2020/08/16 by Harper, Matthew · 1 citation
#17B37 #57K16 #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2008.06983
One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum \mathfraksl2 at a fourth root of unity. We generalize this construction to define a link invariant Δ_\mathfrakg for any semisimple Lie algebra \mathfrakg of rank n, taking values in n-variable Laurent polynomials. Focusing on the case \mathfrakg=\mathfraksl3, we establish a direct relation between Δ_\mathfraksl3 and the Alexander polynomial. We show that certain parameter evaluations of Δ_\mathfraksl3 recover the Alexander polynomial on knots, despite the R-matrix not satisfying the Alexander-Conway skein relation at these points. We tabulate Δ_\mathfraksl3 for all knots up to seven crossings and various other examples, including the Kinoshita-Terasaka knot and Conway knot mutant pair which are distinguished by this invariant.