arXiv research

String topology coproduct and Turaev cobracket computed for surfaces.

Reformulates divergence map for Turaev cobracket in non-commutative geometry.

The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman bracket. In this survey based on the author's joint works with A. Alekseev, Y. Kuno…

Goldman and Turaev constructed a Lie bialgebra structure on the free $\mathbb{Z}$-module generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket $Δ(α)$ is zero if and only if $α$ is a power of a simple class. Chas constructed examples that show Turaev's conjecture is, unfortunate…

We give a tensorial description of the Turaev cobracket on any genus 0 compact surface through the standard group-like expansion, where the Bernoulli numbers appear.

In a previous paper, we defined an operation $μ$ that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding que…

The vector space $\V$ generated by the conjugacy classes in the fundamental group of an orientable surface has a natural Lie cobracket $\mapδ{\V}{\V\times \V}$. For negatively curved surfaces, $δ$ can be computed from a geodesic representative as a sum over transversal self-intersection points. In particular $δ$ is zer…

New combinatorial approach to Goldman-Turaev Lie bialgebra using cyclic word partitions.

In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710…

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6)(1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas (Combinatorial Lie bialgebras of curves on surfaces, Topology 43 (2004) 543), b…

By introducing a refinement of the Goldman-Turaev Lie bialgebra, we interpret the divergence cocycle in the Kashiwara-Vergne problem and the Enomoto-Satoh obstructions for the surjectivity of the Johnson homomorphisms as some part of a regular homotopy version of the Turaev cobracket.

Introduces a new triple coproduct for knots on surfaces, preserving local crossing patterns.

Let $S$ be a closed connected oriented surface of genus $g>0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(π_1(S), \mathrm{GL}(1, \mathbb{R}))/\mathrm{GL}(1, \mathbb{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbb{R})$-bundles over $S$. There is a surjective Lie al…

In this paper, we describe a surprising link between the theory of the Goldman-Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara-Vergne (KV) problem in Lie theory. Let $Σ$ be an oriented 2-dimensional manifold with non-empty boundary and $\mathbb{K}$ a field of characteristic zero. The Goldman-Turaev Lie…

Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinator…

For an oriented 2-dimensional manifold $Σ$ of genus $g$ with $n$ boundary components the space $\mathbb{C}π_1(Σ)/[\mathbb{C}π_1(Σ), \mathbb{C}π_1(Σ)]$ carries the Goldman-Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded (under the natural filtratio…

For a compact oriented surface $Σ$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(Σ)$ spanned by free homotopy classes of loops in $Σ$ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) T…

Functorial approach connects operads to Lie bialgebras.

We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quo…

Given two free homotopy classes $α_1, α_2$ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points $m(α_1, α_2)$ of loops in these two classes. We show that for $α_1\neqα_2$ the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of $α_1$ and $α_2$ i…

We study Poisson symmetric spaces of group type with Cartan subalgebra "adapted" to the Lie cobracket.

We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in surfaces: the homological intersection number, Goldman's Lie bracket, and the author's Lie cobracket. Our approach uses fillings of the surfaces by certain graphs.

We show that for every spherical category $\C$ with invertible dimension, the Turaev-Viro TQFT admits a splitting into blocks which come from an HQFT, called the Turaev-Viro HQFT. The Turaev-Viro HQFT has the classifying space $B\grad$ as target space, where $\grad$ is a group obtained from the category $\C$. This cons…

Study Khovanov homology of Turaev genus one links, finding a trivial summand.

We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative) Hopf algebra of pointed loops on a surface. In the last version I added sections…

The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes a…

Paper finds Hempel pairs distinguishable by Turaev-Viro invariants.

The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kauffman state, the number of positive crossings, and its determinant. We also show that …

New invariants defined for knots and links using Turaev's construction.

Study shows concordance invariants bound Turaev genus.

Study how Turaev-Viro invariants change with cabling operations.

A link is adequate and has Turaev genus one if its Jones polynomial span is one less than its crossing number.

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

Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.

Paper uses Turaev-Viro TQFT to estimate 3-manifold genus.

In this paper, we examine Kitaev's lattice model for an arbitrary complex, semisimple Hopf algebra. We prove that this model gives the same topological invariants as Turaev-Viro theory. Using the description of Turaev-Viro theory as an extended TQFT, we prove that the excited states of the Kitaev model correspond to Tu…

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizo…

Study links' arc index and Turaev genus, proving conjectures.

Turaev-Viro invariants match for certain surface bundles.

The paper finds formulas for a specific invariant order 7.

We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple closed curves on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provid…

Study proves Volume Conjecture for Reshetikhin-Turaev invariants.

We work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Fu…

Among non-alternating knots with $n\le 12$ crossings given in \cite{1} Turaev genus is not known for 191 knot. For 154 of them we show that they are almost alternating, so their Turaev genus is 1.

Homological blocks match Witten-Reshetikhin-Turaev invariants for Seifert fibered 3-spheres.

Recent work extends Turaev's modular categories to non-semisimple settings.

Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.

We derive an explicit formula for the Witten-Reshetikhin-Turaev SO(3)-invariants of lens spaces. We use the representation of the mapping class group of the torus corresponding to the Witten-Reshetikhin-Turaev SO(3)-TQFT to give such formula.