Whitehead's Lemma is a simple result of Stallings folds.
problem None explicitly stated; Whitehead's Lemma is observed as a consequence.
method Observation of Whitehead's Lemma as a consequence of Stallings folds.
result Whitehead's Lemma is a simple result of Stallings folds.
Proves a generalized Whitehead cut vertex lemma for tree groups.
problem Extending Whitehead's cut vertex lemma to tree group conjugacy classes.
method Proves a version of Whitehead's lemma for tree groups.
result Establishes a cut vertex in star graphs for tree group conjugacy classes.
Study Whitehead torsions in inertial h-cobordisms.
problem Understanding Whitehead torsions in specific geometric contexts.
method Analyzing inertial h-cobordisms to identify subsets of the Whitehead group.
result Identified various types of Whitehead torsions representing nested subsets of the Whitehead group.
Defines Whitehead torsion for topological spaces via K-theory.
problem Classical Whitehead torsion for topological spaces.
method Introduces Whitehead categories and torsion cosheaves.
result Establishes a connection between Whitehead torsion and K-theory.
Study on Whitehead doubles and their sliceness properties.
problem Understanding sliceness of Whitehead doubles of knots.
method Survey of techniques to obstruct sliceness and improve bounds on non-orientable genus.
result Improved bounds on non-orientable 4 genus of Whitehead doubles and genus 1 non-orientable cobordisms to cable knots.
We compute the A-polynomial 2-tuple of twisted Whitehead links. As applications, we determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds.
Computes A-polynomials of knots from Whitehead sister link fillings.
problem Computing A-polynomials for knots from Whitehead sister link fillings.
method Using results on A-polynomials of Dehn fillings, formulas are derived for the A-polynomials of knots.
result Formulas to compute A-polynomials for knots from Whitehead sister link fillings.
Character variety of Whitehead link described in detail.
problem Character variety of Whitehead link.
method Nice description of a Zariski open subset.
result Zariski open subset of the character variety described.
Whitehead doubles have matching meridional rank and bridge number.
problem Determining the meridional rank of Whitehead doubles.
method Analyzing algebraically tame knots and their Whitehead doubles.
result Meridional rank and bridge number coincide for Whitehead doubles of prime knots.
Study shows width of Whitehead double knots is four times the original knot's width.
problem Determining the width of Whitehead doubles of nontrivial knots.
method Proved width relationship using knot width function.
result Width of Whitehead double knots is four times the original knot's width.
We prove the volume conjecture for an infinite family of links called Whitehead chains that generalizes both the Whitehead link and the Borromean rings.
We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of Handel-Mosher axis bundles that are not just a single axis, as well as of "nongeneric" behavior in the sense of the "trai…
New knots have unique Whitehead doubles.
problem Understanding independence in knot concordance groups.
method Using 4-dimensional constructions and Whitehead doubles.
result Infinite families of knots with independent Whitehead doubles.
Using geodesic currents, we provide a theoretical justification for some of the experimental results regarding the behavior of Whitehead's algorithm on non-minimal inputs, that were obtained by Haralick, Miasnikov and Myasnikov via pattern recognition methods. In particular we prove that the images of "random" elements…
We show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive…
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
The paper creates spherical CR structures for Whitehead link surgeries.
problem Creating spherical CR structures for Dehn surgeries of the Whitehead link.
method Applying spherical CR Dehn surgery theorem to deform Ford domains.
result Infinitely many Dehn surgeries of the Whitehead link complement with spherical CR structures.
We construct a higher Whitehead torsion map, using algebraic K-theory of spaces, and show that it satisfies the usual properties of the classical Whitehead torsion. This is used to describe a "geometric assembly map" defined on stabilized structure spaces in purely homotopy theoretic terms.
The Whitehead link complement's geometry and topology are studied using ideal triangulations and tropical geometry.
problem Understanding the geometry and topology of the Whitehead link complement and its Dehn surgeries.
method Ideal triangulations, spun-normal surfaces, and tropical geometry.
result All boundary curves of the Whitehead link complement are strongly detected by its character variety.
The article calculates asymptotic expansions for quantum invariants from surgeries on Whitehead link components.
problem Calculating quantum invariants for 3-manifolds resulting from surgeries on Whitehead link components.
method Asymptotic expansion of relative Reshetikhin-Turaev and Turaev-Viro invariants.
result Asymptotic formulas for both invariants are derived.
A technique to calculate the colored Jones polynomials of satellite knots, illustrated by the Whitehead doubles of knots, is presented. Then we prove the volume conjecture for Whitehead doubles of a family of torus knots and show some interesting observations.
The operation of (untwisted) Whitehead doubling trivializes the Alexander module of a knot (and consequently, all known abelian invariants), and converts knots to topologically slice ones. In this note we show that Whitehead doubling does not trivialize the rational function that equals to the 2-loop part of the Kontse…
New knots are found to be non-simple in Legendrian contact geometry.
problem Identifying non-simple Legendrian knots in contact geometry.
method Utilized knot Floer homology and the distinguished surgery triangle.
result Whitehead doubles of the trefoil are Legendrian non-simple.
