Neutron stars are expected to have a tight relation between their moment of inertia ($I$), tidal deformability ($lambda$, which is related to the Love number), and rotational mass quadrupole moment ($Q$) that is nearly independent of the unknown equation of state (EoS) of cold dense matter. These and similar relations are often called universal, and they have been used for various applications including analysis of gravitational wave data. We extend these studies using piecewise polytropic representations of dense matter, including for so-called twin stars that have a second branch of stability at high central densities. The second-branch relations are less tight, by a factor of $sim 3$, than the relations found in the first stable branch. We find that the relations on both branches become tighter when we increase the lower limit to the maximum mass for the EoS under consideration. We also propose new empirical relations between $I$, $lambda$, $Q$, and the complex frequency $omega=omega_R+iomega_I$ of the fundamental axial $w$-mode, and find that they are comparably tight to the I-Love-Q correlations.
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