The paper studies arithmeticity and hidden symmetries in fully augmented pretzel link complements.
problem Determining arithmeticity and commensurability of fully augmented pretzel link complements.
method Careful analysis of geometry, including cusp shapes and totally geodesic surfaces.
result Construction of two infinite families of non-arithmetic fully augmented link complements.
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.
Study fully augmented links using algebraic equations from link diagrams.
problem Understanding the geometry and commensurability of fully augmented links.
method Extend Thistlethwaite and Tsvietkova's method to fully augmented links, define algebraic equations, and relate solutions to cusp shapes.
result Solutions to algebraic equations are linked to cusp shapes of fully augmented link complements.
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformatio…
Flat fully augmented links with homeomorphic complements are equivalent.
problem Determining equivalence of flat fully augmented links based on their complements.
method Careful analysis of totally geodesic surfaces and cusps in link complements and their behavior under homeomorphism.
result Complete classification of flat fully augmented link complements with multiple reflection surfaces and symmetries.
Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.
problem Understanding the geometry of fully augmented links in doubled 3-manifolds.
method Constructing fully augmented links on the reflection surface of doubled 3-manifolds and finding bounds on cusp shapes and volumes.
result Bounds on cusp shapes and volumes of hyperbolic links in doubled 3-manifolds.
Fully augmented links have dense volume densities but discrete in certain ranges.
problem Characterizing the volume density spectrum of fully augmented links.
method Analyzing the ratio of volume to the number of augmentations.
result The set of FAL volume densities is dense in $[2\voct, 10\vtet)$ but discrete in $[\voct,2\voct)$.
A method to convert pretzel links into braids.
problem Representing pretzel links as braids.
method General procedure to convert any pretzel link into a braid.
result A specific braid word corresponding to the pretzel link.
This work classifies belted sum decompositions of fully augmented links.
problem Understanding belted sum decompositions of fully augmented links.
method Explicit classifications of thrice punctured spheres in FAL complements, geometric, combinatorial, and diagrammatic characterizations.
result Every FAL complement canonically decomposes into FALs which are either prime or two-fold covers of the Whitehead link.
We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links b…
The paper connects link symmetries to finite subgroups of O(3).
problem Understanding symmetries of flat fully augmented links.
method Developed a dictionary linking graph automorphisms to link symmetries, constructing infinite link classes.
result Symmetry groups of b-prime flat fully augmented links match finite subgroups of O(3).
Study completes braid index determination for all pretzel links.
problem Determine braid indices of all pretzel links.
method Divided pretzel links into three types and derived formulas for braid indices.
result Close to determining braid indices for all Type 3 pretzel links.
This paper determines the braid indices for non-alternating pretzel links.
problem Determining the braid index of non-alternating pretzel links.
method Classified pretzel links into three types and computed braid indices for Type 1 and Type 2.
result Precise braid indices for all non-alternating Type 1 and Type 2 pretzel links.
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.
This study limits the number of pretzel links with a specific Jones polynomial span.
problem Determining the number of pretzel links with a given Jones polynomial span.
method Developed an algorithm to decide if a knot is pretzel and used it to identify all pretzel knots up to nine crossings.
result Identified all pretzel knots up to nine crossings, proving 812 is not pretzel. Paper generalizes pretzel links using spatial graphs.
problem Classical pretzel links need a generalization.
method Introduces graph-pretzel links based on spatial graph projections.
result Constructs an infinite family of distinct ribbon knots.
The paper classifies pretzel links with 2 components and gives conditions for those with 3 or more.
problem Classifying pretzel links based on their self delta-equivalence.
method Using Conway polynomials to determine self delta-equivalence for links with 2 or more components.
result Necessary and sufficient conditions for self delta-equivalence of pretzel links with 3 or more components.
The study classifies χ−slice pretzel links and Seifert fiber spaces.
problem Understanding χ−slice pretzel links and their properties. method Analyzing the sliceness of pretzel knots and extending results to pretzel links.
result Complete classifications of positive and negative pretzel links that are χ−slice, and partial classifications of 3-stranded and 4-stranded pretzel links. Study supports conjecture about pretzel links' homology.
problem Investigate extreme Khovanov homology of pretzel links.
method Geometric realizations of the independence simplicial complex.
result Supports conjecture that extreme Khovanov homology is torsion-free.
The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a general formula for the unreduced Khovanov homology…
New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. As a consequence, we prove the slice-ribbon conjecture for 4-strand…
We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to…
Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.
problem Calculating colored Jones polynomials for general oriented links is difficult.
method Using Kuperberg's linear skein theory, they focus on one-row polynomials for pretzel links.
result Existence of tails for specific pretzel knots' Jones polynomials is shown.
Knot invariants and quiver stability linked through full twists.
problem Relating knot invariants to quiver stability under twists.
method Using HOMFLY-PT skein relations and linking/unlinking operations on symmetric quivers.
result Full twists on knots correspond to unlinking or linking of augmented symmetric quivers, confirming stable growth in both.