In this paper we study the knot Floer homology invariants of the twisted and untwisted Whitehead doubles of an arbitrary knot K. We present a formula for the filtered chain homotopy type of HFK(D(+,K,t)) in terms of the invariants for K, where D(+,K,t) denotes the t-twisted positive-clasped Whitehead double of K. In pa…
New 3-manifold spines with unique Whitehead graphs identified.
problem Identifying new 3-manifold spines with specific Whitehead graph types.
method Cyclic presentations and conditions on parameters to show spines of 3-manifolds.
result Examples of 3-manifolds with Whitehead graphs of specific types.
Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger…
In this paper we determine topologically the canonical component of the SL2(C) character variety of the Whitehead link complement.
Curious connection found between dynamical systems and 3D manifold fibrations.
problem Existence of fibrations in 3D manifolds and their relation to dynamical systems.
method Established a one-to-one correspondence between fibrations of the Whitehead link complement and `simple orbit pairs` on the torus.
result Found a correspondence between the order of rational numbers and the order of orbits in fibrations.
New invariants for link concordance defined using Whitehead doubles.
problem Defining and analyzing new concordance invariants for links.
method Extending slice-torus invariants to links, proving properties, and using Whitehead doubles.
result New strong concordance invariants for links, independent of slice-torus link invariants.
Classifies tight contact structures on surgeries of the Whitehead link.
problem Classifying tight contact structures on surgeries of the Whitehead link.
method Analyzes various surgeries on the Whitehead link to classify tight contact structures.
result Determines tight contact structures, Stein fillability, and virtually overtwisted properties.
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
Study multiplicity of non-acyclic SL2-representations and L-functions of Whitehead links.
problem Understanding the multiplicity of non-acyclic SL2-representations and their L-functions.
method Geometric interpretation of Reidemeister torsion divisors and application to L-functions.
result Prove multiplicity two for odd-twisted Whitehead links.
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
Uncountably many contractible 3-manifolds with double 3-space property and those without are identified.
problem Characterizing contractible 3-manifolds with specific topological properties.
method Generalizations of the Whitehead Link and interlacing theory.
result Existence of uncountably many contractible 3-manifolds with and without the double 3-space property.
Jones polynomial bounds and crossing numbers of knots.
problem Bounding the degree of colored Jones polynomial in terms of crossing number.
method Sharpness of bounds determined for adequate knots; application to satellite knots.
result Sharpness of bounds for adequate knots; determination of crossing numbers for specific knots.
Study links using Soergel bimodules and Serre duality.
problem Computing the triply-graded homology of the Whitehead link.
method Representability of partial trace functors and diagrammatics for Hochschild cohomology.
result Computed the triply-graded homology of the Whitehead link.
Whitehead link surgeries are not L-spaces if they support taut foliations.
problem Characterizing rational homology spheres as L-spaces.
method Analyzing Whitehead link surgeries and their foliations.
result Whitehead link surgeries are not L-spaces if they support coorientable taut foliations.
Algorithm calculates Hopf invariant for simplicial mappings.
problem Computing Hopf invariant for simplicial mappings.
method Proposed an algorithm based on Whitehead's integral formula.
result Algorithm successfully computes Hopf invariant.
The study characterizes slopes for hyperbolic knots and Whitehead doubles.
problem Identifying characterizing slopes for hyperbolic knots and Whitehead doubles.
method Combining JSJ decompositions, geodesic lengths, and volume inequalities.
result Explicit conditions for characterizing slopes and examples of non-characterizing slopes.
We determine lens surgeries (i.e.\ Dehn surgery yielding a lens space) along the n-twisted Whitehead link. To do so, we first give necessary conditions to yield a lens space from the Alexander polynomial of the link as: (1) n=1 (i.e. the Whitehead link), and (2) one of surgery coefficients is 1, 2 or 3. Our interes…
We describe a family of representations in SL(3,C) of the fundamental group π of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C) and can be seen as factorising through a quotient of π defined by a certain exception…
Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.
problem Behavior of knot invariant θ under satellite operations.
method Proved additivity, introduced computational tool, verified conjecture for 2977 knots.
result Distinguished knots using invariant θ and verified conjecture for 2977 primes.
New examples of manifolds with similar homotopy but different simple homotopy types.
problem Characterizing groups for which high-dimensional manifolds can be homotopy equivalent but not simple homotopy equivalent.
method Construction of doubles of thickenings and use of a formula for Whitehead torsion.
result Examples of high-dimensional manifolds exist for any finitely presented group with a nontrivial Whitehead group involution.
We show that for an alternating pretzel knot K the canonical genera of its Whitehead doubles W(K) are equal to the crossing number c(K) of K, verifying a conjecture of Tripp in the case of these knots.
The paper introduces knot invariants using stratified homotopy groups.
problem Defining knot invariants in stratified spaces.
method Introducing stratified homotopy groups and proving their properties.
result Stratified Whitehead's theorem holds for these groups.
We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of (2,2^n-1) torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank.
The paper calculates the asymptotics of quantum invariants for Whitehead chains.
problem Quantum invariants of Whitehead chains with colored clasps.
method Asymptotic analysis of colored Jones polynomials, considering limiting ratios of sequences.
result The exponential growth rate of invariants matches the hyperbolic volume of link complements.
Study shows bounds on knot groups and nonembeddability of certain open manifolds.
problem Nonembeddability of contractible open manifolds in compact spaces.
method Whitehead doubling and rank estimates of knot groups.
result Infinitely many non-homeomorphic contractible open manifolds cannot embed in compact spaces.