Kauffman-Harary conjecture for alternating virtual knots

TL;DR

This paper confirms the Kauffman-Harary conjecture for alternating virtual knots, extending previous results from classical knots to virtual knots using established proof techniques.

Contribution

It provides a proof of the conjecture for alternating virtual knots, broadening the understanding of Fox p-colorings in virtual knot theory.

Findings

Confirmed the conjecture for all alternating virtual knots.

Extended classical knot results to virtual knots.

Used methods from previous proofs to establish new results.

Abstract

In 1999, Kauffman-Harary conjectured that every non-trivial Fox $p$-coloring of a reduced, alternating knot diagram with prime determinant $p$ is heterogeneous. Ten years later this conjecture was proved by W. Mattman and P. Solis. Mathew Williamson generalized this conjecture to alternating virtual knots and proved it for certain families of virtual knots. In the present note, we use the methods of W. Mattman and P. Solis to give an affirmative answer to the Kauffman-Harary conjecture for alternating virtual knots.