Abstract: The colored Jones polynomial is a generalization of the Jones polynomial from
the representation theory of the quantum group $U_q({\mathfrak sl}_2)$, which revolutionized low-dimensional
topology by giving many new knot and 3-manifold invariants. A key question in the study
of these quantum invariants is how the polynomial relates to other knot and 3-manifold
invariants such as the hyperbolic volume and the boundary slopes of essential surfaces. In
this talk, I will give an overview of related open problems and discuss my recent research
with respect to one of the open questions, the slope conjecture, and related applications to
low-dimensional topology.
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