arXiv research

Algorithm calculates Hopf invariant for simplicial mappings.

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.

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

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

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 formulas link Milnor invariants to Heegaard Floer homology.

J. H. C. Whitehead gave an elegant integral formula for the Hopf invariant H(p) of a smooth map p from the 3-sphere to the 2-sphere. Given an open book structure b on the 3-sphere (or, essentially equivalently, an isolated critical point of a map F from 4-space to the plane), Whitehead's formula can be "integrated alon…

Proves loop coproduct invariance under simple homotopy equivalences.

The Whitehead link exterior lacks most Euler class taut foliations.

New examples of manifolds with similar homotopy but different simple homotopy types.

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

We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length < k are all zero into a link with vanishing Milnor invariants of length < 2k, and we provide formulas for th…

We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.

The paper extends BPS invariants for framed knots and links.

Study flat GL(1|1) connections using fatgraphs and coordinates.

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$ if $n<2τ(K_2)$ holds. We also give a formula of Ozsváth-Szabó's $τ$-invariant as…

Given an $L^2$-acyclic connected finite $CW$-complex, we define its universal $L^2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\operatorname{Wh}^w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré d…

In [8], P. Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [1], M. Bordemann proved this existence using the framework of Thomas-Whitehead connections. In [9], we gave a new proof of the same theorem thanks to the Cartan connections. After these works there was no explicit f…

Formulae for Rasmussen invariant of satellite knots with wrapping number 2 proved.

Defines Whitehead torsion for topological spaces via K-theory.

Study on Whitehead doubles and their sliceness properties.

We find that Koschorke's $β$-invariant and the triple $μ$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit geometric interpretations in terms of Vassiliev's ornaments via new operations akin to the Jin…

Character variety of Whitehead link described in detail.

Whitehead doubles have matching meridional rank and bridge number.

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 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…

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

Formula for satellite operators using knot Floer homology.

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…

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.

New 3-manifold spines with unique Whitehead graphs identified.

Proves a generalized Whitehead cut vertex lemma for tree groups.

New Heegaard Floer homology findings block chirally cosmetic surgeries.

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…

We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative of the twisted Alexander invariant for a hyperbolic knot exterior gives the hyper…

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.

Enhances knot surgery formulae for instanton Floer homology.

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

Jones polynomial bounds and crossing numbers of knots.

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…

Study links using Soergel bimodules and Serre duality.