= Negative continued fraction
{title2=$[a_1,\ldots,a_r]^-$}
The finite negative continued fraction is
$$
[a_1,\ldots,a_r]^-=a_1-\frac1{a_2-\frac1{\cdots-1/a_r}}.
$$
For $a_i\ge2$, it describes a negative-framed chain of unknots: the diagonal framings are $-a_i$ and adjacent linking numbers are one. Repeated <slam-dunk moves> give rational surgery coefficient $-[a_1,\ldots,a_r]^-$. The corresponding chain matrix is negative definite. To check this, the successive determinants of its negative have recurrence $d_j=a_jd_{j-1}-d_{j-2}$, starting with $d_0=1$, $d_1=a_1$; they are positive and increasing, so Sylvester's criterion applies.
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