arXiv subjects

Cristina Ana-Maria Anghel

Publications and source records attributed to Cristina Ana-Maria Anghel.

17 recordsLinked to original sources

arXiv2026

Quantum braid Floer homology

Given a braid whose closure is a knot, we associate to it a doubly pointed Heegaard diagram with additional marked points that we call a quantum Heegaard diagram. This is obtained by surgery from the unified intersection model for the Alexander and Jones polynomials due to the first author. By translating the Alexander and quantum gradings from the disc model to the Heegaard diagram, we obtain a bifiltration on the hat knot Floer complex. The degree-$(0,0)$ component of the differential with respect to this bifiltration defines a triply graded invariant of the braid that we call quantum braid Floer homology. It is generally not invariant under Markov conjugation or stabilisation. It contains a distinguished non-zero element, and admits a spectral sequence to knot Floer homology. Furthermore, its graded Euler characteristic is a two-variable polynomial that specialises to the Alexander and Jones polynomials.

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arXiv2026

Maximal universal invariants from finite quotients of Verma modules

We construct a sequence of new universal quantum knot invariants that are lifts of both the semi-simple and non semi-simple $U_q(sl_2)$ quantum knot invariants. More specifically, for any level $\mathscr N$ we define a ``level $\mathscr N$ universal invariant'' $ \widetildeΩ_{\mathscr N}(L)$ arising from quantum traces on finite quotients of the generic Verma module over certain quotient rings. We show that for $\mathscr N$ prime, this is the maximal invariant that can arise from the $\mathscr N$-part of the Verma module, and it is a specific interpolation between the $\mathscr N^{th}$ coloured Jones and $\mathscr N^{th}$ ADO polynomials. For $\mathscr N$ non prime $ \widetildeΩ_{\mathscr N}(L)(q,s)$ has a richer structure, it recovers the $\mathscr N^{th}$ coloured Jones and $\mathscr N^{th}$ ADO polynomials, but it could contain more information which is not seen in the sequence of coloured Jones and ADO invariants.

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arXiv2025

Geometric universal Jones invariant from configurations on ovals in the disc

We construct geometrically a universal Jones invariant as a limit of invariants given by graded intersections in configuration spaces. For any fixed level $\mathscr N$, we define a new knot invariant, called ``$\mathscr N^{th}$ Unified Jones invariant'' globalising topologically all coloured Jones polynomials at levels less than $\mathscr N$. It is defined via the intersection points between {Lagrangian submanifolds} supported on arcs and ovals in the disc. The geometry of these Lagrangians is novel: previous topological models involved immersed submanifolds rather than embedded ones. We do this by defining a new local system that refines the Lawrence representation, and depends of the distribution of multiplicities of points in the configuration space on the ovals. On the algebraic side, Habiro's famous invariant for knots \cite{H3} is a universal invariant globalising the family of coloured Jones polynomials. He conjectured that this universal invariant recovers also the ADO invariant divided by the Alexander polynomials, which was proved by Willetts \cite{W} in a version of Habiro's ring (\cite{H2}). The universal Jones invariant that we construct belongs to a different ring that comes with a map to Habiro's ring \cite{H2}. We prove that our invariant recovers this version of Habiro's invariant. The difference is that our invariant is given as a limit of new knot invariants, the $\mathscr N^{th}$ unified Jones invariants. These invariants in turn provide a geometrical understanding of sets of all coloured Jones polynomials of bounded colour, collecting more information as we increase the colour.

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arXiv2025

A topological model for the HOMFLY-PT polynomial

We give the first known topological model for the HOMFLY-PT polynomial constructed directly from link diagrams. More precisely, we prove that this invariant is given by graded intersections between explicit Lagrangian submanifolds in a fixed configuration space on a Heegaard surface for the link exterior. The submanifolds are supported on a collection of arcs and ovals on the Heegaard surface. We also obtain two topological models for the Jones polynomial via a Heegaard surface associated to a link diagram. This opens up new avenues for constructing categorifications for the Jones and the HOMFLY-PT polynomials of a geometric nature.

