A functor-valued extension of knot quandles

TL;DR

This paper introduces a functor-valued extension of knot quandles, providing new ways to analyze knots and extend quandle cocycle invariants, thereby enriching knot invariants with functorial structures.

Contribution

It constructs a functor from pointed quandles to quandles in three ways and extends quandle cocycle invariants using these functorial invariants.

Findings

New functorial framework for knot quandles

Extended quandle cocycle invariants for knots

Enhanced understanding of knot invariants through functorial approaches

Abstract

For an oriented knot $K$, we construct a functor from the category of pointed quandles to the category of quandles in three different ways. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariant of knots, and study their properties.