arXiv research

Proves a general connected sum formula for families Seiberg-Witten invariants.

Formula connects knot complements' invariants.

Sum formula for relative Seiberg-Witten invariants in 4-manifolds.

Paper extends trigonometric summation formula with weights.

Proves a formula for a special invariant of 4-manifolds.

The abstract discusses fiber sum formulas for 4-manifolds using topological modular forms.

The paper studies grid homology for spatial graphs and proves a Künneth formula for connected sums.

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…

The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.

We provide a formula for the SU(3) Casson invariant for 3-manifolds given as the connected sum of two integral homology 3-spheres.

Nahm sums are $q$-series of a special hypergeometric type that appear in character formulas in Conformal Field Theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in Quantum Knot Theory, namely we prove the stability of th…

Proofs knot homology connected sums using grid complexes.

The paper studies Gauss sums and their applications in algebra and topology.

In the symplectic category there is a `connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z=X#Y in terms of the relative GW invariants of X and Y. Several appl…

The paper examines $L^2$-torsion in fibration cases relaxing standard conditions.

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori-Hecke algebras of type A.

Derives exact formula for Minkowski sum of ellipsoids in N-space.

We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.

We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with $\underline{d}(Y) \neq d(Y) \neq \overline{d}(Y)$. We also construct a homomorphism from the three-dimensional homolo…

Paper derives explicit formulas for AJ-bracket of tied links.

New proof for knot state-sum formula using bijection between states.

Paper describes a state sum formula for a graph coloring polynomial.

We give a formula for the radial asymptotics to all orders of the special $q$-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in~\cite{CGZ} to prove one direction of Nahm's conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in $K$-theory. The …

We compute the genus zero family Gromov-Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau-Zaslow formula for non-primitive classes of index two.

The formula for the Turaev-Viro invariant of a 3-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. This paper analyzes convergence of this sum when t does not lie on the unit circle, in the presence of an efficient triangulation of the three-manifold. The ter…

We present two range characterizations for the attenuated geodesic X-ray transform defined on pairs of functions and one-forms on simple surfaces. Such characterizations are based on first isolating the range over sums of functions and one-forms, then separating each sub-range in two ways, first by implicit conditions,…

In this paper we determine the integral homology and cohomology groups of a closed 4-manifold X obtained as the generalized fibre sum of two closed 4-manifolds M and N along embedded surfaces of genus g and self-intersection zero. If the homologies of the 4-manifolds are torsion free and the surfaces represent indivisi…

We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, …

Derives curvature formulas for convex metric sums and conditions for positive average variation.

Gromov-Witten invariants of a symplectic manifold are a count of holomorphic curves. We describe a formula expressing the GW invariants of a symplectic sum $X# Y$ in terms of the relative GW invariants of $X$ and $Y$. This formula has several applications to enumerative geometry. As one application, we obtain new relat…

New formulas connect knot invariants with theta functions.

Defines a universal state sum construction for various TQFTs.

We prove a Torres-like formula for the $L^2$-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the $L^2$-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.

We use the Berstein-Hilton invariant to prove the formula $\cat(M_1\sharp M_2)=\max\{\cat M_1, \cat M_2\}$ for the Lustrnik-Schnirelmann category of the connected sum of closed manifolds $M_1$ and $M_2$.

We compute the genus one family Gromov-Witten invariants of K3 surfaces for non-primitive classes. These calculations verify Gottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two topological recursion formula and the symplectic sum formula to establish relationships …

We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casso…

We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere.

We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application, we compute the Eells-Kuiper and t-invariants of certain cohomogeneity one manif…

Study integral kernels on complex symmetric spaces and their Dyson Brownian Motion applications.

New index formula for hypoelliptic operators on manifolds.

We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assu…

Using existing technology, we prove a Masur-Minsky style distance formula for flip- graph distance between two triangulations, expressed as a sum of the distances of the projections of these triangulations into arc graphs of the suitable subsurfaces of S.

Direct formula found for ADO invariants from homological representations.

This paper continues the study of periodic links started in \cite{Politarczyk2}. It contains a study of the equivariant analogues of the Jones polynomial, which can be obtained from the equivariant Khovanov homology. In this paper we describe basic properties of such polynomials, show that they satisfy an analogue of t…

Proves additivity of Casson-Seiberg-Witten invariant for a specific 4-manifold.

Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones-Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which em…

We prove an explicit cabling formula for the colored Jones polynomial. As an application we prove the volume conjecture for all zero volume knots and links, i.e. all knots and links that are obtained from the unknot by repeated cabling and connected sum.

Proves properties of instanton knot Floer homology and connected sum formula.