A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Recent advances in Quantum Topology assign q-series to knots in at least three different ways. The q-series are given by generalized Nahm sums (i.e., special q-hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those q-series that come from planar gra…
The tail of a quantum spin network in the two-sphere is a q-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by 2n. We compute the q-series for an infinite family of quantum spin networks and give the relation between the tail of these networ…
New invariants for 3-manifolds derived from supergroup representations.
problem Developing invariants for 3-manifolds using supergroup analogues.
method Introducing supergroup analogues of 3-manifold invariants for superunitary groups, focusing on SU(2|1). Calculating q-series for specific 3-manifolds and studying their properties.
result Explicit calculation and study of q-series for certain 3-manifolds, providing a formula relating new invariants to quantum invariants.
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…
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 sl2 colored Jones polynomials {Jn(K;q)}n for some link K, for example, an alternating link. For the $\mathf…
Study on Δ invariants of plumbed 3-manifolds, comparing with Heegaard-Floer homology.
problem Analyzing Δ invariants of plumbed 3-manifolds.
method Expressing Δ of Seifert manifolds in terms of singularity theory invariants, providing examples for non-Seifert manifolds, comparing with Heegaard-Floer homology.
result Interesting behavior of Δ invariants for non-Seifert manifolds, comparison with Heegaard-Floer homology.
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…
We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.
It is well-known to the experts that multi-dimensional state integrals of products of Faddeev's quantum dilogarithm which arise in Quantum Topology can be written as finite sums of products of basic hypergeometric series in q=e^{2πiτ} and \tilde{q}=e^{-2πi/τ}. We illustrate this fact by giving a detailed proof for a fa…
We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a…
We show that the Atiyah-Patodi-Singer reduced η-invariant of the twisted Dirac operator on a closed 4m−1 dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight 2m up to an integral q-series. We prove this resu…
In this paper we study new invariants Za(q) attached to plumbed 3-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable q-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed 3-manifold. Here we investigate the series $\wi…
In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in…
In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index…
For a Seifert fibered homology sphere we show that the q-series Z-hat invariant introduced by Gukov, Pei, Putrov and Vafa is a resummation of the Ohtsuki serie. We show that for every even level k there exists a full asymptotic expansion of Z-hat for q tending to a certain k'th root of unity and in particular that the …
The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with r torii boundary components. For a fixed 2r tuple of integers, the index takes values in the set of q-series with integer coefficients. Our goal is to give an axiomatic definition of the tetra…