Negative continued fraction ($[a_1,\dots ,a_r{]}^{-}$)

The finite negative continued fraction isFor $a_i\ge 2$, 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 $-[a_1,\dots ,a_r{]}^{-}$. The corresponding chain matrix is negative definite. To check this, the successive determinants of its negative have recurrence $d_j=a_j{d}_{j-1}-d_{j-2}$, starting with $d_0=1$, $d_1=a_1$; they are positive and increasing, so Sylvester's criterion applies.