Geometric proof confirms link volume conjecture.
problem Volume conjecture for hyperbolic 3-manifolds.
method Topological and skein theoretic techniques.
result Complements of octahedral fully augmented links are isometric to fundamental shadow links.
Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.
problem Understanding polynomial invariants of pretzel links.
method Revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links P(1,1,n). result Reveals properties of Alexander-Conway and Kauffman bracket polynomials for P(1,1,n). Study disproves finiteness conjecture for a specific link's skein module.
problem Finiteness conjecture for a specific link's skein module.
method Analysis of Kauffman bracket skein module over Q(q21). result Skein module is not finitely generated over Q(q21)[t1,t2]. We explicitly calculate the universal character ring of the (-2,2m+1,2n)-pretzel link and show that it is reduced for all integers m and n.
For any knot, a 3-sphere triangulation exists with a knotted edge.
problem Finding triangulations of the 3-sphere with specific knot configurations.
method Constructive proof using fully augmented links.
result Constructs one-vertex triangulations of the 3-sphere with knotted edges.
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…
We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the A-polynomial of the knot. For example, we show that for 2-bridge knots the polynomial…
Study the JSJ-decomposition of a specific 3-manifold.
problem Classify the JSJ-decomposition of a 3-manifold from 0-surgery on a pretzel knot.
method Utilize the classification of exceptional fillings of minimally twisted five-chain links.
result Determine the JSJ-decomposition of the 3-manifold.
A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We classify pretzel knots with any number of tangles which admit L-space surgeries. This rests on Gabai's classification of fibered pretzel links.
New examples show no upper bounds on link volumes on incompressible surfaces.
problem Finding upper bounds on volumes of links on incompressible surfaces.
method Examined weakly generalised alternating and fully augmented links on incompressible surfaces.
result Found infinite families of links on incompressible surfaces with no upper bounds on volume.
In this paper we use Heegaard Floer link homology to determine the dual Thurston polytope for pretzel links of the form P(-2r_1-1, 2q_1, -2q_2, 2r_2+1) where r_i and q_i are positive integers. We apply this result to determine the Thurston norms of spanning surfaces for the individual link components, and we explicitly…
The paper studies symmetries and hidden symmetries of twisted knot complements.
problem Analyzing symmetries and hidden symmetries of (ε,dL)-twisted knot complements. method Analysis via long Dehn fillings on fully augmented links complements.
result No hidden symmetries in (ε,dL)-twisted knot complements, implying at most two commensurability classes. This paper finds bounds on the ribbonlength of knots and links with up to 9 crossings.
problem Finding the infimal folded ribbonlength of knots and links.
method Analyzing pretzel links and constructing knots to find bounds on ribbonlength.
result Uniform bounds on infimal folded ribbonlength for certain link families.
Study shows surgeries on specific links result in L-spaces.
problem Understanding surgeries on specific links and their outcomes.
method Proving surgeries on chainmail links yield L-spaces using alternating surgeries.
result Alternating surgeries on chainmail links result in total L-spaces.
Study on pretzel knots showing cyclic branched covers are L-spaces.
problem Understanding cyclic branched covers of pretzel knots and their properties.
method Analyzing pretzel knots Kk and their n-fold cyclic branched covers for all n≥1. result The n-fold cyclic branched covers of pretzel knots Kk are L-spaces for all n≥1. We compute the reduced Khovanov homology of 3-stranded pretzel links. The coefficients are the integers with the "even" sign assignment. In particular, we show that the only homologically thin, non-quasi-alternating 3-stranded pretzels are P(-p,p,r) with p an odd integer and r greater than or equal to p (these were sho…
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…
New examples of hyperbolic links with generalized torsion elements found.
problem Finding generalized torsion elements in the fundamental groups of hyperbolic links.
method Analyzing the Weeks manifold, figure-eight sister manifold, and Whitehead sister link to identify generalized torsion elements.
result First examples of hyperbolic links with link groups admitting generalized torsion elements.
In this paper we consider some families of links, including (-2,2m+1,2n)-pretzel links and twisted Whitehead links. We calculate the character varieties of these families, and determine the number of irreducible components of these character varieties.
Generalizes Thistlethwaite-Tsvietkova method to 3-manifolds.
problem Determining the hyperbolic structure of 3-manifolds.
method Polyhedral decomposition, PSL(2,ℂ) representations, Thurston equations.
result Solutions to equations correspond to complete hyperbolic structures.
The tail of the colored Jones polynomial of an alternating link is a q-series invariant whose first n terms coincide with the first n terms of the n-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…
New method fractures hyperbolic manifolds using cone singularities.
problem Deforming hyperbolic manifolds with cone singularities.
method Direct manipulation of a fundamental polyhedron to change cone angles.
result Upper unknotting tunnels of highly twisted links can be drilled out.
With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new struc…