arXiv research

New findings on hyperbolicity of augmented links in thickened surfaces.

New tool lassos connects virtual and surface link diagrams, changing primeness rules.

The study determines conditions for hyperbolicity of links in thickened surfaces with boundary.

Essential surfaces in link diagrams on surfaces are crucial for understanding link properties.

The paper presents a chain complex for 3-manifold covers, including surface bundles and surgeries.

Geometrically interprets a duality theorem linking cochain and chain complexes.

TrajectoryNet models dynamic cellular trajectories using optimal transport.

We propose an algorithm to automate fault management in an outdoor cellular network using deep reinforcement learning (RL) against wireless impairments. This algorithm enables the cellular network cluster to self-heal by allowing RL to learn how to improve the downlink signal to interference plus noise ratio through ex…

Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.

Motivation: Understanding functions of proteins in specific human tissues is essential for insights into disease diagnostics and therapeutics, yet prediction of tissue-specific cellular function remains a critical challenge for biomedicine. Results: Here we present OhmNet, a hierarchy-aware unsupervised node feature le…

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are re…

Just as the Temperley-Lieb algebra is a good place to compute the Jones polynomial, the Kauffman bracket skein algebra of a disk with $2k$ colored points on the boundary, each with color $n$, is a good place to compute the $n^{th}$ colored Jones polynomial. Here, this colored skein algebra is shown to be a cellular alg…

Spatial refinement of Bar-Natan homology constructed.

Classifies involutions on alternating prime non-split links.

New quasi-alternating links created from existing ones.

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-web version of Hu--Mathas' graded cellular basis and has two major application…

Extends classical results to virtual links, proving new properties of alternating and semi-alternating virtual links.

Constructs links that are both quasi-alternating and almost alternating.

Study of alternating links on nonorientable surfaces, extending results to nonorientable projections.

This article introduces descriptive cellular homology on cell complexes, which is an extension of J.H.C. Whitehead's CW topology. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. An application of two forms of cellular homology is given in terms of the persistence of shape…

We construct an infinite family of quasi-alternating links from a given quasi-alternating link by replacing a crossing by a product of rational tangles each of which extends that crossing. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family contains all the classes of quasi-…

We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other th…

Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are…

New invariants derived from Seifert graphs help distinguish alternating links.

The study classifies cellular pseudomanifolds and their properties.

Meridian lemma extended to fully alternating links in thickened surfaces.

Extended Thistlethwaite's result on Jones polynomials of quasi-alternating links.

An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, t…

New proof limits Jones polynomial values for quasi-alternating links.

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as …

A link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change. We give formulas for the first two and last two potential coefficients of the Jones polynomial of an almost alternating link. Using these formulas, we show that the Jones …

Quantum cellular automata form a homology theory.

Paper explores conditions for constructing new quasi-alternating links.

Standard position for surfaces extended to weakly generalized alternating links.

Proves alternating quasipositive links are positive and strongly quasipositive.

In this paper, we show that a link which has a positive and almost alternating diagram is alternating, besides that a positive and non-alternating Montesinos link has an almost positive-alternating diagram.

This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link diagram of a given minimal crossing size and by applications of just two operators (T …

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produc…

The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.

We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagra…

Study proves chainmail links are L-space links.

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometr…

Proves certain alternating links have specific geometric properties.

Improved bounds on volume-det inequality for links with many twists.

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

We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. The main theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing numbers of crossings of diagrams in the process. Moreover, we show that almost alter…

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel param…

Study sharpens unlinking number bounds for special alternating links.