A proof of the Kauffman-Harary Conjecture

TL;DR

This paper proves the Kauffman-Harary Conjecture, demonstrating that for prime determinant knots with reduced alternating diagrams, all non-trivial Fox p-colorings assign distinct colors to each arc.

Contribution

The paper provides a rigorous proof of the long-standing conjecture, confirming the coloring property for a broad class of knots.

Findings

Confirmed the conjecture for all prime determinant, reduced alternating knots.

Showed that non-trivial Fox p-colorings distinguish all arcs in the diagram.

Extended understanding of knot colorings and their algebraic properties.

Abstract

We prove the Kauffman-Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.