Curvature and quantized Arnold strangeness

There are three well-known Arnold’s invariants [Formula: see text], [Formula: see text], and St (Arnold strangeness) for plane curves. Two of them, [Formula: see text] and [Formula: see text], have been successfully quantized. However, the quantization of the Arnold strangeness St remained unrealized. In this paper, we will express a quantization of St by integrating curvatures multiplied by nontrivial densities. The key here is a partition function by Shumakovitch which reformulates the Arnold strangeness. Our quantized Arnold strangeness includes the rotation number, the original Arnold strangeness, and the Tabachnikov invariants as coefficients in its Taylor expansion.