Curvature and quantized Arnold strangeness

Abstract

By integrating curvatures multiplied non-trivial densities, we introduce an integral expression of the Arnold strangeness that is a celebrated plane curve invariant. The key is a partition function by Shumakovitch to reformulate Arnold strangeness. Our integrating curvatures suggests a quantized Arnold strangeness which Taylor expansion includes the rotation number and the original Arnold strangeness, and also higher terms are invariants of Tabachnikov. It is an analogue of the quantization by Viro for Arnold $J^-$ and by Lanzat-Polyak for $J^+$.