arXiv research

New inequality for refined knot invariants in a specific space.

Established a stable cohomotopy refinement for Pin(2) monopole invariants.

Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.

We note that our stable homotopy refinements of Khovanov's arc algebras and tangle invariants induce refinements of Chen-Khovanov and Stroppel's platform algebras and tangle invariants, and discuss the topological Hochschild homology of these refinements.

For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the refined combina…

For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even …

Refined 3D index uses surgery and gradings to distinguish 3-manifolds.

This is a short version of math.DG/0505537. For an acyclic representation of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to a unitary representation, we define a refinement of the Ray-Singer torsion associated to this representation. This new invariant can be viewed as an…

Refines Ozsváth-Szabó d-invariants for knot concurrence.

The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. This first of two articles presents the details of the definition of the stable cohomotopy invariant and discusses its relation to the integer valued Seiberg-Witten invariant.

This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable cohomotopy invariants to Seiberg-Witten invariants without restriction on Betti numbers.…

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…

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these…

Refines a tangle invariant using XC-algebras.

We formulate large $N$ duality of $\mathrm{U}(N)$ refined Chern-Simons theory with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the $Ω$-background. This form enables us to relate refined C…

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…

Kronheimer and Mrowka constructed a variant of Seiberg-Witten invariants for a 4-manifold $X$ with contact boundary in 1997. Using Furuta's finite dimensional approximation, we refine this invariant in the case $H^1(X, \partial X; \mathbb{R})=0$.

We first apply the method and results in the previous paper to give a new proof of a result (hold in ${\bf C}/{\bf Z}$) of Gilkey on the variation of h-invariants associated to non self-adjoint Dirac type operators. We then give an explicit local expression of certain h-invariant appearing in recent papers of Braverma…

We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfi…

We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first example is a geometric engineering of five-dimensional U(1) gauge theory with a ma…

Spatial refinement of Bar-Natan homology constructed.

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …

We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for…

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $ρ$-invariants, which we call higher-order signatures. The higher-order genera o…

In the previous article "Refined Analytic Torsion on Manifolds with Boundary" we have presented a construction of refined analytic torsion in the spirit of Braverman and Kappeler, which does apply to compact manifolds with and without boundary. We now derive a gluing formula for our construction, which can be viewed as…

New invariants refine link homology, showing large genus differences.

We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refi…

New bounds for knot distances using Khovanov homology.

We study refined topological string theory in the presence of orientifolds by counting second-quantized BPS states in M-theory. This leads us to propose a new integrality condition for both refined and unrefined topological strings when orientifolds are present. We define the SO(2N) refined Chern-Simons theory which co…

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations …

Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alterna…

In the previous work ([14]) we introduced the well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined…

As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants of a knot in an $R^1$-fibration over a surface $F$. They take values in the free…

Study of 5D SYM theory on toric surfaces yields refined Vafa-Witten invariants.

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quan…

In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator $\mathcal{B}$ in the sense of Se…

New $\hat{Z}$ invariant for plumbed 3-manifolds with detailed computations and conjectures.

We define stable homotopy refinements of Khovanov's arc algebras and tangle invariants.

New field invariant refines real spectrum and relates to absolute Galois group.

Based on work of Rasmussen, we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsvath-Szabo tau invariant.

The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--b…

We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of the Milnor link and of several other links. Moreover, we refine the original Milnor invariants …

For test configurations, the Donaldson-Futaki invariant F_1 is well-known. In this note, its refinement will be discussed. Then we see that Li-Xu's pathology doesn't occur, since their example of a non-normal test configuration, with trivial normalization, actually has non-vanishing F_1 in this refined sense.

An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence…

The refined analytic torsion, defined by M. Braverman and T. Kappeler on closed manifolds, can be viewed as a refinement of the Ray-Singer torsion, since it is a canonical choice of an element with Ray-Singer norm one, in case of unitary representations. The complex phase of the refinement is given by the rho-invariant…

We give a surgery formula for the torsions and Seiberg-Witten invariants associated with $Spin^c$-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.

We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and Watson's result that the rank of knot Floer homology detects the trefoil among knots i…

We prove that the moduli space of holonomy G_2-metrics on a closed 7-manifold is in general disconnected by presenting a number of explicit examples. We detect different connected components of the G_2-moduli space by defining an integer-valued analytic refinement of the nu-invariant, a Z/48-valued defect invariant of …