Caltech/USC Joint Algebra and Geometry Seminar

USC KAP 414

(Joint with P. Orland, T. Saunders-A'Court, and J. Svoboda.) Quantum knot invariants, constructed from the representation theory of quantum groups, are powerful algebraic tools in low-dimensional topology. Yet how these invariants encode geometric information remains a central open question. Our main result shows that the Gukov–Manolescu series, a quantum invariant associated with $U_q(\mathfrak{sl}_2)$ at generic $q$, encodes the Hopf invariant of fibered knots, a natural geometric quantity. This suggests that the GM series may also detect fiberedness. As an application, we obtain an explicit expression of the Hopf invariant in terms of colored Jones polynomials, addressing an open question dating back to the introduction of the Hopf invariant in the 1980s. We also explore a surprising connection between the GM series of certain non-fibered knots and knot Floer homology.