# Quasi-trivial Quandles and Biquandles, Cocycle Enhancements and Link-Homotopy of Pretzel links

## TL;DR

This paper introduces quasi-trivial quandles and biquandles to analyze link-homotopy of pretzel links, providing new algebraic invariants and conditions for triviality under link-homotopy.

## Contribution

It develops the theory of quasi-trivial quandles and biquandles, applying them to link-homotopy classification and enhancing cocycle invariants for links.

## Key findings

- Characterization of trivial pretzel links under link-homotopy
- Generalization of quasi-trivial quandle to biquandle structures
- Enhanced link-homotopy invariants using cocycle methods

## Abstract

We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Ayumu Inoue's article arXiv:1205.5891.