Classical and Quantum Knot Invariants

Murakami showed that the multivariable Alexander polynomial, an invariant from classical algebraic topology, could be recovered from representations of quantum groups using the methods of quantum topology. This thesis focuses on extending Murakami's construction to higher rank quantum groups and exploring their representation theory. This work has culminated in the development of a family of non-abelian knot invariants that generalize the Alexander polynomial. The restricted quantum group of a rank n Lie algebra g admits Verma modules indexed by n-tuples of nonzero complex numbers. These representations determine the polynomial link invariant Δg in n-variables. We prove several results about these representations, including tensor decomposition formulas, which translate into properties of the topological invariants. In the sl₃ case, we show that for any knot K, evaluating Δsl₃ at t₁=1, t₂=1, or t₂ =it₁^-1 recovers the Alexander polynomial of K. This result is not obvious from an examination of the crossing matrix, and it remains an open question whether a similar property holds for other g. We provide a tabulation of Δsl₃ for all knots up to seven crossings along with various other examples. Since Δsl₃ distinguishes mutant knots, and few other invariants can, it is considered a strong invariant and may have applications to other research areas in topology.

Other aspects of this work include descriptions of reducible quantum sl₃ representations and their projective covers. These representations are also related to the Alexander polynomial and allow us to give a more general tensor decomposition theorem. Moreover, we classify all tensor product representations up to isomorphism.

We also develop a diagrammatic calculus of quantum sl₂ representations and simplify the proof of Murakami's work that motivated this thesis. We apply this calculus to prove new theorems in classical topology using quantum methods, generalizing the Seifert-Torres formula for the Alexander polynomial of satellite knots to the multivariable setting.