Alexander module of a knot ($H_1({\tilde{E}}_K;\mathbb{Z})$)

The Alexander module is the first integral homology of the infinite cyclic cover of a knot exterior. Its deck transformation makes it a module over $\mathbb{Z}[t^{\pm 1}]$, and its first elementary ideal determines the Alexander polynomial of a knot.

Boundary-connected-summing Seifert surfaces gives a block-diagonal Seifert matrix, because basis curves in the separated summands have zero linking. The Seifert-matrix presentation of the Alexander module is block diagonal too. Thus the Alexander modules form a direct sum and their Alexander polynomials of a knot multiply up to a unit.