Related papers: DAHA and plane curve singularities
The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character…
We propose a log-concavity conjecture for BPS invariants arising in the enumerative geometry of planar curve singularities, identified with the local Euler obstructions of Severi strata in their versal deformations. We further extend this…
We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The embedding is equivariant with respect to…
In this note, we are interested in the Jacobian Conjecture. Following the results of L.M.~Dru$\dot{\rm z}$kowski, we consider some vector fields depending on a certain \'etale polynomial map. From results of semialgebraic geometry with the…
In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta…
We prove an unexpected general relation between the Jacobian syzygies of a projective hypersurface $V\subset \mathbb{P}^n$ with only isolated singularities and the nature of its singularities. This allows to establish a new method for the…
A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation…
We define generalized double affine Hecke algebras (GDAHA) of higher rank, attached to a non-Dynkin star-like graph D. This generalizes GDAHA of rank 1 defined in math.QA/0406480 and math.QA/0409261. If the graph is extended D4, then GDAHA…
We review the Reshetikhin-Turaev approach to construction of non-compact knot invariants involving R-matrices associated with infinite-dimensional representations, primarily those made from Faddeev's quantum dilogarithm. The corresponding…
Alexander polynomial arises in the leading term of a semi-classical Melvin-Morton-Rozansky expansion of colored knot polynomials. In this work, following the opposite direction, we propose how to reconstruct colored HOMFLY-PT polynomials,…
We establish tools to facilitate the computation and application of the Chekanov-Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots in standard contact three-space. More specifically, we…
We re-build the quantum sl2 unified invariant of knots $F_{\infty}$ from braid groups' action on tensors of Verma modules. It is a two variables series having the particularity of interpolating both families of colored Jones polynomials and…
For a projective hypersurface $Z$ with isolated singularities, we generalize some well-known assertions in the nonsingular case due to Griffiths, Scherk, Steenbrink, Varchenko, and others about the relations between the Steenbrink spectrum,…
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and…
Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $\omega$ on a smooth variety. When $\omega$ is the standard form $dx_1\wedge \dots \wedge dx_n$ on affine…
This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we…
Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…
The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…
The Chekanov-Eliashberg dg-algebra is an algebraic invariant of Legendrian submanifolds of contact manifolds, whose definition recently has been extended to singular Legendrians. We describe a way of constructing simpler models of this…