arXiv research

New definition of twisted 1-loop invariant using Ptolemy coordinates.

New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.

The paper studies the n-loop Kontsevich invariant of knots with same Alexander polynomial.

CLN2INV learns precise loop invariants from program traces.

This paper introduces two virtual knot theory ``analogues'' of a well-known family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop fram…

Proves loop coproduct invariance under simple homotopy equivalences.

The total homology of the loop space of the configuration space of ordered distinct n points in R^m has a structure of a Hopf algebra defined by the 4-term relations if m>2. We describe a relation of between the cohomology of this loop space and the set of finite type invariants for the pure braid group with n strands.…

Paper connects knot invariants and Morse flow loops.

The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.

1-loop invariant equals torsion for 2-bridge knots.

New invariant measures loop iterations in algebraic structures.

New methods detect nontrivial loops of spheres in 4-manifolds.

Researchers prove a conjecture linking 1-loop invariants to torsion for fibered 3-manifolds.

New distances defined on Legendrian spaces without positive loops.

We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree. This implies that no 3-loop finite-type invariant can distinguish between a knot and its inverse.

Researchers compute knot invariants in Seifert manifolds using Wilson loops.

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeis…

Twisted Neumann--Zagier matrices for quantum invariants.

In this note the interrelations between several natural morphisms on the $π_1$ of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (non-linear) Maslov index of loops of quantomorphisms of prequantizations of $\C P^n$ and the Calabi-Weinstein invariant is shown, settling aff…

The loop invariants of Dimofte-Garoufalidis is a formal power series with arithmetically interesting coefficients that conjecturally appears in the asymptotics of the Kashaev invariant of a knot to all orders in $1/N$. We develop methods implemented in SnapPy that compute the first 6 coefficients of the formal power se…

L-CNNs learn gauge invariant quantities on lattices.

New geometric methods solve a conjecture for hyperbolic 3-manifolds.

The paper studies obstructions to homotopy invariance of loop coproducts.

New invariant detects non-homeomorphic arrangements with similar coefficients.

We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the…

Study finds optimal loops in hyperbolic space with Finsler structure.

The paper defines a function for knots in Seifert manifolds and connects it to Witten-Reshetikhin-Turaev invariants.

We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman dia…

The importance of the fusion relation of loops was recognized in the context of spin structures on the loop space by Stolz and Teichner and further developed by Waldorf. On a spin manifold M the equivalence classes of `fusive' spin structures on the loop space LM, incorporating the fusion property, strong regularity an…

Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loo…

Let $F_n$ be the free group on $n$ generators and $Γ_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Γ_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-in…

Stochastic representation for determinants derived from Brownian loop soups.

Motivated by recent developments in the AdS/CFT correspondence, we provide several alternative bulk descriptions of an arbitrary Wilson loop operator in Chern-Simons theory. Wilson loop operators in Chern-Simons theory can be given a description in terms of a configuration of branes or alternatively anti-branes in the …

In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the technique…

Formula derived for Gromov-Witten invariants of smooth curves.

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Cher…

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, wh…

Let Θ(M,K) denote the 2-loop piece of (the logarithm of) the LMO invariant of a knot K in M, a ZHS^3. Forgetting the knot (by which we mean setting diagrams with legs to zero) specialises Θ(M,K) to λ(M), Casson's invariant. This note describes an extension of Casson's surgery formula for his invariant to Θ(M,K). To be …

New invariant links graph structure to tropical curve properties.

We compute the Chern-Simons transgressed forms of some modularly invariant characteristic forms, which are related to the elliptic genera. We study the modularity properties of these secondary characteristic forms and the relations among them. We also compute the Chern-Simons forms of some vector bundles over free loop…

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper w…

Withdrawn by the author in favour of math.GT/0511602

New symmetry found in colored Alexander polynomial.

We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat $n$-dimensional tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singula…

Paper detects non-trivial cycles in embedding spaces using graph integrals.

New invariants explain topological properties of pseudo-Anosov maps.

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops …

The loop space $L\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\mathbb{P}_1$ is an infinite dimensional complex manifold. The loop group $LPGL(2,\mathbb{C})$ acts on $L\mathbb{P}_1$ . We prove that the group of $LPGL(2,\mathbb{C})$ invariant holomorphic …