The fundamental quandle of ribbon concordances

TL;DR

This paper introduces the fundamental quandle for properly embedded surfaces in 3-space cross an interval, providing a presentation via diagrams and exploring its properties under ribbon concordances between knots.

Contribution

It defines the fundamental quandle for surfaces in 3-space cross an interval and relates it to ribbon concordances, including homomorphism properties.

Findings

Fundamental quandle can be presented using motion picture or CH-diagrams.

Ribbon concordance induces injective and surjective quandle homomorphisms.

The study extends quandle theory to surfaces in 4-dimensional topology.

Abstract

We describe the fundamental quandle of a properly embedded surface $F$ (possibly with boundary) in $\mathbb{R} ^{3}\times I$, and derive its presentation in terms of a motion picture diagram or a CH-diagram of $F$. Our study is based on the topological definition of the fundamental quandle. We prove that a ribbon concordance $C$ from a classical knot $K_1$ to $K_0$ gives rise to an injective quandle homomorphism $Q(K_0)\to Q(C)$ and a surjective quandle homomorphism $Q(K_1)\to Q(C)$.