arXiv research

Proves resurgence properties for Habiro elements from radial limits of theta series.

The Habiro ring of a number field uses power series to study algebraic K-theory.

Researchers lift knot coloring polynomial to Habiro ring.

Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.

We use skein theory to compute the coefficients of certain power series considered by Habiro in his theory of sl_2 invariants of integral homology 3-spheres. Habiro originally derived these formulas using the quantum group U_q sl_2. As an application, we give a formula for the colored Jones polynomial of twist knots, g…

New presentation of Goussarov-Habiro Lie algebra using primitive Feynman diagrams.

We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to thi…

New formula and properties of inverted Habiro series derived from GM series.

Study compares WRT and CGP invariants using Habiro's series.

This paper studies the rational homotopy groups of the group $\mathrm{Diff}(S^4)$ of self-diffeomorphisms of $S^4$ with the $C^\infty$-topology. We present a method to prove that there are many `exotic' non-trivial elements in $π_*\mathrm{Diff}(S^4)\otimes \mathbb{Q}$ parametrized by trivalent graphs. As a corollary of…

Invariants of hyperbolic knots connect to quantum modularity.

Habiro gave principal ideals of Z[q,q^{-1}] in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links and Brunnian links are contained in smaller ideals of Z[q,q^{-1}] generated by …

Defines map from skein module to Habiro ring for quantum modularity.

We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollar…

Categorifies a skein relation for links colored by one-column Young diagrams.

We prove that two closed oriented 3-manifolds have isomorphic quintuplets (homology, space of spin structures, linking pairing, cohomology rings, Rochlin function) if, and only if, they belong to the same class of a certain surgery equivalence relation introduced by Goussarov and Habiro.

We define $A_k$-moves for embeddings of a finite graph into the 3-sphere for each natural number $k$. Let $A_k$-equivalence denote an equivalence relation generated by $A_k$-moves and ambient isotopy. $A_k$-equivalence implies $A_{k-1}$-equivalence. Let ${\cal F}$ be an $A_{k-1}$-equivalence class of the embeddings of …

New topological realization of Kontsevich graph complex for large dimensions.

We prove that the quantum SO(3)-invariant of an arbitrary 3-manifold $M$ is always an algebraic integer, if the order of the quantum parameter is co-prime with the order of the torsion part of $H_1(M,\BZ)$. An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-…

The results become part of an upcoming paper.

The universal sl_2 invariant of bottom tangles has a universality property for the colored Jones polynomial of links. Habiro conjectured that the universal sl_2 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra U_h(sl_2) of the Li…

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's $μ$ invariant. These invariants are seen to be naturally r…

We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several d…

New formulas connect knot invariants with theta functions.

Complete classification of links up to specific moves.

The study connects knot complements to 3d theories via half-index calculations.

New geometric invariant from disc intersections captures all coloured Jones polynomials.

For a certain class of compact oriented 3-manifolds, M. Goussarov and K. Habiro have conjectured that the information carried by finite-type invariants should be characterized in terms of ``cut-and-paste'' operations defined by the lower central series of the Torelli group of a surface. In this paper, we observe that t…

Given a rational homology 3-sphere M with the first integral homology of rank b and a link L inside M, colored by odd numbers, we construct a unified invariant I_{M,L} belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev…

We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also continue J. Levine's…

K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of $C_k$-move. In this paper we give another geometric condition in terms of Brunnian local move. The proof is simple and self-contained.

L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).

A formula for the difference of Vassiliev invariants of degree k+1 of two knots all of whose Vassiliev invariants of degree k agree is proven. The proof uses K. Habiro's C-moves and his theorem which relates them to Vassiliev invariants.

In 1993 K. Habiro defined $C_k$-move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by $C_k$-moves if and only if they have the same Vassiliev invariants of order $\leq k-1$. In this paper we define Vassiliev invariant of type $(k_1,...,k_l)$, and show that, for $k=k…

We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations, which can be seen as `higher order Gauss diagrams'. This Arrow calculus is used to d…

The Jones polynomial of a knot in 3-space is a Laurent polynomial in $q$, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing defin…

The $C_k$-equivalence is an equivalence relation generated by $C_k$-moves defined by Habiro. Habiro showed that the set of $C_k$-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order $\leq k-1$. We see that the set of $C_k$-…

We give a proof of the LMO conjecture which say that for any simply connectd simple Lie group $G$, the LMO invariant of rational homology 3-spheres recovers the perturvative invariant $τ^{PG}$. By Habiro-Le theorem, this implies that the LMO invariant is the universal quantum invariant of integral homology 3-spheres.

Hikami observed a discontinuity in a WRT invariant at roots of unity.

In this note we reconsider a familiar result in Vassiliev knot theory - that the coefficients of the Alexander-Conway polynomial determine the top row of the Kontsevich integral - from the point of view of Kazuo Habiro's clasper theory. We observe that in this setting the calculation reflects the topology of the univer…

We prove for the Reidemeister-Turaev torsion of closed oriented three-manifolds some finiteness properties in the sense of Goussarov and Habiro, that is, with respect to some cut-and-paste operations which preserve the homology type of the manifolds. In general, those properties require the manifolds to come equipped w…

Cheptea, Habiro and Massuyeau constructed the LMO functor, which is defined on a certain category of cobordisms between two surfaces with at most one boundary component. In this paper, we extend the LMO functor to the case of any number of boundary components, and our functor reflects relations among the parts correspo…

Constructs universal link invariants from intersections in configuration spaces.

We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally …

In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …

A C_n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a `higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to C_n-moves, under certain restrictions. Namely, we give a classification up to C_4-move…

In 2006 Habiro initiated a construction of generating functions for Witten-Reshetikhin-Turaev (WRT) invariants known as unified WRT invariants. In a series of papers together with Irmgard Buehler and Christian Blanchet we extended his construction to a larger class of 3-manifolds. The unified invariants provide a stron…

In a previous paper [Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243--270, arXiv:math.GT/0010247], a group H_g of homology cylinders over the oriented surface of genus g is defined. A filtration of H_g is defined, using the Goussarov-Habiro notion of finite-type. It is e…