The study of semi-regular biperiodic alternating links and their geometric properties.
problem Understanding the geometric properties of biperiodic alternating links.
method Analyzing semi-regular links in the thickened torus, determining volumes, and relating commensurability and arithmeticity to Euclidean tilings.
result There exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field.
Study shows link determinant densities converge to Mahler measure.
problem Determinant densities of infinite links and their convergence.
method Folner convergence of finite links to infinite biperiodic alternating link L, Mahler measure of characteristic polynomial.
result Determinant densities of finite links converge to Mahler measure of L.
Study fully augmented links in thickened torus, generalizing S3 results.
problem Classify and describe geometric properties of fully augmented links in thickened torus.
method Geometric analysis and decomposition of link complements into ideal right-angled torihedra.
result Proves Volume Density Conjecture for fully augmented links in thickened torus.
A new polynomial invariant for links in a thickened torus exhibits volume conjecture behavior.
problem Defining and characterizing a new polynomial invariant for links in a thickened torus.
method Defining a new invariant JnT, proving properties, and providing constructions. result The invariant JnT exhibits volume conjecture behavior, providing the first example of this in a virtual link. 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.
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.
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.
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.
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.
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.
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.
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-…
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.
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…
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.
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.
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.
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.
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 …
Study of links on surfaces, generalizing Menasco's polyhedral decomposition.
problem Understanding the geometry of links on surfaces embedded in 3-manifolds.
method Decomposing the complement of links into simpler pieces and using hyperbolic geometry.
result Proved various properties of hyperbolic geometry of generalised alternating links.
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.
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.
Characterizes alternating links via definite surfaces, proving conjectures.
problem Understanding alternating links through definite surfaces.
method Characterization of alternating links via definite spanning surfaces.
result Established a new proof of Tait's conjecture and derived a Seifert surface result.
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.
Standard position for surfaces extended to weakly generalized alternating links.
problem Proving standard position for surfaces in a broader class of links.
method Extending techniques from classical alternating links to weakly generalized alternating links.
result All weakly generalized alternating links are prime, and Conway spheres interact with diagrams as in classical setting.
Proves alternating quasipositive links are positive and strongly quasipositive.
problem Characterizing alternating quasipositive links.
method Analyzing alternating diagrams with specific crossing conditions.
result 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 …
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.
problem Understanding alternating links in thickened surfaces and their invariants.
method Using integer flows on Tait graphs and disc mutations, the paper proves invariants and compares link properties.
result Found alternating knots with isometric flow lattices but different linking forms.
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.
problem Understanding L-space links in chainmail links.
method Inspired by alternating links, uses flat augmentation.
result Proves alternating chainmail links are L-space links.
Proves certain alternating links have specific geometric properties.
problem Characterizing alternating links with totally geodesic checkerboard surfaces.
method Analyzes links with two totally geodesic checkerboard surfaces and characterizes them.
result Proves links with two totally geodesic checkerboard surfaces are three specific links.
Improved bounds on volume-det inequality for links with many twists.
problem Vol-Det Conjecture for highly twisted alternating links.
method Improved Burton's and Stoimenow's bounds for links with more than eight twists.
result Enhanced inequalities for volume and determinant of links with many twists.
The study shows involutory quandles of certain links are not left-orderable.
problem Determining left-orderability of involutory quandles of links.
method Using a non-left-orderability criterion for involutory quandles of non-split links, the study improved previous results and introduced new families of links.
result The involutory quandles of non-trivial alternating links and certain augmented alternating links are not left-orderable.
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…
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…
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.
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.
Algorithm for Teichmüller polynomial of certain fibered links.
problem Distinguishing fibered alternating links.
method Algorithm for computing Teichmüller polynomial for fibered alternating links associated with trees.
result Exhibit a mutant pair of links distinguished by Teichmüller polynomial.
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…
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…
The paper derives formulas for braid index of alternating links.
problem Determining the braid index of alternating links.
method Diagrammatic approach based on minimum projections.
result Explicit formulas for braid index of many alternating links.
We prove hyperbolicity for links in thickened surfaces.
problem Proving hyperbolicity for links in thickened surfaces.
method Defining fully alternating links and proving their hyperbolicity; extending results to essential surfaces.
result Prime, fully alternating links in SimesI are hyperbolic. Study extends knot genus results to two-component alternating links.
problem Determining genera of two-component alternating links.
method Unified quantity from splice sequence and correction term.
result Unified quantity completely determines orientable and non-orientable genera.
Researchers classify a specific type of Montesinos links.
problem Classifying quasi-alternating Montesinos links.
method Analyzing properties of Montesinos links and their branched covers.
result Quasi-alternating Montesinos links are identified and linked to specific topological properties.