arXiv research

Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.

Suppose that S is a surface with boundary and that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Let Y_g, Y_h, and Y_{hg} be the three-manifolds with open book decompositions given by (S,g), (S,h), and (S,hg), respectively. We show that the Ozsvath-Szabo contact invariant is natural u…

For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant associated to T_{ws_i} to that associated to T_w, where s_i is one of the standard ge…

We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology …

This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula fo…

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…

Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and c…

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

The study proves nearly Frobenius algebras over certain domains are Frobenius.

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…

The study connects knot Floer homology and Lagrangian cobordisms.

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 …

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.

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…

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

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.

By work of W. Thurston, knots and links in the 3-sphere are known to either be torus links, or to contain an essential torus in their complement, or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic 3…

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.

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

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

The paper proves an asymptotic additivity of Turaev-Viro invariants for a family of 3-manifolds.

Computed minimum crossing numbers for Turaev genus 2 links.

Turaev transformed knot theory and 3-manifold invariants.

Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.

An expository article on Turaev surfaces written for "A Concise Encyclopedia of Knot Theory," to appear.

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

We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $\exp({2π\sqrt{-1}}/{r})$ instead of the standard $\exp({π\sqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $\infty$, these invariants grow exponentially with growt…

New algorithm speeds up knot polynomial calculations.

We survey a geometric approach to the Johnson homomorphisms using the Goldman-Turaev Lie bialgebra.

In 2005 V. Turaev introduced the theory of topology of words and phrases. Turaev defined an equivalence relation on generalized words and phrases which is called homotopy. This is suggested by the Reidemeister moves in the knot theory. Then Turaev gave the homotopy classification of generalized words with less than or …

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…

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…

The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribb…

Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.

We state a simple criterion to prove the infiniteness of the image of Reshetikhin-Turaev irreducible representations of the mapping class groups of surfaces. We use it to study some of the Reshetikhin-Turaev representations associated to the tori with one and two punctures and derive an alternative proof of a result of…