Multicurves and regular functions on the representation variety of a surface in SU(2)

Sign stability of mapping classes on marked surfaces II: general case via reductions

We give a cluster algebraic description of the reduction procedure of mapping classes along a multicurve. Based on this description, we characterize pseudo-Anosov mapping classes on a general marked surface in terms of a weaker version of the uniform sign stability, generalizing the main result in the previous paper [IK20]. Moreover we axiomatize a general reduction procedure of mutation loops parametrized by a rational polyhedral cone in the tropical cluster $\mathcal{X}$-variety, which includes both the reduction along a multicurve and the cluster reduction introduced in [Ish19].

An analytic family of representations for the mapping class group of punctured surfaces

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1…

用微分拓扑证明 genus 二曲面的 SU(2) 特征簇与复射影空间 C P^3 是同胚的

A differential topology proof that the $SU(2)$ character variety of the genus two surface is homeomorphic to ${\mathbb C} P^3$

本文通过微分拓扑证明 genus 二曲面的 SU(2) 特征簇与复射影空间 C P^3 同胚，并探讨了 Lagrangian 浸入与 Heegaard 分解的关系。

The quantum trace as a quantum non-abelianization map

We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov-Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong's quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the S…