arXiv research

Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.

The paper describes handle decompositions and Kirby diagrams for plane algebraic curves.

Algorithm constructs Kirby diagrams for 4D open books.

Note on potential Kirby move type 1 for contact surgery diagrams.

High-dimensional handlebodies are shown to be products of simpler shapes.

The study describes handle decompositions and Kirby diagrams for line arrangements.

Kirby diagrams for exotic R^4's constructed from specific knot complements.

Review of gem theory's interactions with Kirby diagrams and trisections.

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.

The paper connects Kirby diagrams and 5-colored graphs to represent 4-manifolds.

Standard trisection diagrams found for Mazur type 4-manifolds.

Upper bounds for surface-links in the Yoshikawa table are estimated.

Introduces algorithm for Weinstein handlebodies of certain divisors.

Study trisections of non-orientable 4-manifolds with boundary.

Round surgery diagrams represent 3-manifolds in $S^3$.

Study of 3-handle attachments in 4D manifolds using Kirby calculus.

Algorithm decides homeomorphism of 4-manifolds.

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…

New contact Kirby moves complete the set for contact surgery diagrams.

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 $n^2$. In this paper we prove that the number of diffeomorphism classes grows at least as …

We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.

Gordon-Litherland pairing connects combinatorics and topology.

Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.

Survey on knot theory's impact on four-dimensional topology.

The Witten-Reshetikhin-Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used…

Invariants for 4-manifolds from Hopf group-algebras.

We construct a Kirby diagram of the rational homology ball used in "generalized rational blow-down" developed by Jongil Park. The diagram consists of a dotted circle and a torus knot. The link is simpler, but the parameters are a little complicate. Euclidean Algorithm is used three times in the construction and the pro…

We give a solution to a part of Problem 1.60 in Kirby's list of open problems in topology thus answering in the positive the 1987 conjecture by J.Przytycki concerning the existence of knots without matched diagrams.

The complete invariant for gradient like Morse-Smale dynamical systems (vector fields and diffeomorphisms) on closed 4-manifolds are constructed. It is same as Kirby diagram in a case of polar vector field without fixed points of index 3.

We construct two infinite sequences of immersions of the 3-sphere into 4-space, parameterized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We compute their Smale invariants and bordism classes of im…

The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…

According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an …

Software simplifies triangulations of 4-manifolds, revealing exotic structures.

We show that if each of $K_1$ and $K_2$ is a trefoil knot or figure eight knot, the homology 3-sphere defined by the Kirby diagram which is a simple link of $K_1$ and $K_2$ with framing (0, n) is represented by an n-twisted Whitehead double of $K_2$ .

New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.

Study surfaces in 4-manifolds using banded unlink diagrams.

Let $(M,ξ)$ be a contact 3-manifold. We present two new algorithms, the first of which converts an open book $(Σ,Φ)$ supporting $(M,ξ)$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $(M,ξ)$ into a supporting open book decomposition. These constructions lead to a r…

We develop a technique for gluing relative trisection diagrams of $4$-manifolds with nonempty connected boundary to obtain trisection diagrams for closed $4$-manifolds. As an application, we describe a trisection of any closed $4$-manifold which admits a Lefschetz fibration over $S^2$ equipped with a section of square …

New method to encode Weinstein 4-manifolds using multisections with divides.

New 4-manifold invariant defined from trisection diagrams.

We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since humans cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional manifolds. However, one can in fact encode the topology of a surface in a 1-dime…

Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $Σ$, guiding the gluing of the handlebodies. Any morphism $\varphi$ from …

New method to compute homology and intersection form of 4-manifolds.

The twisted face-pairing construction of our earlier papers gives an efficient way of generating, mechanically and with little effort, myriads of relatively simple face-pairing descriptions of interesting closed 3-manifolds. The corresponding description in terms of surgery, or Dehn-filling, reveals the twist construct…

The paper sets lower bounds for a Kirby-Thompson invariant of 4-manifolds.

In this paper, we study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected …

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitel…

Kirby color defined in Khovanov homology for 4D handlebodies.