Curvature and quantized Arnold strangeness

TL;DR

This paper introduces a novel integral expression for Arnold strangeness using curvature and densities, leading to a quantized version that includes classical invariants and higher-order terms, connecting to known quantization methods.

Contribution

It presents a new integral formulation of Arnold strangeness and introduces a quantized version that extends classical invariants with higher-order terms.

Findings

Integral expression of Arnold strangeness as a plane curve invariant

Quantized Arnold strangeness includes rotation number and original strangeness

Higher-order invariants related to Tabachnikov's invariants

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 $J^+$.