arXiv research

Homology handles with trivial Alexander polynomial bound a 3D sphere.

Proves complex homology three-spheres can be bounded by many handles.

Kirby color defined in Khovanov homology for 4D handlebodies.

We give an entirely geometric proof, without recourse to cellular homology, of the fact that $\partial^2=0$ in the chain complex defined by a handle decomposition of a given manifold. Topological invariance of the resulting `handle homology' is a consequence of Cerf theory.

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

New method uses Khovanov homology to distinguish exotic 4-manifolds.

The study distinguishes exotic R^4's using knot Floer homology and Casson handles.

Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…

Study uses twisted Alexander polynomials to link fibered classes in 3-manifolds.

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

Real Milnor fibres become contractible after attaching handles, matching classical results.

Constructs exotic proper 2-knots from open 2-handles.

This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.

Skein lasagna module calculates 4-manifold invariants using handle decompositions.

Two codimension-one submanifolds are cobordant if they have the same homology class.

Fintushel and Stern showed that the Brieskorn sphere $Σ(2,3,7)$ bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to p…

The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.

New insights into knot fusion numbers via cabling.

Khovanov homology fails to differentiate certain slice disks.

The study solves a 2008 problem by Kalai about infinitely many homology-spheres.

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 …

The paper bounds the handle number of sutured manifolds using Morse-Novikov numbers and tunnel numbers.

We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.

We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on …

We give an explicit construction of the Honda--Kazez--Matić gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link F…

Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these b…

Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …

Introduces Floer lasagna modules using link Floer homology.

There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphi…

A new method converts knot Floer homology to immersed curves.

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplec…

Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided $h$-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-…

From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and …

Study Morse complexity of manifolds and homology classes, proving bounds and implications.

New exotic 4D spaces found using knot slicing techniques.

Constructs dg categories from surfaces using Khovanov homology.

We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem …

Link homology theories connect to 4-manifold invariants and TQFTs.

Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of existing Bayesian optimization algorithms can only handle vector data and canno…

Determines conditions for ribbon cobordisms between lens spaces.

The paper calculates homology and intersection pairing of branched covers using disoriented homology.

Develops a method to compute Morse homology for clean but not necessarily transverse intersections.

Study on lens spaces bounding 4-manifolds with specific Betti numbers.

We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ can be used to recover WRT invariants at certain roots of unity, and also recovers the …

We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of $S^3$ by regluing…

The paper introduces new invariants for genus one knots and surfaces.

We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are the maximal and minimal second Betti number divided by $8$ among definite spin boundings of a homology sphere. The similar invariants $g_8$ and $\overline{g_8}$ are defined by the maximal (or minimal) product sum of $E_8$-form of bounding 4-manifold…

Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology f…