arXiv research

New homological results for bordered Floer algebras derived from hypertoric categories.

The paper examines trisection diagrams of spun knots and shows they are standard for certain cases.

We show that if a prime homology sphere has the same Floer homology as the standard three-sphere, it does not contain any incompressible tori.

The paper examines conditions for contact surgeries on rational homology 3-spheres.

We inductively define layers of colorings of knot and knotted surface diagrams using ternary quasigroups. Homological invariants from such systems of colorings use shorter differentials and of higher degree than the standard homology differentials, and give access to typically more complex homology groups.

For an integral homology 3-sphere embedded asymptotically flatly in an Euclidean space, we find a natural framing extending the standard trivialization on the asymptotically flat part.

Witt algebra acts on Khovanov-Rozansky homology of links.

We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show …

New homology $q_2$ from 3D cobordism to integers.

In this paper we prove that for suitable sequences of congruence subgroups of Bianchi groups, including the standard exhaustive sequences of a congruence subgroup, and even symmetric powers of the standard representation of Sl_2(C) the size of the torsion part in the first homology grows exponentially. This extends res…

The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the ho…

We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces (the good pants are pairs of pants whose cuffs have the length nearly equal to some large number R). Combined with our previous work on the Surface Subgroup Theorem, this yields a proof of the Ehrenpre…

We show that if a positive integral surgery on a knot K inside a homology sphere X with Seifert genus g(K) results in an induced knot K_n in X_n(K)=Y which has simple Floer homology, we should have n>=2g(K). Moreover, if X is the standard sphere, the three-manifold Y is a L-space and the Heegaard Floer homology groups …

Constructs flows of tori in sphere perturbations for Morse homology.

In this paper we focus on preprocessing for persistent homology computations. We adapt some techniques which were successfully used for standard homology computations. The main idea is to reduce the complex prior to generating its boundary matrix, which is costly to store and process. We discuss the following reduction…

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parame…

Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.

If a Legendrian knot $Λ$ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface $Σ$ in the standard symplectic 4-ball, then the genus of $Σ$ is equal to the slice genus of (the smooth knot underlying) $Λ$, the sum of the Thurston-Bennequin number of L and the Euler characteristic of $Σ$ is zero …

Enhanced Khovanov TQFT using basepoints for nonorientable surfaces.

We make use of the action of $H_1(Y)$ in Heegaard Floer homology to generalize the Ozsváth-Szabó correction terms for $3$-manifolds with standard $\operatorname{HF}^\infty$. We establish the basic properties of these invariants: conjugation invariance, behavior under orientation reversal, additivity, and spin$^c$ ratio…

Classifies two families of simply connected 7-manifolds with minimal homological complexity.

Mathematical pipeline identifies structural homology of knotted proteins.

Study on Klein bottle's cotangent bundle using contact homology.

Simplified computation of symmetric gl_1 homology for links.

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the exist…

Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut…

New method compares geometric and standard cup products.

We present a braid-theoretic approach to combinatorially computing knot Floer homology. To a knot or link K, which is braided about the standard disk open book decomposition for (S^3,ξ_std), we associate a corresponding multi-pointed nice Heegaard diagram. We then describe an explicit algorithm for computing the associ…

Approaches for approximating persistent homology for large datasets.

Discrete Morse theory simplifies Khovanov homology calculations.

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known in…

We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of $\mathfrak{sl}_N$. In addition, the cons…

Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the …

We have one more look at the (homological) perturbation lemma and we point out some non-standard consequences, including the relevance to deformations.

We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the proof we relate the sutured Floer homology groups associated with a knot complem…

We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N \geq 1$, $\mathcal{H}_N$ is a $\mathbb{Z}_2\o…

As a generalization of Davis-Januszkiewicz theory, there is an essential link between locally standard $(\Z_2)^n$-actions (or $T^n$-actions) actions and nice manifolds with corners, so that a class of nicely behaved equivariant cut-and-paste operations on locally standard actions can be carried out in step on nice mani…

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the…

We consider the universal family $E_n^d$ of superelliptic curves: each curve $Σ_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $Σ_n^d\hookrightarrow E_n^d\to C_n$ is a fibre bundle, where $C_n$ is the configuration space of $n$ distinct points. We fin…

For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant associated to T_{ws_i} to that associated to T_w, where s_i is one of the standard ge…

GRID invariants block certain Lagrangian cobordisms in 3D.

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that t…

Proves surgery exact triangle for monopole Floer homology over integers.

We consider critical points of the global squared $L^2$-norms of the second fundamental form and the mean curvature vector of isometric immersions into a fixed background Riemannian manifold under deformations of the immersion. We use the critical points of the former functional to define canonical representatives of a…

In this note, we combine the recent 4-dimensional light bulb theorem of David Gabai and a recent construction of concordances for knots in $S^2\times S^1$ due to Eylem Zeliha Yildiz to construct a concordance between the standard surface of genus $g$ in $S^2\times S^2$ and any homologous surface.

New geometric variant of factorization homology for conformally flat manifolds.

We construct a new invariant of transverse links in the standard contact structure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift i…

Study exotic tori and their SL_d(Z) actions, proving many do not admit nontrivial actions.