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A quantity which remains unchanged under certain classes of transformations. Invariants are extremely useful for classifying mathematical objects because they usually reflect intrinsic properties of the object of study.
See Adiabatic Invariant, Alexander Invariant, Algebraic Invariant, Arf Invariant, Integral of Motion
References
Hunt, B. ``Invariants.'' Appendix B.1 in The Geometry of Some Special Arithmetic Quotients.
New York: Springer-Verlag, pp. 282-290, 1996.