Véronique Bazier-Matte - Academia.edu (original) (raw)

Papers by Véronique Bazier-Matte

arXiv (Cornell University), May 26, 2024

To every knot (or link) diagram K, we associate a cluster algebra A that contains a cluster xt wi... more To every knot (or link) diagram K, we associate a cluster algebra A that contains a cluster xt with the property that every cluster variable in xt specializes to the Alexander polynomial ∆ K of K. We call xt the knot cluster of A. Moreover, the specialization does not depend on the choice of the cluster variable. Furthermore, there exists a cluster automorphism of A of order two that maps the initial cluster to the cluster xt. We realize this connection between knot theory and cluster algebras in two ways. In our previous work, we constructed indecomposable representations T (i) of the initial quiver Q of the cluster algebra A. Modulo the removal of 2-cycles, the quiver Q is the incidence quiver of the segments in K, and the representation T (i) of Q is built by taking successive boundaries of K cut open at the i-th segment. The relation to the Alexander polynomial stems from an isomorphism between the submodule lattice of T (i) and the lattice of Kauffman states of K relative to segment i. In the current article, we identify the knot cluster xt in A via a sequence of mutations that we construct from a sequence of bigon reductions and generalized Reidemeister III moves on the diagram K. On the level of diagrams, this sequence first reduces K to the Hopf link, then reflects the Hopf link to its mirror image, and finally rebuilds (the mirror image of) K by reversing the reduction. We show that every diagram of a prime link admits such a sequence. We further prove that the cluster variables in xt have the same F-polynomials as the representations T (i). This establishes the important fact that our representations T (i) do indeed correspond to cluster variables in A. But it even establishes the much stronger result that these cluster variables are all compatible, in the sense that they form a cluster. We also prove that the representations T (i) have the following symmetry property. For all vertices i, j of Q, we have dim T (i) j = dim T (j) i. As applications, we show that in the Newton polytope of the F-polynomial of T (i), every lattice point is a vertex. We also obtain an explicit formula for the denominator vector of each cluster variable in the knot cluster. In particular, each entry in the denominator vector is either 0 or 1. Contents 1. Introduction 2 2. Preliminaries 9 3. Recollections from Knot Theory and Cluster Algebras I 12 4. Proof of the main theorem in two examples 14 5. Proof of the main theorem 17 6. Creation of bigons 40 7. Symmetry of dimension 43 8. Applications 47 References 48

Journal of the London Mathematical Society

A new construction of the associahedron was recently given by Arkani‐Hamed, Bai, He, and Yan in c... more A new construction of the associahedron was recently given by Arkani‐Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the ‐polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the ‐vector fan has the property that its nef cone is simplicial.

arXiv (Cornell University), Jan 14, 2022

Advances in Mathematics, Oct 1, 2022

$Research paper thumbnail of Number of Triangulations of a M\"obius Strip$

arXiv (Cornell University), Sep 12, 2020

Consider a Möbius strip with n chosen points on its edge. A triangulation is a maximal collection... more Consider a Möbius strip with n chosen points on its edge. A triangulation is a maximal collection of arcs among these points and cuts the strip into triangles. In this paper, we proved the number of all triangulations that one can obtain from a Möbius strip with n chosen points on its edge is given by 4 n−1 + 2n−2 n−1 , then we made the connection with the number of clusters in the quasi-cluster algebra arising from the Möbius strip. Contents 10 Acknowledgements 15 References 15

arXiv (Cornell University), Sep 6, 2018

A cluster algebra is unistructural if the set of its cluster variables determines its clusters an... more A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis, and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are unistructural. Contents 10 4.2. Consequences for cluster automorphisms 11 Acknowledgements 12 References 12

Proceedings of the American Mathematical Society, Feb 18, 2020

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arXiv (Cornell University), Mar 25, 2019

Boletim da Sociedade Paranaense de Matemática, Apr 1, 2018

arXiv (Cornell University), Nov 28, 2022

We initiate the investigation of representation theory of nonorientable surfaces. As a first step... more We initiate the investigation of representation theory of nonorientable surfaces. As a first step towards finding an additive categorification of Dupont and Palesi's quasi-cluster algebras associated marked nonorientable surfaces, we study a certain modification on the objects of the cluster category associated to the orientable double covers in the unpunctured case. More precisely, we consider symmetric representation theory studied by Derksen-Weyman and Boos-Cerulli Irelli, and lift it to the cluster category. This gives a way to consider 'indecomposable orbits of objects' under a contravariant duality functor. Hence, we can assign curves on a non-orientable surface (S, M) to indecomposable symmetric objects. Moreover, we define a new notion of symmetric extension, and show that the arcs and quasiarcs on (S, M) correspond to the indecomposable symmetric objects without symmetric self-extension. Consequently, we show that quasi-triangulations of (S, M) correspond to a symmetric analogue of cluster tilting objects.

Advances in Mathematics

We establish a connection between knot theory and cluster algebras via representation theory. To ... more We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra is 2n, where n is the number of crossing points in the knot diagram. We then construct 2n indecomposable modules T (i) over the Jacobian algebra of the quiver with potential. For each T (i), we show that the submodule lattice is isomorphic to the corresponding lattice of Kauffman states. We then give a realization of the Alexander polynomial of the knot as a specialization of the F-polynomial of T (i), for every i. Furthermore, we conjecture that the collection of the T (i) forms a cluster in the cluster algebra whose quiver is isomorphic to the opposite of the initial quiver, and that the resulting cluster automorphism is of order two.

We associate a quiver to a quasi-triangulation of a non-orientable marked surface and define a no... more We associate a quiver to a quasi-triangulation of a non-orientable marked surface and define a notion of quiver mutation that is compatible with quasi-cluster algebra mutation defined by Dupont and Palesi [DP15]. Moreover, we use our quiver to show the unistructurality of the quasi-cluster algebra arising from the Möbius strip.

arXiv: Representation Theory, 2018

International Mathematics Research Notices, 2020

We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant as... more We use Khovanov and Kuperberg’s web growth rules to identify the leading term in the invariant associated to an textrmSL_3\textrm{SL}_3textrmSL_3 web diagram, with respect to a particular term order.

Proceedings of the American Mathematical Society, 2020

Boletim da Sociedade Paranaense de Matemática, 2018

Frieze patterns (in the sense of Conway and Coxeter) are related to cluster algebras of type A an... more Frieze patterns (in the sense of Conway and Coxeter) are related to cluster algebras of type A and to signed continuant polynomials. In view of studying certain classes of cluster algebras with coefficients, we extend the concept of signed continuant polynomial to define a new family of friezes, called c-friezes, which generalises frieze patterns. Having in mind the cluster algebras of finite type, we identify a necessary and sufficient condition for obtaining periodic c-friezes. Taking into account the Laurent phenomenon and the positivity conjecture, we present ways of generating c-friezes of integers and of positive integers. We also show some specific properties of c-friezes.

Drafts by Véronique Bazier-Matte

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in c... more A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the F-polynomials of the corresponding cluster algebras.