Abstracts for Conference on "Quantum Topology"

In the last years many interesting connections have been found between quantum invariants of links and three-manifolds, and topological strings on Calabi-Yau threefolds. In this talk I will focus on state integral invariants of hyperbolic knots, their perturbative expansion, and the resurgent structure of the latter.

The $sl_n$-skein algebra of a surface provides a quantization of the $SL_n(\mathbb{C})$ character variety. For surfaces with boundary, this framework extends naturally to the stated skein algebra. We demonstrate how various aspects of quantum groups admit simple and transparent geometric interpretations through the lens of stated skein algebras. In particular, we will present a geometric realization of the dual canonical basis of $\mathcal{O}_q(\mathfrak{sl}_3)$ using skeins.

The $SL_2(C)$-skein modules of closed 3-manifolds were defined by in the 90’s but till recently little was known about their structure. The modules depend on a parameter A and can be considered over $ {\mathbb Z}[A^{\pm 1}]$ or over ${\mathbb Q}(A)$.

The ${\mathbb Q}(A)$-module is known to be finitely generated  while the structure over ${\mathbb Z}[A^{\pm 1}]$ can be complicated.

The Matveev-Piergallini (MP) moves on spines of 3-manifolds are well-known for their correspondence with the Pachner 2-3 moves in dual ideal triangulations. Benedetti and Petronio introduced combinatorial descriptions of closed 3-manifolds and combed 3-manifolds using branched spines and their equivalence relations, which involve MP moves with 16 distinct branching patterns. In this talk, I will demonstrate that these 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves, and their inverses.