**Generalizing the characteristic polynomial via knot theory**

It is well-known that an endomorphism of a finite-dimensional complex
vector space is not completely determined by its eigenvalues or
characteristic polynomial. In other words, not every endomorphism can
be diagonalized. It is also well-known that one can express any endomorphism
in Jordan Normal Form which is not far from diagonal. The characteristic
polynomial can therefore be said to determine the endomorphism
uniquely `up to choices of extensions' (in the sense of short exact
sequences). More generally, if *A* is a commutative ring, Almkvist
proved that an endomorphism of a finitely generated free *A*-module is
determined uniquely up to extensions by its characteristic polynomial.

If *A* is a non-commutative ring then the familiar definitions of
determinant and characteristic polynomial are unavailable. I shall define a
version of the characteristic polynomial which makes sense for any ring *A*
and which determines an endomorphism of a finitely generated free *A*-module
uniquely up to extensions. Generalizing further, I'll give a characteristic
"polynomial" for a system of two or more (non-commuting) endomorphisms
of a free module.

The ideas behind this work come from topology: The relationship between an
endomorphism and its characteristic polynomial can be considered analogous to
the relationship between two classical approaches to knot theory.

*Desmond Sheiham*

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