut graph, Lefschetz fibrations, multisections with divides.
result Definition of Weinstein -invariant and its relation to Kirby-Thompson's invariant.