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COLORING ALGEBRAIC KNOTS AND LINKS

Department of Mathematics and Computer Science, The College of the Holy Cross, 1 College Street, Worcester, MA 01610, USA

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and

ELIZABETH M. SENNOTT

The Graduate School of Architecture Planning and Preservation, Columbia University, 1172 Amsterdam Avenue, New York, NY 10027, USA

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https://doi.org/10.1142/S0218216508006324Cited by:0 (Source: Crossref)

Abstract

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.

Keywords:

AMSC: Primary 57M27, Secondary 57M25, Secondary 57M05

References

M. M. Asaeda, J. H. Przytycki and A. S. Sikora, J. Knot Theory Ramifications 13(4), 467 (2004), DOI: 10.1142/S0218216504003251. Link, Web of Science, Google Scholar
G. Burde and H. Zieschang , Knots ( Walter de Gruyter & Co. , Berlin , 1985 ) . Google Scholar
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf.) pp. 329–358. Google Scholar
R. H. Crowell and R. H. Fox , Introduction to Knot Theory ( Springer-Verlag , New York , 1963 ) . Google Scholar
R. H. Fox , Topology of 3-Manifolds and Related Topics ( Prentice-Hall , Englewood Cliffs , 1962 ) . Google Scholar
R. H. Fox, Canad. J. Math. 22, 193 (1970). Crossref, Web of Science, Google Scholar
J. R. Goldman and L. H. Kauffman, Adv. Appl. Math. 18, 300 (1997), DOI: 10.1006/aama.1996.0511. Crossref, Web of Science, Google Scholar
F. Harary and L. H. Kauffman, Adv. Appl. Math. 22, 329 (1999), DOI: 10.1006/aama.1998.0634. Web of Science, Google Scholar
L. H. Kauffman and S. Lambropoulou, Adv. Appl. Math. 33, 199 (2004), DOI: 10.1016/j.aam.2003.06.002. Crossref, Web of Science, Google Scholar
A. Kawauchi , A Survey of Knot Theory ( Birkhäuser Verlag , Boston , 1996 ) . Google Scholar
W. B. R. Lickorish , An Introduction to Knot Theory ( Springer-Verlag , New York , 1997 ) . Crossref, Google Scholar
J. M. Montesinos, Revetements ramifies des noeuds, Espaces fibres de Siefert et scindements de Heegaard, Publicaciones del Seminario Mathematico Garcia de Galdeano, Serie II, Seccion 3 (1984) . Google Scholar
K. Murasugi , Knot Theory and Its Applications ( Birkhäuser Verlag , Boston , 1996 ) . Google Scholar
D. Rolfsen , Knots and Links ( American Mathematical Society , Providence , 2003 ) . Crossref, Google Scholar