Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does not depend on $m,$ where adding a multiple of a knot means doing a Dehn surgery with the indicated slope. My question is: with the standard normalization, when is the difference actually equal to $\pm 1?.$

**EDIT** In the case I was specifically looking at, the complement of the knot was fibered, so the Alexander polynomial was the (normalized to be a symmetric Laurent series) characteristic polynomial of the monodromy action on homology. However, it is not clear what the deep significance (from the algebraic standpoint) is of taking the second derivative of this object and evaluating it at one (the evaluating at $1$ part is natural, since the homology of a mapping torus is equal to the cokernel of $f_* - I,$ but the second derivative is a little mysterious).