相关论文: The ADO Invariants are a q-Holonomic Family

One of the first key examples of a quantum modular form, which unifies the Witten-Reshetikhin-Turaev (WRT) invariants of the Poincar\'e homology sphere, appears in work of Lawrence and Zagier. We show that the series they construct is one…

We show that the $p$-adic KZ connection associated with the family of curves $y^q=(t-z_1)\dots (t-z_{qg+1})$ has an invariant subbundle of rank $g$, while the corresponding complex KZ connection has no nontrivial proper subbundles due to…

For a classical link, Milnor defined a family of isotopy invariants, called Milnor $\overline{\mu}$-invariants. Recently, Chrisman extended Milnor $\overline{\mu}$-invariants to welded links by a topological approach. The aim of this paper…

Polynomial invariants constitute a dynamic and essential area of study in the mathematical theory of knots. From the pioneer Alexander polynomial, the revolutionary Jones polynomial, to the collectively discovered HOMFLYPT polynomial, just…

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket…

The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the…

We will announce some results on the values of quantum sl_2 invariants of knots and integral homology spheres. Lawrence's universal sl_2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the…

Using Atiyah-Bott localization on the space of stable maps to the stack quotient $[\mathbb{P}^1/\mathbb{Z}_2]$, we find recursions that determine all Hodge integrals with descendent insertions at one marked point on the hyperelliptic locus…

The purpose of this paper is to present a certain combinatorial method of constructing invariants of isotopy classes of oriented tame links. This arises as a generalization of the known polynomial invariants of Conway and Jones. These…

In this article, the 2-iterated q-Appell family is introduced. Certain 2-iterated q-Appell and mixed type q-special polynomials are considered as members of this family. The numbers related to these polynomials are obtained. The determinant…

The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]^N…

We give a new, elementary proof that Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$--coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that $\delta$--graded knot…

We study $q$-series-valued invariants of 3-manifolds that depend on the choice of a root system $G$. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057]…

We construct link invariants using the $D_{2n}$ subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These…

The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $\Delta_L(t)$ and has been shown to share some of its most geometric features in several…

R.~Lawrence has conjectured that for rational homology spheres, the series of Ohtsuki's invariants converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant. We prove this conjecture for Seifert rational homology spheres. We also…

We prove that the HOMFLYPT polynomial of a link, colored by partitions with a fixed number of rows is a $q$-holonomic function. Specializing to the case of knots colored by a partition with a single row, it proves the existence of an…

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of $\mathbb{Q}$ with unitriangular Galois groups, and contain the Legendre and Redei symbols as special…

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…

We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of…