Kauffman-Harary Conjecture holds for Montesinos knots

Jozef H. Przytycki

Knots in Washington XVI; Conference on Knot Theory and its Ramifications
May 5-7, 2003
University of Maryland
College Park, MD, USA

Organizers
Marta M. Asaeda (UMD), William M. Goldman (UMD), John J. Millson (UMD), Jozef H. Przytycki (GWU)

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Kauffman-Harary Conjecture holds for Montesinos knots
by
Jozef H. Przytycki
George Washington University
Coauthors: Marta M. Asaeda (UMD), Adam S. Sikora (IAS)

We consider the conjecture by Kauffman and Harary on Fox colorings of alternating diagrams. We prove the conjecture for Montesinos knots. We propose a generalization of this conjecture to alternating links formulated in terms of homology of the double branched cover of S³ branched along a link. We prove this homology conjecture for Montesinos links. We speculate about the relation of the conjecture to a question on incompressible surfaces in the exterior of alternating links.

Kauffman-Harary conjecture: Consider a knot K with the determinant, D(K), equal to a prime number p. Then for any alternating diagram of K with no nugatory crossings, every non trivial Fox p-coloring of the diagram colors different arcs with different colors.

Homology Conjecture: If K is an alternating diagram of a prime knot (more generally prime link) without a nugatory crossing then different arcs represent different elements of H₁(M_K⁽²⁾, Z), where M_K⁽²⁾ is the double branched cover of S³ branched along K.

Theorem: The Homology Conjecture holds for all alternating Montesinos links.

Date received: April 29, 2003