A combinatorial approach to knot recognition

TL;DR

This paper explores a combinatorial method for knot recognition using algebraic colorings with quandles, emphasizing computational techniques like SAT solving to determine knot colorability and recognition.

Contribution

It introduces a computational framework for knot recognition based on quandle colorings and demonstrates how SAT solving can be applied to determine knot colorability.

Findings

SAT solving effectively determines knot colorability

The method provides a practical approach to computational knot recognition

Complexity analysis supports the feasibility of the approach

Abstract

This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in a compact, accessible manner, and to show how to use it for computational purposes. In particular, we address how to determine colorability of a knot, and propose to use SAT solving to search for colorings. The computational complexity of the problem, both in theory and in our implementation, is discussed. In the last part, we explain how coloring can be utilized in knot recognition.