arXiv research

By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…

A theorem of Furuta and Fintushel-Stern provides a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In this article we use these results and some 4-dimensional constructions to produce infinite families of positive torus knot…

We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the co…

The paper proves a Whitehead theorem for fine shape spaces.

Study algebraic K-theory of 3-manifold groups using Farrell-Jones isomorphism and geometrization.

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to…

The paper describes projective structures with torsion using formal frames and Thomas-Whitehead connections.

Mazur's knot exterior is described by a single regular ideal octahedron, leading to hyperbolic structures related to the Whitehead link.

H. Masur and J. Smillie proved precisely which singularity index lists arise from pseudo-Anosov mapping classes. In search of an analogous theorem for outer automorphisms of free groups, Handel and Mosher ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible outer auto…

The study computes trace fields and minimal polynomials for specific knots and links.

We show how to construct, for each $r \geq 3$, an ageometric, fully irreducible $φ\in Out(F_r)$ whose ideal Whitehead graph is the complete graph on $2r-1$ vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal …

Defines Whitehead torsion for topological spaces via K-theory.

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.

Study on Whitehead doubles and their sliceness properties.

Study satellite operators expanding concordance groups and their applications.

Computes A-polynomials of knots from Whitehead sister link fillings.

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.

Character variety of Whitehead link described in detail.

Whitehead doubles have matching meridional rank and bridge number.

Let X be a closed manifold of dimension 2m >= 6 with torsion-free middle-dimensional homology. We construct metrics on X of arbitrarily small volume, such that every middle-dimensional submanifold of less than unit volume necessarily bounds. Thus, Loewner's theorem has no higher-dimensional analogue.

We prove the volume conjecture for an infinite family of links called Whitehead chains that generalizes both the Whitehead link and the Borromean rings.

Let $φ$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_φ = F_n \rtimes_φ \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite graph $K_{3,3}$ into $\partial G_φ$. To do so, we define the directional Whitehead graph and …

We define the LS-category cat_g by means of covers of a space by general subsets, and show that this definition coincides with the classical Lusternik-Schnirelmann category for compact metric ANR spaces. We apply this result to give short dimension theoretic proofs of the Grossman-Whitehead theorem and Dranishnikov's t…

We study the Whitehead torsions of inertial h-cobordisms, and identify various types representing a nested sequence of subsets of the Whitehead group. A number of examples are given to show that these subsets are all different in general.

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…

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…

A fundamental theorem in the study of Dunwoody manifolds is a classification of finite graphs on $2n$ vertices that satisfy seven conditions (concerning planarity, regularity, and a cyclic automorphism of order $n$). Its significance is that if the presentation complex of a cyclic presentation is a spine of a 3-manifol…

Paper confirms Whitehead's conjecture for aspherical 2-complexes.

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.

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…

The article calculates asymptotic expansions for quantum invariants from surgeries on Whitehead link components.

The paper proves a generalized inverse function theorem for curved $L_\infty$ spaces.

We observe that Whitehead's lemma is an immediate consequence of Stallings folds.

New knots are found to be non-simple in Legendrian contact geometry.

A formula calculates the Euler class of foliations using dual graphs.

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.

Proves a generalized Whitehead cut vertex lemma for tree groups.

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…

Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then use this the…

In this paper we determine topologically the canonical component of the $SL_2(\mathbb{C})$ character variety of the Whitehead link complement.

Classifies tight contact structures on surgeries of the Whitehead link.

The Whitehead link exterior lacks most Euler class taut foliations.

Study multiplicity of non-acyclic SL2-representations and L-functions of Whitehead links.

Let $M^n$ be a closed, connected $n$-manifold. Let $\mtm$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $\mtm$ is homotopy equivalent to the Spanier-Whitehead dual of $M$ with a disjoint basepoint, $M_+$. This dual can be viewed as the function spectrum, $F(M, S)$, whe…

Jones polynomial bounds and crossing numbers of knots.