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arXiv2025

Universal geometrical link invariants

We construct geometrically two universal link invariants: universal ADO invariant and universal Jones invariant, as limits of invariants given by graded intersections in configuration spaces. More specifically, for a fixed level $\mathscr N$, we define new link invariants: ``$\mathscr N^{th}$ Unified Jones invariant'' and ``$\mathscr N^{th}$ Unified Alexander invariant''. They globalise topologically all coloured Jones polynomials for links with multi-colours bounded by $\mathscr N$ and all ADO polynomials with bounded colours. These invariants both come from the same weighted Lagrangian intersection supported on configurations on arcs and ovals in the disc. The question of providing a universal non semi-simple link invariant, recovering all the ADO polynomials was an open problem. A parallel question about semi-simple invariants for the case of knots is the subject of Habiro's famous universal knot invariant \cite{H3}. Habiro's universal construction is well defined for knots and can be extended just for certain classes of links. Our universal Jones link invariant is defined for any link and recovers all coloured Jones polynomials, providing a new semi-simple universal link invariant. The first non semi-simple universal link invariant that we construct unifies all ADO invariants for links, answering the open problem about the globalisation of these invariants.

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arXiv2022

A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space

We consider two Laurent polynomials in two variables associated to a braid, given by {\em graded intersections} between {\em fixed Lagrangians in configuration spaces}. In order to get link invariants, we notice that we have to quotient by a quadratic relation. Then we prove by topological tools that this relation is sufficient and the first graded intersection gives an invariant which is the Jones polynomial. This shows a {\em topological model for the Jones polynomial} and a direct {\em topological proof}\hspace{0.4mm} that it is a well-defined invariant. The other intersection model in the quotient turns out to be an invariant globalising the Jones and Alexander polynomials. This globalisation in the quotient ring is given by a {\em specific interpolation between the Alexander and Jones polynomials}.

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arXiv2022

$U_q(sl(2))-$quantum invariants unified via intersections of embedded Lagrangians

In this paper we prove a unified model for $U_q(sl(2))$ quantum invariants through intersections of embedded Lagrangians in configuration spaces. More specifically, we construct a {\em state sum of Lagrangian intersections in the configuration space in the punctured disc}, which is a polynomial in three variables. It {\em recovers the coloured Jones polynomial and the coloured Alexander polynomial} through specialisations of coefficients. This formula works for oriented links coloured with the same representation of the quantum group and can be evaluated at roots of unity. As a corollary, the Jones and Alexander polynomials come both as {\em specialisations of an intersection pairing between embedded Lagrangians} in configuration spaces, which is suitable for computations. In particular, we obtain the {\em first intersection model for the Jones polynomial} from intersections between submanifolds which are given by {\em arcs and circles} in the punctured disc.

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arXiv2022

$U_q(sl(2))$-quantum invariants from an intersection of two Lagrangians in a symmetric power of a surface

In this paper we show that coloured Jones and coloured Alexander polynomials can both be read off from the same picture provided by two Lagrangians in a symmetric power of a surface. More specifically, the $N^{th}$ coloured Jones and $N^{th}$ coloured Alexander polynomials are specialisations of a graded intersection between two explicit Lagrangian submanifolds in a symmetric power of the punctured disc. The graded intersection is parametrised by the intersection points between these Lagrangians, graded in a specific manner using the diagonals of the symmetric power. As a particular case, we see the original Jones and Alexander polynomials as two specialisations of a graded intersection between two Lagrangians in a configuration space, whose geometric supports are Heegaard diagrams.

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arXiv2021

Relative (pre)-modular categories from special linear Lie superalgebras

We examine two different m-traces in the category of representations over the quantum Lie superalgebra associated to $\mathfrak{sl}(m|n)$ at root of unity. The first m-trace is on the ideal of projective modules and leads to new Extended Topological Quantum Field Theories. The second m-trace is on the ideal of perturbative typical modules. We consider the quotient with respect to negligible morphisms coming from this m-trace and show that in the case of $\mathfrak{sl}(2|1)$ this quotient leads to 3-manifolds invariants. We conjecture that the quotient category of perturbatives over quantum $\mathfrak{sl}(m|n)$ leads to 3-manifold invariants and more generally ETQFTs.

