  We have examined the existence of the $N^*(1685)$ resonance, which is
recently listed by the Particle Data Group as a one-star nucleon resonance, by
using a covariant isobar model for kaon photoproduction and assuming that the
resonance has $J^p=1/2^+$, in accordance with our previous finding. After the
inclusion of this resonance the changes in the total $\chi^2$ show two clear
minima at $M_{N^*}=1650$ and 1696 MeV, which correspond to two different
resonance states. The former corresponds to the narrow nucleon resonance found
in our previous investigation, whereas the latter corresponds to a new
resonance found as we increase the resonance width. From the latter we derive
the mass and width relation of the $N^*(1685)$ resonance. We observe that the
properties of both the $N^*(1685)$ and $N^*(1710)P_{11}$ resonances are
strongly correlated. Although the best fit of the present work yields
$M_{N^*}=1696$ MeV and $\Gamma_{N^*}=76$ MeV, the apparently small
$N^*(1710)P_{11}$ coupling constant to the $K^+\Lambda$ channel found in
previous investigations suggests that $\Gamma_{N^*}\lesssim 35$ MeV, which,
according to the mass and width relation, corresponds to $M_{N^*}\lesssim 1680$
MeV.
