Jones polynomial coefficients of almost alternating links are nontrivial and have alternating signs.
problem Understanding the Jones polynomial of almost alternating links.
method Formulas for Jones polynomial coefficients, crossing change analysis, and diagram comparison.
result Jones polynomial coefficients of almost alternating links alternate in sign and are nontrivial.
Constructs links that are both quasi-alternating and almost alternating.
problem Creating links that are both quasi-alternating and almost alternating.
method Using dealternator extension to replace dealternator with rational tangle.
result All not alternating and quasi-alternating Montesinos links can be obtained.
Paper explores conditions for constructing new quasi-alternating links.
problem Applying construction method to non-alternating and opposite type tangles.
method Substituting quasi-alternating crossings with rational tangles of same type.
result Identifies conditions and specific tangles for construction validity.
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 quasi-alternating links created from existing ones.
problem Creating new quasi-alternating links from existing ones.
method Extending the construction of quasi-alternating links by replacing a crossing with an alternating tangle of the same type.
result Jones polynomial of new quasi-alternating links has no gap if the original link has no gap.
Classifies involutions on alternating prime non-split links.
problem Classifying involutions on alternating prime non-split links.
method Using the category of flypes between reduced alternating diagrams.
result Quotient of an alternating periodic link is alternating; freely 2-periodic alternating links have an even number of components.
Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alter…
Odd primes act on alternating knots via flypes.
problem Understanding odd order group actions on alternating knots.
method Investigation of flypes between reduced alternating diagrams.
result Odd prime actions on alternating knots are isotopic through a single flype.
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.
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-…
Jones polynomial gaps removed for quasi-alternating links.
problem Proving Jones polynomial gaps for quasi-alternating links.
method Proving Jones polynomial properties through link twisting and conjecturing for prime quasi-alternating links.
result Jones polynomial gaps removed for certain quasi-alternating links.
Study shows bounds on volumes of weakly generalised alternating knots.
problem Volume bounds for weakly generalised alternating knots.
method Analysis of weakly generalised alternating knots in 3-manifolds.
result Upper volume bound does not hold for weakly generalised alternating knots.
Study of alternating links on nonorientable surfaces, extending results to nonorientable projections.
problem Generalizing hyperbolic geometry results to nonorientable surfaces.
method Extending results from orientable to nonorientable surfaces.
result Klein-bottly alternating links in prism manifolds have hyperbolic geometry.
The study calculates and analyzes alternating surgeries for various knots.
problem Identifying and understanding alternating surgeries on knots.
method Algorithmic computation and structural analysis of alternating surgery slopes.
result The set of alternating surgery slopes is algorithmically computable and exhibits interesting phenomena.
Decomposes prime alternating links into prime tangles.
problem Understanding how to decompose prime alternating links into prime tangles.
method Refined results from Menasco and Thistlethwaite, focusing on alternating property and pseudo-Montesinos links.
result If a prime alternating link can be decomposed into two prime tangles, it must be visible in an alternating link diagram or the link is pseudo-Montesinos.
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 …
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…
Extends classical results to virtual links, proving new properties of alternating and semi-alternating virtual links.
problem Classical results for virtual links, focusing on alternating and semi-alternating links.
method Inequality relating link determinant and crossing number, matrix-tree theorem, Tait conjectures for virtual and welded links.
result Alexander polynomial of almost classical alternating virtual links is alternating.
Proves special alternating knots cannot be decomposed as non-trivial band sums.
problem Understanding the structure and properties of special alternating knots.
method Analyzes concordance properties and uses band sums to restrict knot decompositions.
result Conjecture that special alternating knots are ribbon concordance minimal is verified in many cases.
Proves Girth Alternative for some HNN extensions, finds counterexamples.
problem Girth Alternative for HNN extensions of finitely generated groups.
method Proves Girth Alternative for a specific class of HNN extensions.
result Girth Alternative holds for some HNN extensions but fails in general.
The study examines quasi-alternating surgeries on knots and their properties.
problem Understanding quasi-alternating surgeries and their relation to L-space knots.
method Exploration of quasi-alternating surgeries, classification of knots, and analysis of slopes.
result All SnapPy census L-space knots except two admit quasi-alternating surgeries.
New proof limits Jones polynomial values for quasi-alternating links.
problem Limits on Jones polynomial values for quasi-alternating links.
method Proved finitely many values of Jones polynomial for quasi-alternating links of a given determinant.
result Only finitely many quasi-alternating links have a given Jones polynomial.
Characterizes strongly quasipositive quasi-alternating and Montesinos links.
problem Characterizing strongly quasipositive links and detecting new classes of Montesinos links.
method Analyzing quasi-alternating links and their crossings, and using properties of Seifert surfaces.
result Strongly quasipositive links have specific properties related to their crossings and Seifert surfaces.
