My research interests lie at the the interface between knot theory, algebraic topology and representation theory. More precise, my interests cover:
Vassiliev invariants
braid-like groups
knotted objects and generalizations
polynomial (quantum) invariants and their categorifications (knot homologies)
topological complexity
Below you will find a list of my current projects and here a list of my papers on the arXiv.
An Alexander type invariant for doodles, with B. Cisneros, M. Flores and J. Juyumaya. Submitted (arXiv:2005.06290)
Planar pure braids on six strands, with Jacob Mostovoy. J. Knot Theory Ramifications 29 (2020), no. 1, 1950097, 11 pp. (arXiv:1905.08326)
Linear motion planning with controlled collisions and pure planar braids, with Jesús González and José L. León-Medina. Homology Homotopy Appl. 23 (2021), no. 1, pp. 275-296. (arXiv:1902.0619)
Motion planning in tori revisited, with J. González, B. Gutiérrez, A. Guzmán, C. Hidber, M. Mendoza. Morfismos 19 (2015), no. 1, pp. 7-18.
Quantum invariants for planar knotted objects. In analogy of the Reshetekhin-Turaev construction of knot invariants from the category of tangles to the category of representations of quantum groups, we study the category of plane tangles for an ad hoc category for functorial representations of this category of planar knotted objects.
Categorification of the Burau type representation of planar braids. In a recent work, we construct a linear representation of the planar braid group (also called twin group) as a deformation of its Tits representation as a Coxeter group. It is an analogue of the reduced Burau representation for braid groups. Motivated by Khovanov and Seidel's work on the categorification of the Burau representation, we plan the categorification of this new representation.
A Kohno-Kontsevich integral for the planar pure braid group. Motivated by the construction of the Kontsevich integral for knots and the theory of finite type invariants for braids by Kohno using iterated Chen integrals, we hope to construct a universal finite type invariant for the planar pure braid group with explicit computations.