# Knot quandle decompositions

## TL;DR

This paper introduces a functorial approach to fundamental quandles in knot theory, enabling the decomposition of complex links into simpler tangles and deriving presentations for various classes of knots.

## Contribution

It establishes a functor from the tangle category to a quandle category and uses it to decompose and analyze the fundamental quandle of complex links.

## Key findings

- Fundamental quandle defines a functor from tangle category to quandle category.
- Decomposition of link quandles from tangle quandles.
- Derived presentations for periodic, composite, and satellite knots.

## Abstract

We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link $L$, the fundamental quandle of $L$ may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the fundamental quandle of periodic links, composite knots and satellite knots.