Introduces alternators for modeling sequences, outperforming baselines.
problem Modeling complex sequential data with stability and efficiency.
method Two neural networks (OTN and FTN) alternate between outputting samples in observation and feature spaces, learned via cross-entropy criterion.
result Alternators outperform strong baselines in various domains (Lorenz equations, Neuroscience, Climate Science).
Explains alternating knots for a concise encyclopedia.
problem Understanding alternating knots.
method Expository writing.
result Clarifies the concept of alternating knots.
We extend Howie's characterization of alternating knots to give a topological characterization of toroidally alternating knots, which were defined by Adams. We provide necessary and sufficient conditions for a knot to be toroidally alternating. We also give a topological characterization of almost-alternating knots whi…
We prove that the degree of the Brandt-Lickorish-Millet polynomial of any quasi-alternating link is less than its determinant. Therefore, we obtain a new and a simple obstruction criterion for quasi-alternateness. As an application, we identify some knots of 12 crossings or less and some links of 9 crossings or less th…
It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alt…
We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alte…
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…
This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime alternating knots of n crossings, the set of all prime alternating knots of n+1 crossing…
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
Extended Thistlethwaite's result on Jones polynomials of quasi-alternating links.
problem Characterizing Jones polynomials of quasi-alternating links.
method Analyzing structure and properties of Jones polynomials for quasi-alternating links.
result Jones polynomials of prime quasi-alternating links have no gaps.
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 …
Proves strong Tits alternative for 3D automorphisms over zero char fields.
problem Tits alternative for 3D tame automorphisms over zero char fields.
method Proved strong Tits alternative using 3D tame automorphisms over zero char fields.
result Strong Tits alternative proven for 3D tame automorphisms over zero char fields.
Study bounds on cusp volumes of alternating knots on surfaces.
problem Bounding cusp volumes of knots on surfaces.
method Analyzing hyperbolic knots with alternating projections on embedded surfaces.
result Two-sided bounds on cusp area in terms of twist number and surface genus.
Improved clustering algorithm for large datasets.
problem Finding alternative partitions in large datasets.
method Iterative Spectral Method (ISM) for alternative clustering.
result Significantly improved scalability and computation time.
New invariants derived from Seifert graphs help distinguish alternating links.
problem Distinguishing alternating links from each other.
method Introducing new quantities derived from Seifert graphs of reduced alternating link diagrams and proving they are link invariants.
result These new invariants can easily distinguish many different alternating links, even large and complicated ones.
Generalizing Howie and Greene's characterization of alternating knots, we give a topological characterization of almost alternating knots.
Characterizes alternating link exteriors using cubed complexes.
problem Understanding the structure of alternating link exteriors.
method Introducing signed BW cubed-complexes and characterizing their homeomorphism to link exteriors.
result Characterization of alternating link exteriors in terms of cubed complexes.
Study sharpens unlinking number bounds for special alternating links.
problem Determining the exact unlinking number for special alternating links.
method Analyzes links in the 3-sphere, focusing on special alternating links and their crossing changes.
result Sharp lower bounds for unlinking number realized by crossing changes in alternating diagrams.
Alternating knots follow a pattern theorem, making them rarer than previously thought.
problem Understanding the scarcity of alternating knots.
method Developed a pattern theorem for alternating knots and used it to prove a conjecture about their rarity.
result Alternating knots are rarer than previously believed.
The paper proves periodic projections of alternating knots using Flyping theorem.
problem Proving the existence of periodic projections of alternating knots.
method Flyping theorem and symmetry analysis.
result Existence of q-periodic alternating projections of prime alternating q-periodic knots. The paper tabulates and computes the number of alternating pretzel links up to a given crossing number.
problem Computing the total number of alternating pretzel links for a given crossing number.
method Derived a closed formula to compute the total number of alternating pretzel links, P(c), for any given crossing number c. result The number of alternating pretzel links grows exponentially with the crossing number.
New findings on hyperbolicity of augmented links in thickened surfaces.
problem Proving hyperbolicity of links in thickened surfaces.
method Extending hyperbolicity results to generalized augmented cellular alternating links.
result Generalized augmented cellular alternating links in thickened surfaces are hyperbolic.
Algorithm finds ribbon disks for alternating knots, resolving sliceness for most prime knots.
problem Finding ribbon disks for alternating knots to determine sliceness.
method Algorithm based on Donaldson's diagonalization theorem, computer implementation.
result Successfully finds ribbon disks for many prime knots, resolves sliceness for most.
We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for…
Automatic computation speeds up crosscap number calculation for alternating knots.
problem Computing crosscap numbers for alternating knots efficiently.
method Introduced an automatic computation with complexity O(E3). result Crosscap numbers of alternating knots can be computed in O(E3) time.