arXiv research

Study fully augmented links in thickened torus, generalizing $S^3$ results.

Flat fully augmented links with homeomorphic complements are equivalent.

Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.

We study the geometry of fully augmented link complements in $S^3$ by looking at their link diagrams. We extend the method introduced by Thistlethwaite and Tsvietkova to fully augmented links and define a system of algebraic equations in terms of parameters coming from edges and crossings of the link diagrams. Combinin…

Fully augmented links have dense volume densities but discrete in certain ranges.

This work classifies belted sum decompositions of fully augmented links.

The paper connects link symmetries to finite subgroups of O(3).

This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant …

Geometric proof confirms link volume conjecture.

For any knot, a 3-sphere triangulation exists with a knotted edge.

New examples show no upper bounds on link volumes on incompressible surfaces.

Study shows surgeries on specific links result in L-spaces.

Generalizes Thistlethwaite-Tsvietkova method to 3-manifolds.

New method fractures hyperbolic manifolds using cone singularities.

In this paper we analyze symmetries, hidden symmetries, and commensurability classes of $(ε, d_L)$-twisted knot complements, which are the complements of knots that have a sufficiently large number of twists in each of their twist regions. These knot complements can be constructed via long Dehn fillings on fully augmen…

New findings on hyperbolicity of augmented links in thickened surfaces.

We study the fibration of augmented link complements. Given the diagram of an augmented link we associate a spanning surface and a graph. We then show that this surface is a fiber for the link complement if and only if the associated graph is a tree. We further show that fibration is preserved under Dehn filling on cer…

We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different types. We also study properties of the augmentation representation…

Augmented alternating links are links obtained by adding trivial components that bound twice-punctured disks to non-split reduced non-2-braid prime alternating projections. These links are known to be hyperbolic. Here, we extend to show that generalized augmented alternating links, which allow for new trivial component…

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We prese…

The paper studies hyperbolic geometry of links formed by adding trivial components to knots.

Satellite links of fully positive braids are characterized.

Study proves chainmail links are L-space links.

Study of Legendrian links using Floer theory and cluster varieties.

We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating.

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…

The study analyzes consistency-based SSL methods and proposes a new framework.

DAERNN models censored data using neural networks with data augmentation.

Meridian lemma extended to fully alternating links in thickened surfaces.

Associated to Legendrian links in the standard contact three-space, Ruling polynomials are Legendrian isotopy invariants, which also compute augmentation numbers, that is, the points-counting of augmentation varieties for Legendrian links (up to a normalized factor) \cite{HR15}. In this article, we generalize this pict…

The paper connects Legendrian links to cluster theory and exact Lagrangian fillings.

Hyperbolic links in thickened torus decompose into angled tetrahedra.

We describe a presentation for the augmented fundamental rack of a link in the lens space $L(p,1)$. Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in $L(p,1)$. In this case, the counting rack invariants also include the informatio…

We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.

A gamma process dynamic Poisson factor analysis model is proposed to factorize a dynamic count matrix, whose columns are sequentially observed count vectors. The model builds a novel Markov chain that sends the latent gamma random variables at time $(t-1)$ as the shape parameters of those at time $t$, which are linked …

For families of knots and links given in Conway notation we compute lower maximal and upper minimal bound of hyperbolic volume by using source links and augmented links.

Paper constructs a spectral sequence linking annular Khovanov homology to reduced Khovanov homology.

Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in $S^3$. We prove a similar result for links in closed thickened surfaces $S \times I$. We define a link to be fully alternating if it has an alternating projection from $S\times I$ to $S$ where the interior of every complemen…

The study shows involutory quandles of certain links are not left-orderable.

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang-Baxter homology with a combinatorial formulation for the boundary map and specializes to $N$-reduced rack homology when the birack is a rack. We introduce augmente…

Given a Legendrian link in $\#^k(S^1\times S^2)$, we extend the definition of a normal ruling from $J^1(S^1)$ given by Lavrov and Rutherford and show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over $\mathbb{Z}[t,t^{-1}]$ is equivalent to the existence of a …

New link types derived from old links.

A novel fully asynchronous scheme for distributed reinforcement learning over networks.

We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also impl…

Augments graph node features to improve GNN performance.

Augmentations and sheaves linked for Legendrian graphs.

COTA learns abstraction maps from data without complete SCM knowledge.

Data augmentation has been widely applied as an effective methodology to improve generalization in particular when training deep neural networks. Recently, researchers proposed a few intensive data augmentation techniques, which indeed improved accuracy, yet we notice that these methods augment data have also caused a …