arXiv research

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

Classifies torus bundles bounding 4-manifolds with rational homology.

Classifies surgeries on torus knots and cables that bound rational homology balls.

Lower bounds on rational slice genus using Heegaard Floer invariants.

New 3-manifolds bound rational 4-balls through specific operations.

New knot invariant λ bounds rational unknotting.

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…

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…

In this paper, we introduce a rational $τ$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsváth-Szabó contact invariants. Such an invariant is an upper bound for the sum of rational Thurston-Bennequin invariant and the rational rotation number of the Legendrian representatives o…

Study shows Seifert fibered spaces don't bound rational homology balls.

We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.

Local knots can't bound smaller surfaces in rational homology 3-spheres.

New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.

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…

New findings on knots that are both topologically and rationally slice.

In this paper the theory of semi-bounded rationality is proposed as an extension of the theory of bounded rationality. In particular, it is proposed that a decision making process involves two components and these are the correlation machine, which estimates missing values, and the causal machine, which relates the cau…

Study shows surgeries on certain knots bound rational homology 4-balls.

The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.

For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,…

The abstract discusses connections between K-stability, heights, and rational points on Fano varieties.

The paper calculates bounds for unknotting rational tangles using knot Floer homology.

New knots show linear independence in slice concordance.

This work explains crises in markets without external news using bounded rational agents.

Knots generating infinite subgroup bound rational homology balls.

Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.

Using the Heegaard Floer homology of Ozsvath and Szabo we investigate obstructions to definite intersection pairings bounded by rational homology spheres. As an application we obtain new lower bounds for the four-ball genus of Montesinos links.

The paper proves that rational concordance of double twist knots is reciprocal.

Paper finds new 3D shapes that can be inside a 4D space.

New 4-manifold accounts for rationally slice knots.

We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology $S^1\times S^2$'s bound rational homology $S^1\times D^3$'…

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.

New rational curvature measures for 2-complexes.

We call an integral homology sphere $\textit{non-trivially}$ bounds a rational homology ball if it is obstructed from bounding an integral homology ball. After Fintushel and Stern's well-known example $Σ(2,3,7)$, Akbulut and Larson recently provided the first infinite families of Brieskorn spheres non-trivially boundin…

The study sets constraints on 4-manifold forms linked to specific invariants.

Rational neural networks approximate functions more efficiently with less depth.

The article classifies cubiquitous sublattices and applies them to branched covers.

Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

Bounded rationality, that is, decision-making and planning under resource limitations, is widely regarded as an important open problem in artificial intelligence, reinforcement learning, computational neuroscience and economics. This paper offers a consolidated presentation of a theory of bounded rationality based on i…

We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory.

It is known that Garside groups are strongly translation discrete. In this paper, we show that the translation numbers in a Garside group are rational with uniformly bounded denominators and can be computed in finite time. As an application, we give solutions to some group-theoretic problems.

We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsváth and Szabó's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on p…

New model for rational tropical points using $\mathfrak{sp}_4$-webs and measures.

Prime power fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. In this paper, we give a new construction of non-slice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander p…

In this note we study the Seifert rational homology spheres with two complementary legs, i.e. with a pair of invariants whose fractions add up to one. We give a complete classification of the Seifert manifolds with 3 exceptional fibers and two complementary legs which bound rational homology balls. The result translate…

We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.

Researchers study rational and pretzel knots using affine group representations.

Optimal bounds on rational points on algebraic curves established.