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We generalize the classical study of Alexander polynomials of smooth or PLlocally-flat knots to PL knots that are not necessarily locally-flat. Weintroduce three families of generalized Alexander polynomials and study theirproperties. For knots with point singularities, we obtain a classification ofthese polynomials that is complete except for one special low-dimensional case.This classification extends existing classifications for PL locally-flat knots.For knots with higher-dimensional singularities, we further extend thenecessary conditions on the invariants. We also construct several varieties ofsingular knots to demonstrate realizability of certain families of polynomialsas generalized Alexander polynomials. These constructions, of independentinterest, generalize known knot constructions such as frame spinning and twistspinning.

Alexander polynomials of non-locally-flat knots