On the nonexistence of good involutions of symplectic quandles

TL;DR

This paper explores the conditions under which good involutions exist in symplectic quandles, focusing on their nonexistence in certain cases, and provides a detailed mathematical analysis.

Contribution

It establishes necessary and sufficient conditions for the existence of good involutions in symplectic quandles and proves their nonexistence in specific instances.

Findings

Identifies conditions for the existence of good involutions

Proves nonexistence of good involutions in certain symplectic quandles

Provides a mathematical framework for symplectic quandle involutions

Abstract

We investigate the necessary and sufficient condition for the existence of good involutions of symplectic quandles, which are defined on free $R$-modules with an antisymmetric bilinear form. In particular, we discuss the nonexistence of good involutions of symplectic quandles.