Title: The Alexander Polynomial is a Quantum Invariant
Speaker: Matthew Harper (Ohio State University)
Abstract: J. W. Alexander discovered his polynomial, the first polynomial knot invariant, in 1923. It has since been fundamental in the study of knots and has motivated other ideas within topology. In this introductory talk, we will show that Alexander's "classical" invariant is a quantum invariant -- it can be determined from the representation theory of a Hopf algebra. In this case, the Hopf algebra is the unrolled quantum group $\overline{U_\xi(\mathfrak{sl_2})}$. Throughout our discussion we will make comparisons between the quantum group representations and those of the Lie algebra $\mathfrak{sl}(2)$, using the classical algebra as our "easy example." Once we have the representation theory in place, we make the jump to topology by associating maps of vector spaces to tangles. From here, it is a nice verification that the Alexander polynomial comes from the representations of the quantum group.