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arXiv2021

Witten-Reshetikhin-Turaev invariants for 3-manifolds from Lagrangian intersections in configuration spaces

In this paper we construct a topological model for the Witten-Reshetikhin-Turaev invariants for $3$-manifolds coming from the quantum group $U_q(sl(2))$, as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level $\cN \in \N$ we show that the level $\cN$ WRT invariant for a $3-$manifold is a state sum of Lagrangian intersections in a covering of a {\bf fixed} configuration space in the punctured disk. This model brings a new perspective on the structure of the level $\cN$ Witten-Reshetikhin-Turaev invariant, showing that it is completely encoded by the intersection points between certain Lagrangian submanifolds in a fixed configuration space, with additional gradings which come from a particular choice of a local system. This formula provides a new framework for investigating the open question about categorifications of the WRT invariants.

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arXiv2020

ADO invariants directly from partial traces of homological representations

The ADO invariants are a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. Ito showed that these invariants are sums of traces of quotients of homological representations of braid groups (truncated Lawrence representations). In this paper we show a direct homological formula for the ADO invariants, as sums of partial traces of Lawrence type representations, without further truncations.

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arXiv2020

Lawrence-Bigelow representations, bases and duality

We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general account of the fundamental relationships (non-degenerate pairings, embeddings, isomorphisms) between the many different flavours of homological representations. Our motivating examples are the Lawrence-Bigelow representations of the braid groups, which are of central importance in the study of the braid groups themselves, as well as their connections with quantum invariants of knots and links.

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arXiv2020

Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces

Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by Lagrangian submanifolds. The main result proves that the $N^{th}$ coloured Jones polynomial and $N^{th}$ coloured Alexander polynomial come as different specialisations of an intersection pairing of the same homology classes over two variables, with extra framing corrections in each case. The first corollary explains Bigelow's picture for the Jones polynomial with noodles and forks from the quantum point of view. Secondly, we conclude that the $N^{th}$ coloured Alexander polynomial is a graded intersection pairing in a $ \mathbb Z \oplus \mathbb Z_N$-covering of the configuration space in the punctured disc.

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arXiv2019

A topological model for the coloured Jones polynomials

In this paper we will present a homological model for Coloured Jones Polynomials. For each colour $N \in \mathbb {N}$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homology classes in a covering of the configuration space on the punctured disk. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups.

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arXiv2019

A topological model for the coloured Alexander invariants

Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. This sequence recovers the original Alexander polynomial as the first term. We give a topological model for this invariants, showing that they can be obtained as graded intersection pairings between homology classes in a covering of the configuration space in the punctured disc.

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arXiv2019

A combinatorial description of the centralizer algebras connected to the Links-Gould Invariant

In this paper we study the tensor powers of the standard representation of the quantum super-algebra $U_q(sl(2|1)$, focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by $LG_n$. Their dimensions were conjectured by I. Marin and E. Wagner \cite{MW}. We prove this conjecture, describing the intertwiner spaces from a semi-simple decomposition as sets consisting of certain paths in a planar lattice with integer coordinates. Using this model, we present a matrix unit basis for the centraliser algebra $LG_n$, by means of closed curves in the plane, which are included in the lattice with integer coordinates.

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arXiv2017

Modified Turaev-Viro Invariants from quantum sl(2|1)

The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: 1) specialize q to a root of unity; this forces some modules to have zero quantum dimension, 2) quotient by morphisms of modules with zero quantum dimension, 3) show the resulting category is finite and semi-simple. In this paper we show an analogous construction works in the context of U_q(sl(2|1)) by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize q to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative G-spherical categories. As a consequence we obtain invariants of 3-manifold with additional structures.

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