The generalized Kauffman–Harary conjecture istrue

For a reduced alternating diagram of a knot with a prime determinant p, the Kauffman-Harary conjecture states that every nontrivial Fox p-coloring of the knot assigns different colors to its arcs.We prove a generalization of the conjecture, stated nineteen years ago by Asaeda, Przytycki and Sikora: for every pair of distinct arcs in the reduced alternating diagram of a prime link with determinant , there exists a Fox -coloring that distinguishes them.57K10; 57M12 1 History of the alternation conjecture In 1998, Louis H Kauffman and Frank Harary formulated the following conjecture [7]: Alternation conjecture Let D be a reduced , alternating diagram of a knot K having determinant p, where p is prime.Then every nontrivial p-coloring of D assigns different colors to different arcs.This conjecture is now known as the Kauffman-Harary conjecture.It was proved for rational knots [9; 13], Montesinos knots [1], some Turk's head knots [6], and algebraic knots [5].In 2009, Thomas W Mattman and Pablo Solis proved this conjecture using the notion of pseudocolorings.A generalization of this conjecture, known as the generalized Kauffman-Harary (GKH) conjecture, was formulated by Marta M Asaeda, Adam S Sikora and the fifth author in 2004 [1].They proved this conjecture for Montesinos links in the same paper.In this paper, we prove it in full generality.The paper is structured as follows.In the next section we introduce the GKH conjecture and we prove it in Section 3. In Section 4, we reformulate and prove the conjecture for nonprime alternating links.We illustrate the results with some examples in Section 5.In the last section, we discuss pseudocolorings, followed by some open questions. PreliminariesIn this section, we state the original GKH conjecture and then some alternative versions.The difference between the original and generalized Kauffman-Harary conjecture is that the former is restricted to links with prime determinant.Note that the only link whose determinant is prime is the Hopf link.