  We investigate the strong and radiative decay widths of the narrow nucleon
resonances $N^*(1685)$ and $N^{\ast}(1726)$ within the framework of the SU(3)
chiral quark-soliton model. All the relevant parameters are taken from those
used to describe the properties of the baryon octet and decuplet in previous
works. The masses of the antidecuplet nucleon and the eikosiheptaplet (27-plet)
nucleon with spin 3/2 are determined respectively to be $(1690.2\pm 10.5)\,
\mathrm{MeV}$ and $(1719.6\pm7.4)\,\mathrm{MeV}$. The decay width for
$N^{\ast}(1685)\to \eta + N$ is found to be approximately three times larger
than that for $N^{\ast}(1685)\to \pi +N$. The width of the decay
$N^{\ast}\left(1726\right)3/2^+\to \eta +N$ is even about 31 times larger than
that of $N^{\ast}\left(1726\right)3/2^+\to \pi + N$. The ratio of the radiative
decays for $N^*(1685)$ is obtained to be
$\Gamma_{nn^*(1685)}/\Gamma_{pp^*(1685)}=8.62\pm3.45$ which explains very well
the neutron anomaly. In contrast, we find
$\Gamma_{pp^*(1726)}/\Gamma_{nn^*(1726)}=3.72\pm0.64$, which indicates that the
production of $N^*(1726)$ is more likely to be observed in the proton channel.
We also examined the decay modes of these narrow nucleon resonances with the
strangeness hadrons involved.
