Year: 2025
Type: article
Abstract: The volume conjecture relates the quantum invariant and the hyperbolic geometry.Bonahon, Wong and Yang introduced a new version of the volume conjecture by using the intertwiners between two isomorphic irreducible representations of the skein algebra.The intertwiners are built from surface diffeomorphisms; they formulated the volume conjecture when diffeomorphisms are pseudo-Anosov.We explicitly calculate all the intertwiners for the closed torus using an algebraic embedding from the skein algebra of the closed torus to a quantum torus, and show the limit superior related to the trace of these intertwiners is zero.Moreover, we consider the periodic diffeomorphisms for surfaces with negative Euler characteristic, and conjecture the corresponding limit is zero because the simplicial volume of the mapping tori for periodic diffeomorphisms is zero.For the once punctured torus, we make precise calculations for intertwiners and prove our conjecture.14H10, 14H30, 14H45, 14H50, 14L10 1 .S/ to SL.2; C/ such that ' and have the same character, where ' is the isomorphism from 1 .S/ to 1 .S/ induced by ').Suppose the quantum parameter for the skein algebra is q n D e 2 i=n with .qn / 1=2 D e i=n and n odd.For each puncture v of S , we choose a complex number p v such that p v D p '.v/ and T n .pv / D Trace..˛v//,where T n is the n th Chebyshev polynomial of the first type and ˛v is the element in 1 .S/ going around puncture v.According to Theorem 2.1, we know and p v uniquely determine an irreducible representation of the skein algebra.Let ' ] be the isomorphism from the skein algebra of S to itself induced by '.Since both and p v are '-invariant, we have and ' ] are isomorphic according to Theorem 2.2.Thus there exists the intertwiner ƒ q n '; between these two isomorphic irreducible representations.We normalize it such that jdet.ƒq n '; /j D 1. Then we formulate the following conjecture, please refer to Conjecture 5.4 for a more detailed version.
Source: Algebraic & Geometric Topology
Author: Zhihao Wang
Institution: University of Groningen
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4413469032