  We introduce the inversion polynomial for Dedekind sums $f_b(x)=\sum
x^{\operatorname{inv}(a,b)}$ to study the number of $s(a,b)$ which have the
same value for given $b$. We prove several properties of this polynomial and
present some conjectures. We also introduce connections between Kloosterman
sums and the inversion polynomial evaluated at particular roots of unity.
Finally, we improve on previously known bounds for the second highest value of
the Dedekind sum and provide a conjecture for a possible generalization.
Lastly, we include a new restriction on equal Dedekind sums based on the
reciprocity formula.
