COLORING ALGEBRAIC KNOTS AND LINKS

• DOI: 10.1142/s0218216508006324
• Publication Date: 2008-05-20T09:05:42Z
• Authors: DAVID B. DAMIANO; ELIZABETH M. SENNOTT
• Keywords: 0101 mathematics; 01 natural sciences

Let K be a knot or link and let p = det (K). Using integral colorings of rational tangles, Kauffman and Lambropoulou showed that every rational K has a mod p coloring with distinct colors. If p is prime this holds for all mod p colorings. Harary and Kauffman conjectured that this should hold for prime, alternating knot diagrams without nugatory crossings for p prime. Asaeda, Przyticki and Sikora proved the conjecture for Montesinos knots. In this paper, we use an elementary combinatorial argument to prove the conjecture for prime alternating algebraic knots with prime determinant. We also give a procedure for coloring any prime alternating knot or link diagram and demonstrate the conjecture for non-algebraic examples.