arXiv research

New techniques prove quantum modularity for various functions.

New indefinite false theta functions match homological blocks for a specific 3-manifold.

Hikami observed a discontinuity in a WRT invariant at roots of unity.

The paper calculates colored Jones polynomials for specific link configurations.

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…

Formula for arborescent link tails using theta functions.

The tail of a sequence $\{P_n(q)\}_{n \in \mathbb{N}}$ of formal power series in $\mathbb{Z}[[q]]$ is the formal power series whose first $n$ coefficients agree up to a common sign with the first $n$ coefficients of $P_n$. This paper studies the tail of a sequence of admissible trivalent graphs with edges colored $n$ o…

Quantum modularity proven for specific theta series.

New resurgent analysis reveals dual $q$-series for Chern-Simons theory crossing natural boundaries.

Proves resurgence properties for Habiro elements from radial limits of theta series.

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathf…

We calculate the homological blocks for Seifert manifolds from the exact expression for the $G=SU(N)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds obtained by Lawrence, Rozansky, and Mariño. For the $G=SU(2)$ case, it is possible to express them in terms of the false theta functions and their derivatives. …

The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynom…

New formulas connect knot invariants with theta functions.

New formula and properties of inverted Habiro series derived from GM series.

In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced by Dufresne to computing the negative moments of Yor's accumulation processes. For these he has given functional recursion rules whose probabilistic structure has been the object of intensive recent studies of Yor an…

Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.

New method uses resurgent analysis to determine growth rate of quantum field theory coefficients.

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false thet…

We study generalized special cycles on Hermitian locally symmetric spaces $Γ\backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R})$ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally symmetric spaces associated to subgroups of $G$ which are of the same type. Using oscillator…

Study compares two methods to extend $\widehat{Z}$ invariants, finding incompatibility for Brieskorn spheres.

Proves conjecture linking WRT invariants and homological blocks for plumbed 3-manifolds.

Study on planar graph braid groups' second homology.

New rules control false discoveries in online anomaly detection for time series data.

Meta-learning predicts optimal ensemble size and methods for time series forecasting.

New method controls false discoveries in financial asset pricing.

We use superconnections to define and study some natural differential forms on period domains $\mathbb{D}$ that parametrize polarized Hodge structures of given type on a rational quadratic vector space $V$. These forms depend on a choice of vectors $v_1,\ldots,v_r \in V$ and have a Gaussian shape that peaks on the locu…

New findings on isospectral tori and harmonic maps between flat tori.

Study theta-curves on torus in 3-sphere, classifying them.

New method identifies causes in time series with latent variables.

Transformer learns representations from time series data for money laundering detection.

ModelRadar evaluates forecasting models across multiple aspects.

Determinants of theta curves and symmetric graphs are studied.

Proves prime theta-curves for knots on minimal genus surfaces.

We prove a folklore theorem of W. Thurston which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot U is prime if and only if lifting the third arc of the theta-curve to the double branched cover over U …

The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by…

In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar …

The probability density function for the visible sector of a Riemann-Theta Boltzmann machine can be taken conditional on a subset of the visible units. We derive that the corresponding conditional density function is given by a reparameterization of the Riemann-Theta Boltzmann machine modelling the original probability…

Proves existence of flat connection on theta functions for G-bundles.

Study uses RNN to detect CPs with SI to control false positives.

We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that the extrem…

Detects data drift in deep learning models using neural embeddings.

Autoencoder detects subtle changes in time series data.

Study theta functions and adiabatic curvature on Abelian varieties.

We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.

The paper proves rigidity of certain Dirac operators using theta functions.

The paper constructs a canonical connection on bundles over Riemann surfaces and relates it to the theta divisor.

We establish an existence and uniqueness theorem for prime decompositions of theta-curves in $3$-manifolds.