arXiv subjects

Daniel López Neumann

Publications and source records attributed to Daniel López Neumann.

7 recordsLinked to original sources

arXiv2026

Genus bounds for knot polynomials of Lie superalgebras

Knot polynomials colored by typical representations of Lie superalgebras of type I (except $\mathfrak{psl}(n|n)$) have two variables $q$ and $t$, the latter corresponding to the complex-valued weight of the distinguished odd root. We prove that for every typical representation of a Lie superalgebra of type I, the $t$-degree of the knot polynomial is at most the number of odd roots times the genus of the knot. A complimentary bound being at least the number of odd roots times degree of the Alexander polynomial can be obtained from a specialization at $q=1$. These two bounds become equalities when the Alexander polynomial detects the genus of the knot, as is the case for alternating knots and fibered knots.

math.GT↗

arXiv2024

Non-semisimple $\mathfrak{sl}_2$ quantum invariants of fibred links

The Akutsu-Deguchi-Ohtsuki (ADO) invariants are the most studied quantum link invariants coming from a non-semisimple tensor category. We show that, for fibered links in $S^3$, the degree of the ADO invariant is determined by the genus and the top coefficient is a root of unity. More precisely, we prove that the top coefficient is determined by the Hopf invariant of the plane field of $S^3$ associated to the fiber surface. Our proof is based on the genus bounds established in our previous work, together with a theorem of Giroux-Goodman stating that fiber surfaces in the three-sphere can be obtained from a disk by plumbing/deplumbing Hopf bands.

math.QA↗

arXiv2023

Genus bounds from unrolled quantum groups at roots of unity

For any simple complex Lie algebra $\mathfrak{g}$, we show that the degrees of the "ADO" link polynomials coming from the unrolled restricted quantum group $\overline{U}^H_q(\mathfrak{g})$ at a root of unity give lower bounds to the Seifert genus of the link. We give a direct simple proof of this fact relying on a Seifert surface formula involving universal $\mathfrak{u}_q(\mathfrak{g})$-invariants, where $\mathfrak{u}_q(\mathfrak{g})$ is the small quantum group. We give a second proof by showing that the invariant $P_{\mathfrak{u}_q(\mathfrak{b})}^θ(K)$ of our previous work coincides with such ADO invariants, where $\mathfrak{u}_q(\mathfrak{b})$ is the Borel part of $\mathfrak{u}_q(\mathfrak{g})$. To prove this, we show that equivariantizations of relative Drinfeld centers of crossed products essentially contain unrolled restricted quantum groups, a fact that could be of independent interest.

math.QA↗

arXiv2023

Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles

In this paper, we consider the Reshetikhin-Turaev invariants of knots in the three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or equivalently, the relative Drinfeld center of the crossed product $\text{Rep}(H)\rtimes\text{Aut}(H)$. These are quantum invariants of knots endowed with a homomorphism of the knot group to $\text{Aut}(H)$. We show that, at least for knots in the three-sphere, these invariants provide a non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants of sutured manifolds introduced previously by the author, which are only defined for involutory Hopf algebras. In particular, we describe the $SL(n,\mathbb{C})$-twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant.

math.QA↗

arXiv2022

Genus bounds for twisted quantum invariants

By twisted quantum invariants we mean polynomial invariants of knots in the three-sphere endowed with a representation of the fundamental group into the automorphism group of a Hopf algebra $H$. These are obtained by the Reshetikhin-Turaev construction extended to the $\mathrm{Aut}(H)$-twisted Drinfeld double of $H$, provided $H$ is finite dimensional and $\mathbb{N}^m$-graded. We show that the degree of these polynomials is bounded above by $2g(K)\cdot d(H)$ where $g(K)$ is the Seifert genus of a knot $K$ and $d(H)$ is the top degree of the Hopf algebra. When $H$ is an exterior algebra, our theorem recovers Friedl and Kim's genus bounds for twisted Alexander polynomials. When $H$ is the Borel part of restricted quantum $\mathfrak{sl}_2$ at an even root of unity, we show that our invariant is the ADO invariant, therefore giving new genus bounds for these invariants.

math.QA↗

arXiv2021

Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.

math.GT↗

arXiv2020

Kuperberg invariants for balanced sutured 3-manifolds

We construct quantum invariants of balanced sutured 3-manifolds with a $Spin^{c}$ structure out of an involutive (possibly non-unimodular) Hopf superalgebra $H$. If $H$ is the Borel subalgebra of $U_{q}(\mathfrak{gl}(1|1))$, we show that our invariant is computed via Fox calculus and it is a normalization of Reidemeister torsion. The invariant is defined via a modification of a construction of G. Kuperberg, where we use the $Spin^{c}$ structure to take care of the non-unimodularity of $H$ or $H^{*}$.

math.GT↗