An invariant is a property that is always the
same in two equivalent knots. Note that this does not mean that all
knots whose invariants are the same are equivalent knots. This leads
to questions like what are some examples of invariants, and how do you
prove that particular characteristics of knots are indeed invariants?
There are many other invariants besides the ones
I mentioned. Some interesting ones can be found in Murasugi's book
and in Colin C. Adams' book, The Knot Book. Knot Theorists
are constantly working to find more invariants that are easier to calculate,
and that are more conclusive in determining equivalence of knots.
Deciding whether two knots are equivalent sometimes makes it easier to
study other aspects of knots because a simpler variant of a knot can be
chosen to study. An example of such an aspect is knot composition,
which leads into the next topic I will examine.